TITLE: Rate of tumour increasing, exponential growth
QUESTION [1 upvotes]: Suppose the volume, $V$, of a spherical tumour with a radius of $r = 2\,\textrm{cm}$ uniformly grows at a rate of $dV/dt=0.3\,\textrm{cm}^3/\textrm{day}$, where $t$ is the time in days. At what rate is the surface area of the tumour increasing. The volume of a sphere is given by $V=\dfrac43\pi r^3$ and the surface area is given by $A=4\pi r^2$.

REPLY [1 votes]: You can find $r$ as a function of $V$:
$$r=\sqrt[3]{\dfrac{3V}{4\pi}},$$
then
$$A=4\pi r^2= 4 \pi \left(\dfrac{3V}{4\pi}\right)^{2/3},$$
the last step is derive
$$\dfrac{dA}{dt}={2\pi} \left(\dfrac{3V}{4\pi}\right)^{-1/3}\dfrac{dV}{dt}.$$
Now you can put the numbers in the last equation.


text:
\vspace{-3mm}
\section{Problem Formulation}
\label{sec:formulation}
\vspace{-2mm}
We begin with the standard statistical learning framework for empirical risk minimization, where the objective is to minimize the following loss function:
\begin{align}
f(\w)= \frac{1}{n} \sum_{j=1}^n \ell_j(\w^T\x_j) + \frac{\lambda}{2}\|\w\|^2,   \label{prob}
\end{align}
where, the loss functions $\ell_j : \mathbb{R} \rightarrow \mathbb{R}$, $j \in [n]$ are \emph{convex, twice differentiable and  smooth}. Moreover, $\x_1,\x_2,\ldots ,\x_n \, \in \mathbb{R}^d$ denote the input feature vectors and $y_1,y_2,\ldots ,y_n \in \real $  denote the corresponding responses. Furthermore, we assume that the function $f$  is \emph{strongly convex}, implying the existence of a unique minimizer of \eqref{prob}. We denote this minimizer by $\w^*$. Note that the response  $\{y_j\}_{j=1}^n$ is captured by the corresponding loss function $\{\ell_j\}_{j=1}^n$. Some examples of $\ell_j$ are
\begin{align*}
\text{logistic loss: }\,\, \ell_j(z_j) =\log(1- \exp(-z_jy_j)), \quad
\text{squared loss: }\,\, \ell_j(z_j) = \frac{1}{2}(z_j-y_j)^2
\end{align*}
We consider the framework of distributed optimization with $m$ worker machines, where the feature vectors and the loss functions $(\x_1,\ell_1),\ldots,(\x_n,\ell_n) $ are partitioned homogeneously among them. Furthermore, we assume that $\alpha$
fraction of the worker machines are Byzantine for some $\alpha < \frac{1}{2}$.  The Byzantine machines, by nature, may send any arbitrary values to the center machine. Moreover, they can even collude with each other and plan malicious attacks with complete information of the learning algorithm.

config: arxiv


TITLE: what is the intuitive explanation of the truth of the statement if 0=1, then 1=2?
QUESTION [3 upvotes]: Consider the statement
if 0=1, then 1=2
This statement is weird but since the hypothesis (if 0=1) is false, then according to propositional logic the entire statement is true, although the conclusion (then 1=2) is false. I just cannot understand how can we consider the entire statement to be true even though our conclusion is wrong. Can you provide another examples where such situation is easy to grasp? Maybe I can get it with concrete examples.

REPLY [1 votes]: The word "if" means something different in mathematical English than it means in ordinary English. In mathematical English, the word "if" is an abbreviation for a particular statement involving the words "not" and "or."
Specifically, when a mathematician writes a sentence like

if p, then q,

then what they are actually saying is that

either it's false that p, or it's true that q.

So, when a mathematician writes the sentence "if $0 = 1$, then $1 = 2$," then what the mathematician is actually claiming is:

either it's false that $0 = 1$, or it's true that $1 = 2$.

This claim is, of course, true.


text:
\begin{document}

\keywords{Admissibility, Dirichlet process, gamma process, kernel mixture, predictive density}

\maketitle

\begin{abstract}
We consider nonparametric Bayesian estimation and prediction
for nonhomogeneous Poisson process models with unknown intensity functions.
We propose a class of improper priors for intensity functions.
Nonparametric Bayesian inference with kernel mixture based on the class improper priors is shown to be useful,
although improper priors have not been widely used for nonparametric Bayes problems.
Several theorems corresponding to those for finite-dimensional independent Poisson models
hold for nonhomogeneous Poisson process models with infinite-dimensional parameter spaces.
Bayesian estimation and prediction based on the improper priors
are shown to be admissible under the Kullback--Leibler loss.
Numerical methods for Bayesian inference based on the priors are investigated.
\end{abstract}

\section{Prediction based on models with finite dimensional parameter}

Statistical modeling and data analysis based on nonhomogeneous Poisson point processes have various applications
\citep[e.g.][]{SM91, Streit10}.
We consider nonparametric Bayesian inference with kernel mixture
for nonhomogeneous Poisson processes from the viewpoint of predictive density theory.
In the present paper,
it is shown that Bayesian procedures based on a class of improper priors provide reasonable estimation and prediction.
Several theorems concerning admissibility
corresponding to those for finite-dimensional independent Poisson models are shown to hold
for the nonhomogeneous Poisson process model.

In this section, we summarize the basic framework of predictive density theory mainly studied
for finite-dimensional models.
Suppose that we have an observation $x$
from a probability density $p(x \mid \theta)$ that belongs to a parametric model
$\{p(x \mid \theta) \mid \theta \in \Theta \in \mathbb{R}^d \}$.
The objective is to predict an unobserved random variable $y$
distributed according to $p(y \mid \theta)$ using a predictive density $q(y;x)$.

We adopt the Kullback--Leibler loss

\begin{align*}
D\{p(y \mid \theta),q(y;x)\}=\int p(y \mid \theta)
\log \frac{ p(y \mid \theta)}{q(y;x)} \dd y.
\end{align*}

from the true density $p(y \mid \theta)$ to a predictive density $q(y;x)$.
A predictive density $q(y ; x)$ is said to dominate another predictive density $q'(y ; x)$
if the risk of $q(y ; x)$ is not greater than that of $q'(y ; x)$ for all $\theta$ and
the strict inequality holds
for at least one point $\theta$ in the parameter space.
A predictive density $q(y ; x)$ is said to be admissible
if there does not exist a predictive density dominating $q(y;x)$.
See \citet{Berger85} and \citet{Schervish95}
for basic frameworks of statistical decision theory concerning admissibility and its relationship with Bayes theory.

Many studies recommend Bayesian predictive densities

\[
p_{\pi}(y \mid x)
= \frac{\int p(y \mid \theta)p(x \mid \theta)\pi(\theta) \dd \theta}
{\int p(x \mid \overline{\theta})\pi(\overline{\theta}) \dd \overline{\theta}}
\]

based on a prior $\pi(\theta)$ rather than plug-in densities
$p(y \mid \widehat{\theta}(x))$
\citep[e.g.][]{AD75,Geisseer93,K:Biometrika1996},
where $\widehat{\theta}$ is an estimated value of $\theta$.
This is because there exist Bayesian predictive densities
dominating plugin densities in many examples.

It is important to construct prior distributions for Bayesian inference
when we do not have specific prior information about unknown parameters.
Such a prior is called a noninformative prior or an objective prior.
The Jeffreys prior is a theoretically important objective prior.
Let $\theta = (\theta_1, \ldots, \theta_d)$ be an unknown parameter
of a finite dimensional parametric statistical model and $I(\theta)$ be the corresponding Fisher information matrix.
Then, the Jeffreys prior is defined by

\begin{align*}
 \pi_\mathrm{J}(\theta) \dd \theta_1 \cdots \dd \theta_d
= |I(\theta)|^{1/2} \dd \theta_1 \cdots \dd \theta_d,
\end{align*}

where $|I(\theta)|$ is the determinant of $I(\theta)$.
Noninformative priors such as the Jeffreys prior often becomes improper, i.e.\ the total mass of the prior is infinite.
The Bayes theorem cannot be directly applied to improper priors because they are not probability distributions, whose total masses are $1$.
However, formal application of the Bayes theorem to improper priors sometimes gives useful prediction and estimation.
Statistical decision theory concerning admissibility provides a basis of such generalized Bayesian procedure based on improper priors
\citep[e.g.][]{Berger85}.

Bayesian predictive densities based on shrinkage priors often dominate
the Bayesian predictive densities based on the Jeffreys prior
when the dimension of the parameter space is large.
Shrinkage priors assign more weight to parameter values close to
a subset in the parameter space than the Jeffreys prior.
In particular, Bayesian prediction based on shrinkage priors for finite-dimensional models such as
the multivariate Normal model
and the multidimensional independent Poisson model have been investigated.
See \citet{FSW:book18} for recent developments of parameter estimation theory based on shrinkage priors.

First, consider the $d$-dimensional Normal model.
Suppose that $x_i$ $(i=1,\ldots,d)$ are independently distributed
according to $\mbox{N}(\mu_i, \sigma^2)$
and that $y_i$ $(i=1,\ldots,d)$ are independently distributed
according to $\mbox{N}(\mu_i, \tau^2)$,
where $\mbox{N}(\mu_i,\sigma^2)$ is the Normal distribution with mean $\mu_i$ and variance $\sigma^2$.
Here, $\mu := (\mu_1, \ldots, \mu_d)$ is the unknown parameter and $\sigma$ and $\tau$ are known positive constants.
We consider prediction of $y = (y_1,y_2,\ldots,y_d)$
using $x=(x_1,x_2,\ldots,x_d)$
under the Kullback--Leibler loss.
The Bayesian predictive density $p_\mathrm{S}(y \mid x)$
based on the prior
$\pi_\mathrm{S}(\mu) = \| \mu \|^{-(d-2)} = (\sum_i {\mu_i}^2)^{-(d-2)/2}$
introduced by \citet{Stein74}
dominates the Bayesian predictive density $p_\mathrm{J}(y \mid x)$
based on the Jeffreys prior $\pi_\mathrm{J}(\mu)=1$ \citep{K:Biometrika2001}.
This corresponds to the widely known result that
the generalized Bayes estimator based on Stein's prior $\pi_\mathrm{S}(\mu)$
dominates the best invariant estimator $\widehat{\mu} = x$ when $d \geq 3$.
See \cite{GLX06} for sufficient conditions for general priors other than the Stein prior
and \cite{BGX:AS08} for admissible predictive densities for Normal models.
The asymptotics of minimax risk of predictive density estimation for non-parametric regression
is studied by \citet{XuLiang10}.

Next, consider the $d$-dimensional Poisson model, which is closely related to the nonhomogeneous Poisson process models
considered herein.
Intuitively speaking, nonhomogeneous Poisson process models are infinite-dimensional Poisson models.
Suppose that $x_i$ $(i=1,\ldots,d)$ are independently distributed
according to $\mbox{Po}(s \lambda_i)$
and that $y_i$ $(i=1,\ldots,d)$ are independently distributed
according to $\mbox{Po}(t \lambda_i)$, where $\mbox{Po}(s \lambda_i)$ is the Poisson distribution with mean $s \lambda_i$,
$\lambda := (\lambda_1, \ldots, \lambda_d)$ is the unknown parameter, and $s$ and $t$ are known positive constants.
We consider prediction of $y = (y_1,y_2,\ldots,y_d)$ using $x=(x_1,x_2,\ldots,x_d)$
under the Kullback--Leibler loss.

A natural class of priors including the Jeffreys prior
$\pi_\mathrm{J} (\lambda) \dd \lambda_1 \dotsb \dd \lambda_d
= \lambda_1^{-\frac{1}{2}} \dotsb \lambda_d^{-\frac{1}{2}} \dd \lambda_1 \dotsb \dd \lambda_d$ is
$\pi_\alpha (\lambda) \dd \lambda_1 \dotsb \dd \lambda_d
:= \lambda_1^{\alpha_1-1} \dotsb \lambda_d^{\alpha_d-1} \dd \lambda_1 \dotsb \dd \lambda_d$,
where $\alpha_i > 0$ $(i=1,\ldots,d)$.
A class of improper prior densities

\begin{align}
\pi_{\alpha, \gamma}(\lambda) \dd \lambda_1 \dd \lambda_2 \cdots \dd \lambda_d
&= \frac{\lambda_1^{\alpha_1-1} \lambda_2^{\alpha_2-1} \cdots \lambda_d^{\alpha_d-1}}
{(\lambda_1+\lambda_2+\cdots+\lambda_d)^\gamma}
\dd \lambda_1 \dd \lambda_2 \cdots \dd \lambda_d
\label{eq:class}
\end{align}

with $\sum_i \alpha_i - \gamma > 0$ and $\alpha_i > 0$ $(i=1,2,\ldots,d)$ is investigated
and Theorems 1 and 2 below are obtained \citep{K:AS2004}.

\vspace{0.4cm}

\begin{theorem}[\citealp{K:AS2004}]
When
$\sum_i \alpha_i - \gamma > 1 ~~\mbox{and}~~ \alpha_i > 0 ~~~ (i=1,2,\ldots,d)$,
the Bayesian predictive density
$p_{\alpha, \gamma}(y \mid x) ~~{\mbox{based on}}~~ \pi_{\alpha,\gamma}(\lambda)$
is dominated by the Bayesian predictive density
$p_{\talpha, \tilde{\gamma}}(y \mid x)~~\mbox{based on}~~ \pi_{\talpha, \tilde{\gamma}}(\lambda)$,

where $\tilde{\gamma} := \sum_i \alpha_i -1$ and
$\talpha = (\talpha_1, \talpha_2, \ldots, \talpha_d) := (\alpha_1, \alpha_2, \ldots, \alpha_d)$.

\end{theorem}

\begin{theorem}[\citealp{K:AS2004}]
For every $d \geq 1$, the Bayesian predictive densities based on the priors in the class
$\{\pi_{\alpha,\gamma}(\lambda): 0 < \sum_{i=1}^d \alpha_i - \gamma \leq 1, \alpha_i > 0~(i=1,2,\ldots,d)\}$
defined by (\ref{eq:class}) are admissible under the Kullback--Leibler loss.
\end{theorem}

\vspace{0.4cm}

In particular,
when $d \geq 3$,
the Bayesian predictive density $p_{\pi_\mathrm{S}}(y \mid x)$ based on the shrinkage prior
$\pi_\mathrm{S}(\lambda)
:= \pi_{\alpha = (\frac{1}{2},\dots,\frac{1}{2}),\, \gamma = \frac{d}{2}-1}(\lambda)$
dominates the Bayesian predictive density $p_{\pi_\mathrm{J}}(y \mid x)$ based on the Jeffreys prior
and is admissible under the Kullback--Leibler loss.
Parameter estimation
can be regarded as infinitesimal prediction under the Kullback--Leibler loss
in the multivariate Poisson model \citep{K:JMVA2006}.
Intuitively, infinitesimal prediction means prediction for infinitely near future.

In the present paper, we generalize the results for finite-dimensional independent Poisson models to the results for
nonhomogeneous Poisson models.
The remainder of the present paper paper is organized as follows.
In Section 2, a class of improper shrinkage priors for
nonparametric Bayesian inference with kernel mixtures for nonhomogeneous Poisson models is introduced.
Several theorems concerning admissibility of Bayesian predictive densities and Bayes estimators
for nonhomogeneous Poisson models
corresponding to those for finite-dimensional models are proved.
In Section 3, numerical methods to evaluate Bayesian predictive densities and Bayes estimators are investigated.
Finally, conclusions are presented in Section 4.

\section{Nonparametric Bayesian inference for nonhomogeneous Poisson processes}

We consider nonhomogeneous Poisson processes on a region $U$ in the Euclidean space $\mathbb{R}^d$.
The results in the following can be generalized to those for nonhomogeneous Poisson processes
on general spaces such as a Polish space.

Basic properties of nonparametric inference of nonhomogeneous Poisson processes
using gamma process priors are given by \cite{Lo82} and \cite{LoWeng89}.
Corresponding results for probability density estimation are given by \cite{Lo84}.
Various estimation methods of intensity functions
of nonhomogeneous Poisson process models have been studied \citep[e.g.][]{BBMT81, Kolaczyk1999}.

Conventional statistical decision theory mainly deals with problems with finite dimensional parameters.
In the present paper, we show that the framework of statistical decision theory is effectively applied to
prediction and estimation for nonhomogeneous Poisson process models with infinite dimensional parameter spaces.
The results suggest that decision theoretic approach is also useful for other infinite dimensional problems.

\subsection{Bayes estimators and Bayesian predictive densities}

Let $\lambda(u)$ be a positive function of $u \in U$ satisfying $\int_U \lambda(u) \dd u < \infty$.
We observe $\xx = (\xnum, x_1, x_2, \ldots, x_\xnum)$
distributed according to the nonhomogeneous Poisson process $\Po(s\lambda)$ with intensity function $s \lambda$ ($s>0$).
Here, $\xnum$ is the number of the observed points and $x_1,\ldots,x_\xnum$ are observed points in $U$.
We introduce a known constant $s>0$ for later use.
Let $X(B) := \# \{ x_i \mid x_i \in B\ (1 \leq i \leq N)  \}$ and $\lambda(B) := \int_{B} \lambda(u) \dd u$
for $B \subset U$.
If $X(B_i)$  $(i=1,\ldots,k)$ are independently distributed according to the Poisson distribution with mean
$s\lambda(B_i)$ for an arbitrary partition $(B_1,\ldots,B_k)$ of $U$,
$\xx$ is said to be distributed according to the nonhomogeneous Poisson distribution
with intensity function $s \lambda$.
The function $\lambda$ is the unknown parameter.

The likelihood for the nonhomogeneous Poisson process model is given by

\begin{align}
\label{likelihood}
\biggl\{ \prod_{i=1}^\xnum & s \lambda(x_i) \biggr\}
\exp \biggl\{ - s \int \lambda(x) \dd x \biggr\}
=
\bigl\{ \prod_{i=1}^\xnum \blambda(x_i) \bigr\}
(s \masslambda)^\xnum \exp \bigl( - s \masslambda \bigr)  \\
&\propto
\biggl\{ \prod_{i=1}^\xnum \blambda(x_i) \biggr\}
\frac{(s \masslambda)^\xnum}{\xnum!} \exp \bigl( - s \masslambda \bigr)
\label{ldecomp}
=: p_\lambda(\xx),
\end{align}

where
$w = |\lambda| := \int_U \lambda(u) \dd u$ and $\overline{\lambda} := \lambda/w$
\citep[see e.g.][p.\ 22]{DV:book03}.
We identify $\lambda$ with $(\masslambda, \blambda)$.
The probability density \eqref{ldecomp} of the observed points
$(x_1,\ldots,x_\xnum)$ multiplied by $\xnum!$ coincides with \eqref{likelihood}.
These two representations do not make essential differences in the following discussions.
We mainly use \eqref{ldecomp} in the following.

Let $\yy = (\ynum, y_1, y_2, \ldots, y_\ynum)$ be
a sample independent of $\xx$ from the nonhomogeneous Poisson process $\mathcal{P\hspace{-1.7pt}o}(t \lambda)$,
where $t>0$ is a known constant.
We investigate estimation of $\lambda$ and prediction of $\yy$ using $\xx$.
In Subsection \ref{infinitesimal prediction}, estimation is formulated as a limit of prediction.

First, we consider estimation of $\lambda$.
From \eqref{ldecomp},
the Kullback--Leibler divergence from the probability density $p_\lambda(\yy)$
corresponding to the intensity $t\lambda$
to another probability density  $p_{\lambda'}(\yy)$
corresponding to the intensity $t\lambda'$ is

\begin{align}
D( p_\lambda(\yy), p_{\lambda'}(\yy))
=&\ \E_\lambda \log
\frac{\biggl\{\displaystyle \prod_{i=1}^\ynum \blambda(y_i) \biggr\}
\frac{(t \masslambda)^\ynum}{\ynum!} \exp \bigl( -t \masslambda \bigr)}
{\displaystyle \biggl\{ \prod_{i=1}^\ynum \overline{\lambda'}(y_i) \biggr\}
\frac{(t \masslambda')^\ynum}{\ynum!} \exp \bigl( -t \masslambda' \bigr)} \notag \\
=&\
\E_\lambda \left[ t \masslambda' - t \masslambda + \ynum \log \frac{\masslambda}{\masslambda'}
+ \sum_{i=1}^\ynum \log \frac{\blambda(y_i)}{\overline{\lambda'}(y_i)} \right] \notag \\
=&\
\label{eq:KLeat0}
t \masslambda \left( \frac{\masslambda'}{\masslambda} - 1 - \log \frac{\masslambda'}{\masslambda}
+ \int \blambda(y) \log \frac{\blambda(y)}{\overline{\lambda'}(y)} \dd y \right) \\
=&\ t \int
\biggl(\lambda'(y) - \lambda(y)
+ \lambda(y) \log \frac{\lambda(y)}{\lambda'(y)} \biggr) \dd y.
\label{eq:KLeat}
\end{align}

Suppose that a prior density $\pi(\dd \lambda)$ is adopted and observation $\xx$ is given.
Let $p_\pi(\dd \lambda \mid \boldsymbol{x})$ be the posterior distribution.
If the posterior mean of $\lambda$ has a density $\lambda_{\pi,\xx}(u)$ with respect to the Lebesgue measure on $U$,
the posterior mean of the Kullback--Leibler loss of an intensity estimator $\widehat{\lambda}$ is

\begin{align*}
\int
D( p_\lambda(\yy), p_{\lambda'}(\yy))\, p_\pi(\dd \lambda \mid \xx)
&= t \iint
\biggl(\hlambda(y) - \lambda(y)
+ \lambda(y) \log \frac{\lambda(y)}{\hlambda(y)} \biggr) \dd y \, p_\pi(\dd \lambda \mid \boldsymbol{x}) \\
&= t \iint \biggl\{
\biggl(\widehat{\lambda}(y) - \lambda_{\pi,\xx}(y) + \lambda_{\pi,\xx}(y)
\log \frac{\lambda_{\pi,\xx}(y)}{\widehat{\lambda}(y)}\biggr) \\
&\;\;\;\;\;\;+ \biggl(\lambda_{\pi,\xx}(y) - \lambda(y) + \lambda(y) \log \frac{\lambda(y)}{\lambda_{\pi,\xx}(y)}\biggr)
\biggr\} p_\pi(\dd \lambda \mid \boldsymbol{x}) \dd y
\end{align*}

and is minimized when $\widehat{\lambda} = \lambda_{\pi,\xx}$.
Here, we assume that the integral exists.
Thus, the Bayes estimator of $\lambda$
is the posterior mean $\lambda_{\pi,\xx}$ given observation $\xx$.

If the posterior is decomposed as $p_\pi(\dd w \, \dd \blambda \mid \xx) = p_\pi(\dd w \mid \xx) p_\pi(\dd \blambda \mid \xx)$,
then the Bayes estimators of $w$, $\blambda$, and $\lambda$ based on the prior $\pi$ are given by

\begin{align}
w_{\pi,\xx} &= \int w \, p_\pi(\dd w \mid \xx),\ \
\blambda_{\pi,\xx} = \int \blambda \, p_\pi(\dd \blambda \mid \xx), \ \mathrm{and} \ \
\lambda_{\pi,\xx} = w_{\pi,\xx} \blambda_{\pi,\xx}, \notag
\end{align}

respectively.

Next, we consider predictive densities of $\yy$.
The Kullback--Leibler divergence from the probability density $p_\lambda(\yy)$
corresponding to the intensity $t\lambda$
to another probability density $q(\yy) = q(\ynum,y_1,\ldots,y_\ynum) = q(\ynum)q(y_1,\ldots,y_\ynum \mid \ynum)$ is

\begin{align}
D(p_\lambda(\yy),q(\yy))
&= \E_\lambda \left[ \log \frac
{p^\mathrm{Po}_{t\masslambda}(\ynum) \prod_{i=1}^\ynum \overline{\lambda}(y_i)}
{q(\ynum, y_1,\ldots,y_\ynum)} \right],
\end{align}

where $p^\mathrm{Po}_{t\masslambda}(\ynum) := \{(tw)^M/M!\}\exp(-tw)$
denotes the Poisson probability density with mean $t\masslambda$.

Since the posterior mean of the Kullback--Leibler divergence is

\begin{align}
&\int D(p_\lambda(\yy),q(\yy;\xx)) p_\pi(\dd \lambda \mid \xx) \notag \\
&=
\iint \sum_{\ynum=0}^\infty \idotsint p^\mathrm{Po}_{t \masslambda}(\ynum)
\Bigl\{\prod_{i=1}^\ynum \overline{\lambda}(y_i)\Bigr\}
\log \frac{p^\mathrm{Po}_{t \masslambda}(\ynum) \prod_{i=1}^\ynum \overline{\lambda}(y_i)}
{q(\ynum,y_1,\ldots,y_\ynum)} \dd y_1 \cdots \dd y_\ynum
p_\pi(\dd w \, \dd \blambda \mid \xx),
\label{pm-KL}
\end{align}

the Bayesian predictive density minimizing \eqref{pm-KL} is

\begin{align}
p_\pi(\ynum,y_1,\ldots,y_\ynum \mid \xx) =
\iint p^\mathrm{Po}_{t \masslambda}(\ynum) \Bigl\{\prod_{i=1}^\ynum \overline{\lambda}(y_i)\Bigr\}
p_\pi(\dd w \, \dd \blambda \mid \xx). \notag
\end{align}

If the posterior is decomposed as $p_\pi(\dd w \, \dd \blambda \mid \xx) = p_\pi(\dd w \mid \xx) p_\pi(\dd \blambda \mid \xx)$,
then the Bayesian predictive density based on the prior $\pi$ is given by

\begin{align}
p_\pi(\ynum,y_1,\ldots,y_\ynum \mid \xx) = p_\pi(\ynum \mid \xx) p_\pi(y_1,\ldots,y_\ynum \mid \ynum, \xx),
\end{align}

where

\begin{align}
p_\pi(\ynum \mid \xx)
&= \int p^\mathrm{Po}_{t \masslambda}(\ynum) p_\pi(\dd w \mid \xx) \notag
\end{align}

and

\begin{align}
p_\pi(y_1,\ldots,y_\ynum \mid \ynum, \xx) =
\int \Bigl\{\prod_{i=1}^\ynum \overline{\lambda}(y_i)\Bigr\}
p_\pi(\dd \blambda \mid \xx). \notag
\end{align}

\subsection{Kernel mixture models}\label{Kernel}
We need to adopt a prior for $\lambda$ in order to use Bayesian methods.
First, we consider the gamma process prior $\Ga(\alpha, \beta)$,
where $\alpha(u)$ is a positive density function of $u$ and $\beta$ is a positive scalar that does not depend on $u$.
Then, for an arbitrary partition $(B_1,\ldots,B_k)$ of $U$,
$\mu_i := \mu(B_i)$ $(i=1,\ldots,k)$ are independently distributed according to the gamma distributions
$\mathrm{Ga}(\alpha_i,\beta)$ with densities
$\{1/\Gamma(\alpha_i)\} (\mu_i^{\alpha_i-1}/\beta^{\alpha_i}) \exp(-\mu_i/\beta)$,
where $\alpha_i := \alpha(B_i) = \int_{B_i} \alpha(u) \dd u$.
The mixture of the nonhomogeneous Poisson process $\Po(t \lambda)$
with respect to the prior
$\mathcal{G \hspace{-1.7pt} a}(\alpha, \beta)$ for $\lambda$ is
the negative binomial process $\mathcal{N\hspace{-2.75pt}e\hspace{-1.7pt}B\hspace{-1.7pt}i}(\alpha, t \beta/(1 + t \beta))$.
For $\mathcal{N\hspace{-2.75pt}e\hspace{-1.7pt}B\hspace{-1.7pt}i}(\alpha, t \beta/(1 + t \beta))$
and an arbitrary partition $(B_1,\ldots,B_k)$ of $U$,
the numbers $N_i$ of points in $B_i$ $(i=1,\ldots,k)$ are independently distributed according
to the negative binomial distribution with density ${N_i + \alpha_i - 1 \choose \alpha_i}
 \{t \beta/(1 + t \beta)\}^{N_i} \{1/(1 + t \beta)\}^{\alpha_i}$.
The posterior with respect to the prior $\mathcal{G \hspace{-1.7pt} a}(\alpha, \beta)$ and observation $\xx$ is
$\Ga(\alpha + \sum_i \delta_{x_i}, 1/(s+1/\beta))$ \citep[see][]{LoWeng89}.
Then, the posterior means of $w$, $\blambda$, and $\lambda$ are

\begin{align*}
 w_{\alpha,\beta,\xx} = \frac{\beta}{1+s\beta} (|\alpha| + \xnum),~
 \blambda_{\alpha,\beta,\xx} = \frac{\alpha + \sum_i \delta_{x_i}}{|\alpha| + \xnum},~
\mbox{~and~~~}~
 \lambda_{\alpha,\beta,\xx} = \frac{\beta}{1+s\beta} (\alpha + \sum_i \delta_{x_i}),
\end{align*}

respectively,
where $\delta_{x_i}$ denotes the Dirac measure at $x_i$.
Thus, the Bayesian predictive density based on the gamma process prior $\Ga(\alpha, \beta)$ and observation $\xx$ is

\begin{align*}
\NeBi \biggl( \alpha + \sum_i \delta_{x_i}, \frac{(s + 1/\beta)^{-1} t}{1 + (s + 1/\beta)^{-1} t} \biggr)
&= \NeBi \biggl( \alpha + \sum_i \delta_{x_i}, \frac{t \beta}{ 1 + (s + t) \beta} \biggr).
\end{align*}

Although the gamma process prior is a conjugate prior for the nonhomogeneous Poisson model,
it is not natural to use this prior directly for $\lambda$
because the measure $\alpha + \sum_i \delta_{x_i}$ is not absolutely continuous with respect to the intensity measure $\lambda$.
In fact,
the Kullback--Leibler loss of the posterior mean $\lambda_{\alpha,\beta,\xx}$ with respect to
observation $\xx$ and the gamma prior $\Ga(\alpha,\beta)$ is

\begin{align*}
D(t \lambda, t \lambda_{\alpha,\beta,\xx})
=&\
t \masslambda \left\{ \frac{\masslambda_{\alpha,\beta}}{\masslambda} - 1 - \log \frac{\masslambda_{\alpha,\beta}}{\masslambda}
+ \int \blambda(y) \log \frac{\blambda(y)}{\overline{\lambda}_{\alpha,\beta,\xx}(y)} \dd y \right\} \\
=&\
t \masslambda \left\{ \frac{\masslambda_{\alpha,\beta}}{\masslambda} - 1 - \log \frac{\masslambda_{\alpha,\beta}}{\masslambda}
+ \int \blambda(y) \log \frac{\blambda(y)}{\overline{\alpha}(y)} \dd y + \log \biggl( 1+ \frac{\xnum}{|\alpha|} \biggr) \right\},
\end{align*}

where $|\alpha| := \int_U \alpha(u) \dd u$ and $\overline{\alpha} := \alpha/|\alpha|$.
The amount of information concerning $\lambda$ included in $\xx$ increases as $s$ increases.
However, when $s$ goes to infinity
and $t$ is fixed, $\xnum$ and the divergence $D(t \lambda, t \lambda_{\alpha,\beta})$ diverges to infinity.
Here, $w_{\alpha, \beta}$ converges to $w$ with probability $1$.
However, $\overline{\lambda}_{\alpha,\beta,\xx}$ does not converge to $\blambda$ in the Kullback--Leibler sense,
although $\overline{\lambda}_{\alpha,\beta,\xx}$ weakly converges to $\blambda$.

In order to overcome the difficulty caused by the fact that $\alpha + \sum_i \delta_i$ is not absolutely continuous
with respect to $\lambda$, kernel mixture models are widely used.
A nonnegative function $k: U \times U \rightarrow \mathbb{R}_{\geq 0}$
satisfying $\int k(y,u) \dd y = 1$ is called a kernel function.
Suppose that an intensity function $\lambda$ is represented by

\begin{align*}
\lambda(y) &= k(y,\mu) := \int k(y,u) \mu(\dd u),
\end{align*}

where $\mu(\dd u)$ is a finite measure on $U$.
Then,

\begin{align*}
\masslambda &= \abslambda
:= \int \lambda(y) \dd y = \iint k(y,u) \mu(\dd u) \dd y = \int \mu(\dd y) = |\mu|.
\end{align*}

We identify $\mu$ with $(\masslambda, \bmu)$, where $\overline{\mu} := \mu/|\mu|$.
Then, we have

\begin{align*}
\blambda(y) &= \frac{\lambda(y)}{\abslambda} = k(y,\overline{\mu}) = \int k(y,u) \overline{\mu}(\dd u).
\end{align*}

We hereinafter consider kernel mixture models because these models are reasonable under the Kullback--Leibler loss.
The results in the following can be generalized to the setting in which the kernel function has an unknown parameter
by introducing a prior for the unknown parameter.

\vspace{0.4cm}

\noindent
Example. The Gaussian kernel

\begin{align*}
 k(y,u) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \biggl\{ - \frac{1}{2 \sigma^2} (y-u)^2 \biggr\} \hspace{8mm} (y,u \in \mathbb{R})
\end{align*}

is frequently used in applications.
For simplicity, we assume that $\sigma > 0$ is known.
\qed

\vspace{0.4cm}

Assume a gamma process prior $\Ga(\alpha,\beta)$ for $\mu$.
Then, $\overline{\mu} := \mu/\masslambda$ is distributed according to the Dirichlet process $\Di(\alpha)$
\citep[see e.g.][p.~96]{GR03}.
For every partition $(B_1,\ldots,B_k)$ of $U$,
$(\overline{\mu}(B_1),\ldots,\overline{\mu}(B_k))$
is distributed according to the $k$-dimensional Dirichlet distribution $\mathrm{Di}(\alpha(B_1),\ldots,\alpha(B_k))$.
The weight parameter $\masslambda$ is distributed according to the gamma distribution
$\mathrm{Ga}(|\alpha|,\beta)$ independently of $\overline{\mu}$.
Thus, the simultaneous distribution of $w$ and $\overline{\mu}$ is given by

\begin{align}
 p^{\Di}_\alpha(\dd \overline{\mu}) \, p^\mathrm{Ga}_{|\alpha|,\beta}(\masslambda) \, \dd \masslambda,
\label{gammaprior}
\end{align}

where $p^{\Di}_\alpha(\dd \overline{\mu})$ denotes the Dirichlet process measure and
$p^\mathrm{Ga}_{|\alpha|,\beta}(\masslambda) = \{1/\Gamma(|\alpha|)\} (w^{|\alpha|-1}/\beta^{|\alpha|}) \exp (-w/\beta)$
is the gamma probability density.

\subsection{Improper priors}
It is difficult to determine the scale parameter $\beta$ from the viewpoint of objective Bayes.
Let $c$ be an arbitrary positive constant for time scale change.
Then, $\Po(s\lambda) = \Po(\tilde{s}\tilde{\lambda})$,
where $\tilde{s} := cs$ and $\widetilde{\lambda} := \lambda/c$.
Inference for $\lambda$ is equivalent to inference for $\widetilde{\lambda}$
because the time scale change does not affect the essence of the problem.
Thus, the objective prior should be (relatively) invariant with respect to the time scale change.
However, the gamma process prior $\Ga(\alpha,\beta)$ is not relatively invariant if $\beta$ is finite.
One method by which to construct an invariant posterior is to adopt the improper prior

\begin{align}
\pi_\alpha(\mu) \dd \mu = p^{\Di}_\alpha(\dd \overline{\mu}) \, p^\mathrm{Ga}_{|\alpha|,\infty}(\masslambda) \, \dd \masslambda,
\label{gammalimit}
\end{align}

which could be intuitively denoted by $\mu^{\alpha-1} \dd \mu$,
that is obtained by taking the limit $\beta \rightarrow \infty$.
The posterior based on the prior $\pi_\alpha$ is invariant with respect to the time scale change.
The prior \eqref{gammalimit} is a natural generalization of the improper prior
$\prod_{i=1}^d (\mu_i^{\alpha_i-1} \dd \mu_i)$ discussed by \cite{K:AS2004}
for the finite-dimensional independent Poisson model.

Here, we consider a generalization of the gamma process prior \eqref{gammaprior}.
Let

\begin{align}
\pi_{\alpha,\beta,\gamma}(\dd \mu)
:= p^{\Di}_\alpha(\dd \overline{\mu}) \, p^\mathrm{Ga}_{|\alpha|-\gamma,\beta}(\masslambda) \, \dd \masslambda,
\label{extgammaprior}
\end{align}

where $\gamma < |\alpha|$.
We consider

\begin{align}
\label{ourpriormu}
 \pi_{\alpha,\gamma}(\dd \mu) := \pi_{\alpha,\beta=\infty,\gamma}(\dd \mu)
=  p^{\Di}_\alpha(\dd \overline{\mu}) \, p^\mathrm{Ga}_{|\alpha|-\gamma,\infty}(\masslambda) \, \dd \masslambda,
\end{align}

which is a generalization of \eqref{gammalimit},
by taking the limit $\beta \rightarrow \infty$.
We denote the distributions of $\lambda(u) = \int k(u,v) \mu(\dd v)$
corresponding to \eqref{extgammaprior} and \eqref{ourpriormu} by $\pi_{\alpha,\beta,\gamma}(\dd \lambda)$
and $\pi_{\alpha,\gamma}(\dd \lambda)$, respectively,
by abuse of notation without confusion.

We investigate Bayesian inference based on the prior $\pi_{\alpha,\beta,\gamma}$.
From \eqref{ldecomp} and \eqref{extgammaprior},
the posterior distribution of $\mu$ with respect to prior $\pi_{\alpha,\beta,\gamma}$
and observed data $\xx$ is proportional to

\begin{align}
\biggl\{ \prod_{i=1}^\xnum & k(x_i,\overline{\mu}) \biggr\}
p^{\Di}_\alpha(\dd \overline{\mu})
\frac{(s\masslambda)^\xnum}{\xnum!} \exp \bigl( -s\masslambda \bigr)
p^\mathrm{Ga}_{|\alpha|-\gamma,\beta}(\masslambda) \, \dd \masslambda \notag \\
&\propto\
\biggl\{ \prod_{i=1}^\xnum k(x_i,\overline{\mu}) \biggr\}
p^{\Di}_\alpha(\dd \overline{\mu})
p^\mathrm{Ga}_{|\alpha|-\gamma+\xnum,\beta/(s\beta+1)}(\masslambda) \, \dd \masslambda.
\label{posterior}
\end{align}

Thus, the posterior and the Bayesian predictive density for the kernel mixture models
have more complex forms than those for the simple Gamma--Poisson processes.

The posterior mean of $\blambda$ given $\xx$ is

\begin{align}
\blambda_{\alpha, \boldsymbol{x}}(y)
= \frac{\lambda_{\alpha, \beta, \gamma, \xx}(y)}{|\lambda_{\alpha, \beta, \gamma, \boldsymbol{x}}|}
=&\ \frac{\int k(y,\overline{\mu})
\bigl\{ \prod_{i=1}^\xnum k(x_i,\overline{\mu}) \bigr\} p^{\Di}_\alpha(\dd \overline{\mu}) }
{\int \bigl\{ \prod_{i=1}^\xnum k(x_i,\overline{\mu}) \bigr\} p^{\Di}_\alpha(\dd \overline{\mu}) },
\end{align}

not depending on $\beta$, $\gamma$, or $s$.
For precise treatment of quantities related to
posterior distributions based on disintegration, see e.g. \citet{AS:James2005}.

The posterior means of $w$ and $\lambda$ are

\begin{align}
\masslambda_{\absalpha-\gamma,\beta,\xnum}
= \lvert \lambda_{\alpha,\beta,\gamma, \boldsymbol{x}} \rvert
=&\ \frac{1}{s+1/\beta}(\absalpha - \gamma + \xnum)
\label{pomeanw}
\end{align}

and

\begin{align}
\lambda_{\alpha,\beta,\gamma, \boldsymbol{x}}
&= \masslambda_{\absalpha-\gamma,\beta,\xnum} \overline{\lambda}_{\alpha,\xx}
= \frac{1}{s+1/\beta}(\absalpha - \gamma + \xnum)
\blambda_{\alpha, \boldsymbol{x}},
\label{pomeanlambda}
\end{align}

respectively.
Since the posterior density of $w$ depends on $\alpha$ and $\gamma$ only through $\absalpha - \gamma$
and on $\xx=(N,x_1,\ldots,x_N)$ only through $\xnum$, we denote the posterior mean of $w$ by
$\masslambda_{\absalpha - \gamma,\beta,\xnum} := \lvert \lambda_{\alpha,\beta,\gamma, \boldsymbol{x}} \rvert$.

From \eqref{posterior},
the Bayesian predictive density is given by

\begin{align*}
p^\mathrm{NB}_{t/(t+s+1/\beta),\absalpha-\gamma+\xnum}&(\ynum) p_\alpha(y_1,\ldots,y_\ynum \mid \ynum, \xx),
\end{align*}

where

\begin{align}
p_\alpha(y_1,\ldots,y_\ynum \mid \ynum, \xx)
=&\ \frac{\int \bigl\{ \prod_{j=1}^\ynum k(y_j,\overline{\mu}) \bigr\}
\bigl\{ \prod_{i=1}^\xnum k(x_i,\overline{\mu}) \bigr\} p^{\Di}_\alpha(\dd \overline{\mu})}
{\int \bigl\{ \prod_{i=1}^\xnum k(x_i,\overline{\mu}) \bigr\} p^{\Di}_\alpha(\dd \overline{\mu})}.
\label{preddensity}
\end{align}

and

\[
p^\mathrm{NB}_{t/(s+t+1/\beta),\absalpha-\gamma+\xnum}(\ynum) =
\frac{\Gamma(\ynum+\xnum+\absalpha-\gamma)}{\ynum! \Gamma(\xnum+\absalpha-\gamma)}
\left(\frac{t}{t+s+1/\beta}\right)^\ynum
\left(\frac{s+1/\beta}{t+s+1/\beta}\right)^{\absalpha-\gamma+\xnum}.
\]

Here, $p^\mathrm{NB}_{t/(s+t+1/\beta),\absalpha-\gamma+\xnum}(\ynum)$
is the negative binomial distribution
with success probability $t/(t+s+1/\beta)$ and number of failures $\absalpha-\gamma+\xnum$.
We can evaluate the posterior mean \eqref{pomeanlambda} of $\lambda$
and the predictive density \eqref{preddensity} using
Markov chain Monte Carlo (MCMC) methods as discussed in Section \ref{numerical}.

By taking the limit $\beta \rightarrow \infty$,
we obtain the Bayes estimate and the Bayesian predictive density based on the improper prior $\pi_{\alpha,\gamma}$.
From \eqref{posterior}, the posterior with respect to the improper prior $\pi_{\alpha,\gamma}$ and observation $\xx$ is
proportional to

\begin{align*}
\biggl\{ \prod_{i=1}^\xnum & k(x_i,\overline{\mu}) \biggr\}
p^{\Di}_\alpha(\dd \overline{\mu})
\frac{(s\masslambda)^\xnum}{\xnum!} \exp \bigl( -s\masslambda \bigr)
p^\mathrm{Ga}_{|\alpha|-\gamma,\infty}(\masslambda) \, \dd \masslambda \\
&\propto
\biggl\{ \prod_{i=1}^\xnum k(x_i,\overline{\mu}) \biggr\}
p^{\Di}_\alpha(\dd \overline{\mu})
p^\mathrm{Ga}_{|\alpha|-\gamma+\xnum,1/s}(\masslambda) \, \dd \masslambda.
\end{align*}

The posterior means of $\blambda$, $w$ and $\lambda$ with the improper prior $\pi_{\alpha,\gamma}$ are

\begin{align}
\blambda_{\alpha, \boldsymbol{x}}(y)
=&\ \frac{\int k(y,\overline{\mu})
\bigl\{ \prod_{i=1}^\xnum k(x_i,\overline{\mu}) \bigr\} p^{\Di}_\alpha(\dd \overline{\mu}) }
{\int \bigl\{ \prod_{i=1}^\xnum k(x_i,\overline{\mu}) \bigr\} p^{\Di}_\alpha(\dd \overline{\mu}) },
\label{posteriormean}
\end{align}

\begin{align}
\masslambda_{\absalpha-\gamma,\xnum} =&\ \lvert \lambda_{\alpha,\gamma, \boldsymbol{x}} \rvert
= \frac{1}{s} (\absalpha - \gamma + \xnum),
\label{pomeanw2}
\end{align}

and

\begin{align}
\lambda_{\alpha,\gamma, \boldsymbol{x}}
= \masslambda_{\absalpha-\gamma,\xnum} \blambda_{\alpha, \boldsymbol{x}}
= \frac{1}{s} (\absalpha - \gamma + \xnum) \blambda_{\alpha, \boldsymbol{x}},
\label{pomeanlambda2}
\end{align}

respectively.

The Bayesian predictive density with the improper prior $\pi_{\alpha,\gamma}$ is given by

\begin{align*}
p^\mathrm{NB}_{t/(t+s),\absalpha-\gamma+\xnum}&(\ynum) p_\alpha(y_1,\ldots,y_\ynum \mid \ynum, \xx),
\end{align*}

where

\begin{align*}
p^\mathrm{NB}_{t/(t+s),\absalpha-\gamma+\xnum}(\ynum) &=
\frac{\Gamma(\ynum+\xnum+\absalpha-\gamma)}{\ynum! \Gamma(\xnum+\absalpha-\gamma)}
\left(\frac{t}{t+s}\right)^\ynum
\left(\frac{s}{t+s}\right)^{\absalpha-\gamma+\xnum}.
\end{align*}

\subsection{Estimation as infinitesimal prediction}\label{infinitesimal prediction}

For the finite-dimensional independent Poisson model,
Bayesian parameter estimation under the Kullback--Leibler loss
is formulated as infinitesimal Bayesian prediction \citep{K:JMVA2006,K:JMVA15}.
We derive the corresponding results for nonhomogeneous Poisson processes.
Using this formulation,
an integral representation of the Kullback--Leibler risk of a predictive density is obtained.
This representation provides a unified framework for prediction and estimation and is a basis for later discussions.
Intuitively, for the nonhomogeneous Poisson model, estimation is infinitesimal prediction and
prediction is cumulative estimation.

Let
$\boldsymbol{z} = (\xnum+\ynum,x_1,\ldots,x_\xnum,y_1,\ldots,y_\ynum)$.
Then, the density of $\boldsymbol{z}$ is

\begin{align}
p_\lambda(\zz) &= \biggl\{\displaystyle \prod_{i=1}^{\xnum+\ynum} \blambda(z_i) \biggr\}
\frac{\{(s+t)\masslambda\}^{\xnum+\ynum}}{(\xnum+\ynum)!} \exp \bigl\{ - (s+t) \masslambda \bigr\}.
\end{align}

Since the conditional density

\begin{align}
p_\lambda(\boldsymbol{x} \mid \boldsymbol{z})
&= \frac{p_\lambda(\xx,\zz)}{p_\lambda(\zz)}
= \frac{p_\lambda(\xx)p_\lambda(\yy)}{p_\lambda(\zz)}
={\xnum+\ynum \choose \xnum} \biggl(\frac{s}{s+t}\biggr)^\xnum \biggl(\frac{t}{s+t}\biggr)^\ynum
\end{align}

does not depend on $\lambda$,
$\zz$ is a sufficient statistic when $\xx$ and $\yy$ are observed.
We denote $p_\lambda(\xx \mid \zz)$ by $p(\xx \mid \zz)$.
Let
$p_\pi(\xx, \zz) := \int p_\lambda(\xx, \zz) \pi(\dd \lambda)$ and
$p_\pi(\zz) := \int p_\lambda(\zz) \pi(\dd \lambda)$,
where $\pi(\dd \lambda)$ is a prior.
Then,
we have

\begin{align}
p_\pi(\yy \mid \xx) = p_\pi(\zz \mid \xx)
= \frac{p(\xx \mid \zz) p_\pi(\zz)}{p_\pi(\xx)}.
\label{eq:sufficiency1}
\end{align}

We consider a nonhomogeneous Poisson process $\zz_\tau = (N_\tau, z_1, \ldots, z_{N_\tau})$ distributed
according to $\Po(\tau \lambda)$ on $U$ with time $\tau \geq 0$.
When we need to explicitly specify time $\tau$, $z_i$ is denoted by $z_{\tau,i}$.
For $a,b > 0$, the difference between $\zz_{a+b} = (N_{a+b}, z_1, \ldots, z_{N_{a+b}})$
and $\zz_a = (N_a, z_1, \ldots, z_{N_a})$ is defined by
$\zz_{a+b} - \zz_a := (N_{a+b}-N_a, z_{N_a+1}, \ldots, z_{N_{a+b}})$.
For all $a,b > 0$, $\zz_a$ and $\zz_{a+b}-\zz_a$ are independently distributed according to $\Po(a \lambda)$
and $\Po(b \lambda)$,
respectively.
This spatial point process $\zz_\tau$ on $U$ is called a pure immigration process.
Then, the simultaneous distribution of $\xx$ and $\yy$ is identical to that of $\zz_s$ and $\zz_{s+t} - \zz_{s}$.
The probability density of $\zz_\tau$ is denoted by $p_{\lambda,\tau}(\zz_\tau)$.
From \eqref{eq:sufficiency1},
the Kullback--Leibler risk is represented by

\begin{align}
\E_\lambda \left[ \log \frac{p_\lambda(\yy \mid \xx)}{p_\pi(\yy \mid \xx)} \right]
&= \E_\lambda \left[ \log \frac{p_{\lambda,s+t}(\zz_{s+t})}{p_{\pi,s+t}(\zz_{s+t})} \right]
   - \E_\lambda \left[ \log \frac{p_{\lambda,s}(\zz_s)}{p_{\pi,s}(\zz_s)} \right] \notag \\
\label{KLintegral}
&= \int_s^{s+t} \frac{\partial}{\partial \tau}
\E_\lambda \left[ \log \frac{p_{\lambda,\tau}(\zz_\tau)}{p_{\pi,\tau}(\zz_\tau)} \right]
\dd \tau,
\end{align}

where $p_{\pi,\tau}(\zz_\tau) = \int p_{\lambda,\tau}(\zz_\tau) \pi(\dd \lambda)$.
Here, we assume that the expectation
$\E_\lambda \left[ \log \{p_{\lambda,\tau}(\zz_\tau)/p_{\pi,\tau}(\zz_\tau)\} \right]$
exists and is differentiable with respect to $\tau \in [s,s+t]$.

In order to evaluate the integrand of \eqref{KLintegral},
we prepare Lemma \ref{fielddifferential}, which is a generalization of
Lemma \ref{1dimdifferential} below,
which is essentially used in \citet{K:JMVA15}.

\begin{lemma}
\label{1dimdifferential}
Suppose that $N_\tau$ $(\tau \geq 0)$ is distributed according to a Poisson distribution with mean $\tau \masslambda$,
where $w$ is a fixed positive real number.
Let $h$ be a function from the nonnegative integers to the real numbers.
Assume that $\sum_{n=0}^\infty |h(n)| (\theta^n / n!) \exp(-\theta) < \infty$ for all $\theta > 0$.
Then,

\begin{align*}
\frac{\dd}{\dd \tau} \E \Bigl[ h(N_\tau) \Bigr]
&= \E \left[ \Bigl( \frac{N_\tau}{\tau}-\masslambda \Bigr) h(N_\tau) \right]
= \masslambda \, \E \Bigl[ h(N_\tau+1) - h(N_\tau) \Bigr].
\end{align*}

\end{lemma}

\vspace{0.4cm}

The proof is straightforward.

\begin{lemma}
\label{fielddifferential}

Suppose that
$\zz_\tau = (N_\tau,z_1,\ldots,z_{N_\tau})$
is distributed according to the nonhomogeneous Poisson process $\Po(\tau \lambda)$ $(\tau \geq 0)$.

Let $h_n$ $(n=0,1,2,\ldots)$ be functions from $U^n \times \mathbb{R}_{\geq 0}$ to $\mathbb{R}$,
where $U^0 := \emptyset$ and $h_0$ is a constant.
Every function $h_n(z_1,\ldots,z_n,\tau)$ is symmetric with respect to $z_1,\ldots,z_n$
and is differentiable with respect to $\tau \geq 0$ for every fixed $z_1,\ldots,z_n$.
Let $h$ be a function from $(\cup_{n=0}^\infty U^n) \times \mathbb{R}_{\geq 0}$ to $\mathbb{R}$
defined by $h(\zz,\tau) = h(n,z_1,\ldots,z_n,\tau) := h_n(z_1,\ldots,z_n,\tau)$, where $\zz=(n,z_1,\ldots,z_n)$.
Assume that $f(r,s) := \E [h(\zz_r,s)]$ exists and is differentiable with respect to $(r,s) \in \mathbb{R}^2_{>0}$
in a neighborhood of every $r=s>0$
and that $\frac{\partial}{\partial s}\E[h(\zz_r,s)] = \E[\frac{\partial}{\partial s} h(\zz_r,s)]$.

Then,

\begin{align}
\frac{\dd}{\dd \tau} \E \bigl[ h(\zz_\tau,\tau) \bigr]
=&\ \int_U \lambda(y) \E \bigl[ h(\zz_\tau + \delta_y) - h(\zz_\tau) \bigr] \dd y
+ \E \left[ \frac{\partial h}{\partial r} (\zz_\tau,r) \biggr|_{r=\tau} \right],
\label{PoissonFieldStein}
\end{align}

where $\zz_\tau + \delta_y := (N_\tau+1,z_1,\ldots,z_{N_\tau},y)$.

\end{lemma}

\vspace{0.4cm}
The proof is given in Appendix A.

Now, we have an integral representation of the risk of a Bayesian predictive density.

\vspace{0.4cm}

\begin{theorem}\label{PRF}

Suppose that
$\zz_\tau = (N_\tau,z_1,\ldots,z_{N_\tau})$
is distributed according to the nonhomogeneous Poisson process $\Po(\tau \lambda)$ $(\tau \geq 0)$.

Assume that $f(r,s) := \E [\log \{p_{s,\lambda}(\zz_r)/p_{s,\pi}(\zz_r)\}]$ exists
and is differentiable with respect to $(r,s) \in \mathbb{R}^2_{>0}$
in a neighborhood of every $r=s>0$
and that $\frac{\partial}{\partial s}\E[\log \{p_{s,\lambda}(\zz_r)/p_{s,\pi}(\zz_r)\}]
= \E[\frac{\partial}{\partial s} \log \{p_{s,\lambda}(\zz_r)/p_{s,\pi}(\zz_r)\}]$.

Then,

\begin{align}
\E_\lambda \left[ \log \frac{p_\lambda(\yy \mid \xx)}{p_\pi(\yy \mid \xx)} \right]
=&\ \int_s^{s+t} \E_\lambda \left[ w_{\pi,\zz_\tau,\tau} - w
- w \log \frac{w_{\pi,\zz_\tau,\tau}}{w}
+ w \int_U \blambda (y) \log \frac{\blambda(y)}{\blambda_{\pi,\zz_\tau,\tau}(y)} \dd y \right] \dd \tau \notag \\
\label{integralrepresentation}
=&\ \int_s^{s+t} \E_\lambda \left[ \int_U \biggl\{
\lambda_{\pi,\zz_\tau,\tau}(y) - \lambda(y)
+ \lambda(y) \log \frac{\lambda(y)}{\lambda_{\pi,\zz_\tau,\tau}(y)} \biggr\} \dd y \right] \dd \tau.
\end{align}

\end{theorem}

\vspace{0.4cm}

The proof is given in Appendix A.

Since the integrand \eqref{integralrepresentation}
multiplied by $t$ coincides with the Kullback--Leibler loss \eqref{eq:KLeat}
for the intensity estimator $\lambda_{\pi,\zz,\tau}$,
Bayes estimation under the Kullback--Leibler loss for the nonhomogeneous Poisson model
can be regarded as infinitesimal Bayesian prediction
as in the finite-dimensional independent Poisson model.

When the prior is decomposable as $\pi(\dd \lambda) = \pi(\dd w) \pi(\dd \overline{\lambda})$,
the assumption in Theorem \ref{PRF} and Theorem \ref{domination} in the next subsection
can be easily verified in many problems.
If $\pi(\dd \lambda) = \pi(\dd w) \pi(\dd \overline{\lambda})$, then

\begin{align}
f(r,s)&=
\E_\lambda \left[
\log
\frac
{\prod^{N_r}_{i=1} \lambda (z_{r,i}) \exp ( -s \masslambda )}
{\int \prod^{N_r}_{j=1} \tlambda (z_{r,j}) \exp ( -s \tmasslambda ) \pi ( \dd \tlambda)}
\right] \notag \\
&= \E_\lambda \left[
\log
\frac
{\prod^{N_r}_{i=1} \masslambda^{N_r} \exp ( -s \masslambda )}
{\int \prod^{N_r}_{j=1} \tmasslambda^{N_r} \exp ( -s \tmasslambda ) \pi ( \dd \tmasslambda)}
\right]
+
\E_\lambda \left[
\log
\frac
{\prod^{N_r}_{i=1} \blambda (z_{r,i})}
{\int \prod^{N_r}_{j=1} \btlambda (z_{r,j}) \pi ( \dd \btlambda)}
\right].
\label{decomposition}
\end{align}

The first term in \eqref{decomposition} can be explicitly evaluated for our priors.
The second term in \eqref{decomposition} is represented by

\begin{align}
h(r) := \E_\lambda \left[\log \biggl\{
\frac{\prod^n_{j=1} \overline{\lambda}(z_j)}
{\int \prod^n_{k=1} \overline{\lambda}_{\overline{\mu}} (z_k) \pi ( \dd \overline{\mu} )} \biggr\} \right]
= \exp (- rw) \sum^\infty_{n = 0} \frac{(rw)^n}{n!} c(n),
\label{power1}
\end{align}

where

\[
c(n) := \idotsint
\biggl\{ \prod^n_{i=1} \overline{ \lambda } ( z_i) \biggr\}
\log \biggl\{
\frac{\prod^n_{j=1} \overline{\lambda}(z_j)}
{\int \prod^n_{k=1} \overline{\lambda}_{\overline{\mu}} (z_k) \pi ( \dd \overline{\mu} )} \biggr\}
\dd z_1 \dotsb \dd z_n.
\]

Here, $c(n)$ is the cumulative Kullback--Leibler risk when we predict
$z_1,\ldots,z_n$ independently distributed according to a probability density $\blambda$
using the simultaneous Bayesian predictive density
$\int \prod^{N_r}_{j=1} \btlambda (z_{r,j}) \pi ( \dd \btlambda)$.

Therefore, in order to verify the assumption in Theorem \ref{PRF}, it is sufficient
to show that \eqref{power1} converges for every $0 \leq r < \infty$.

In many problems, we can verify that \eqref{power1} converges for every $0 \leq r < \infty$
using the inequality

\begin{align}
0 \leq c(n) \leq n \biggl\{ \sup_{\overline{\mu} \in B} D(\overline{\lambda},\overline{\lambda}_{\overline{\mu}})
+ \int_B \pi(\dd \overline{\mu}) \biggr\},
\label{resolv}
\end{align}

where $B$ is an arbitrary measurable set in the space of intensity functions.
The inequality \eqref{resolv} is used to define the index of resolvability
by \citet{B:BS99} \citep[see also][]{BC:IEEEIT91}.
Thus, a sufficient condition is that there exists $B$ such that
$\sup_{\overline{\mu} \in B} D(\overline{\lambda},\overline{\lambda}_{\overline{\mu}}) < \infty$ and $\pi(B) > 0$.
For example, the nonhomogeneous Poisson model with the gamma prior and without a kernel discussed in Subsection \ref{Kernel}
does not satisfy this condition.

If the power series of $r$ in \eqref{power1} converges for all $0 \leq r < \infty$, we have

\begin{align}
 \frac{ \dd}{ \dd r}  h(r)& = -w \exp ( -rw) \sum^\infty_{ n=0} \frac{ (rw)^n}{n!} c (n)
+ \exp ( -rw) \sum^\infty_{n=1} \frac{ w ( r w )^{n-1}}{ ( n -1) ! } c (n) \notag \\
&= w \exp ( -rw) \sum^\infty_{n=0}  \frac{ ( r w )^{n-1}}{n!} \{ c ( n+1) - c(n) \}
\label{power2} \\
&= w \E_\lambda \left[
\int \overline{\lambda}(y)
\log
\frac
{\overline{\lambda}(y)}
{\overline{\lambda}_{\pi,\mathbf{z}_r}(y)}
\dd y
\right], \notag
\end{align}

because

\begin{align}
c(n+1) - c(n)
&= \int \overline{ \lambda } (y)  \idotsint \Bigl\{ \prod^n_{i=1} \overline{\lambda}(z_i) \Bigr\}
\log
\left\{
\frac
{\overline{\lambda}(y) \prod^n_{ i=1} \overline{ \lambda } ( z_i ) }
{\int \overline{\lambda}_{\overline{\mu}} (y) \prod^n_{ i=1} \overline{ \lambda }_{ \overline{\mu} } (z_i)
\pi (\dd \overline{\mu})}
\right\}
\dd z_1 \dotsb \dd z_n \dd y \notag \\
&\ \ \ \, -  \idotsint
\Bigl\{ \prod^n_{ i=1}  \overline{ \lambda }(z_i) \Bigr\}
 \log \left\{
 \frac{\prod^n_{i=1}\overline{\lambda}(z_i)}
 {\prod^n_{i=1} \int \overline{ \lambda }_{ \overline{\mu} } (z_i) \pi(\dd \overline{\mu})}
 \right\}
\dd z_1 \dotsb \dd z_n \notag \\
&= \idotsint \Bigl\{ \prod^n_{ i=1} \overline{ \lambda } (z_i) \Bigr\}
\int \overline{ \lambda } (y)
\log \frac
{\overline{ \lambda } (y)}
{\overline{ \lambda }_{ \pi , z^n} (y)}
\dd y \dd z_1 \dotsb \dd z_n,
\label{individual}
\end{align}

where

\begin{align*}
\overline{ \lambda }_{\pi , z^n} (y)
= \frac{\int \prod^n_{ i=1}
\overline{ \lambda }_{ \overline{\mu} } (z_i)
\overline{ \lambda }_{ \overline{\mu} } (y) \pi ( \dd  \overline{\mu} ) }
{\int \prod^n_{j=1}
\overline{ \lambda }_{ \overline{\mu} } (z_j) \pi ( \dd  \overline{\mu} )}.
\end{align*}

Thus, the difference \eqref{individual} between $c(n+1)$ and $c(n)$ is the individual Kullback--Leibler risk
of the Bayesian predictive density for predicting $z_{n+1}$ using $z_1,\ldots,z_n$,
where $z_1,\ldots,z_{n+1}$ are independently distributed according to probability density $\blambda$.
The power series of $r$ in \eqref{power1} converges for $0 \leq r < \infty$
if and only if the power series in \eqref{power2} converges for $0 \leq r < \infty$.

\subsection{Shrinkage priors dominating the prior $\pi_\alpha$ and their admissibility}

We propose shrinkage priors dominating $\pi_\alpha(\dd \mu) = \mu^{\alpha-1} \dd \mu$ when $|\alpha| > 1$.
We prove admissibility of Bayes estimators and Bayesian predictive densities based on the shrinkage priors.

Although few studies on admissibility concerning infinite-dimensional models
have been carried out,
we show that Blyth's method with a convex loss is useful for our infinite-dimensional problem
because the method works even when the support of the prior is a small subset of the whole parameter space.
The key idea is to decompose the problem into two sub-problems:
a one-dimensional problem with an improper prior, and an infinite-dimensional problem with a proper prior.

\vspace{0.4cm}

\begin{theorem}\label{domination}~~

Suppose that
$\zz_\tau = (N_\tau,z_1,\ldots,z_{N_\tau})$
is distributed according to the nonhomogeneous Poisson process $\Po(\tau \lambda)$ $(\tau \geq 0)$.

Assume that $f(r,s) := \E [\log \{p_{s,\lambda}(\zz_r)/p_{s,\pi}(\zz_r)\}]$ exists and is differentiable with respect to $(r,s) \in \mathbb{R}^2_{>0}$
in a neighborhood of every $r=s>0$
and that $\frac{\partial}{\partial s}\E[\log \{p_{s,\lambda}(\zz_r)/p_{s,\pi}(\zz_r)\}]=
\E[\frac{\partial}{\partial s} \log \{p_{s,\lambda}(\zz_r)/p_{s,\pi}(\zz_r)\}]$.

\begin{enumerate}
\item
When $|\alpha|-\gamma > 1$,
the Bayesian predictive density $p_{\alpha, \gamma}(\yy \mid \xx)$
based on
$\pi_{\alpha,\gamma}(\dd \mu)$
is dominated by the Bayesian predictive density $p_{\alpha, \tgamma}(\yy \mid \xx)$
based on $\pi_{\alpha, \tgamma}(\dd \mu)$,
where $\tgamma := |\alpha| -1$.
\vspace{0.3cm}
\item When $|\alpha|-\gamma > 1$,
the generalized Bayes estimator of $\lambda$
based on
$\pi_{\alpha,\gamma}(\dd \mu)$
is dominated by the generalized Bayes estimator of $\lambda$
based on $\pi_{\alpha, \tgamma}(\dd \mu)$,
where $\tgamma := |\alpha| -1$.
\end{enumerate}
\end{theorem}

\vspace{0.4cm}

The proof is given in Appendix A.

In particular,
if $|\alpha| > 1$, the Bayesian predictive density based on
$\pi_{\alpha}(\dd \mu) = \pi_{\alpha,\gamma=0}(\dd \mu) = \mu^{\alpha-1} \dd \mu$
is dominated by the Bayesian predictive density based on $\pi_{\alpha,\tgamma=|\alpha|-1}(\dd \mu)$.

\vspace{0.4cm}
\begin{lemma}\label{1dimPoisson}
Let $X$ be a Poisson random variable with mean $\theta \geq 0$, and let $c$ be an arbitrary positive constant.
Then,

\begin{align*}
\theta \, \E_\theta& \biggl[
\log (X+1+c) - \log (X+1) \biggr]
\leq c - c \exp(-\theta) < c.
\end{align*}

\label{Poissonineq}
\end{lemma}

\vspace{0.4cm}

The proof of Lemma \ref{1dimPoisson}, which is used in \citet{K:AS2004}, is given in Appendix A. for self-containedness.

Next, we prove admissibility of Bayes estimators and Bayesian predictive densities based on our shrinkage priors.

Suppose that the parameter space $\Lambda$ is a set of finite measures
that are mutually absolutely continuous with respect to the Lebesgue measure on $U \subset \mathbb{R}^d$.
We assume that if $\lambda = (w,\overline{\lambda}) \in \Lambda$, then $(w',\overline{\lambda}) \in \Lambda$ for all $w' > 0$.
Thus, $\Lambda = \mathbb{R}^+ \times \overline{\Lambda}$,
where $\mathbb{R}^+$ is the set of positive real numbers
and $\overline{\Lambda}$ is a set of probability densities
that are mutually absolutely continuous with respect to the Lebesgue measure.

Note that the support of a prior for the kernel mixture model only covers a small subset
of the whole parameter space $\Lambda$ if $\Lambda$ is a large set
as in ordinary settings for nonparametric inference.

Let $\mathcal{A}$ be the space of all finite measures on $U$ such that the measure is mutually absolutely continuous
with respect to the Lebesgue measure.
For estimation of $\lambda$, we choose an intensity estimate from $\mathcal{A}$.
Let $\mathcal{P}$ be the space of all probability measures
on $\cup_{m=0}^\infty U^m$
such that the marginal probability $P(m)$ of $P \in \mathcal{P}$
is positive (i.e.\;not equal to $0$) for every nonnegative integer $m$ and
the conditional probability $P(\cdot \mid m)$ on $U^m$ has density with respect to
the Lebesgue measure on $U^m \subset \mathbb{R}^m$ for every $m$.
For prediction problem, we choose a probability density from $\mathcal{P}$.

We assume that $\int k(y,u) \overline{\mu}(\dd u) \in \overline{\Lambda}$ with probability 1
if $\bmu$ is distributed according to $\Di(\alpha)$.
This condition is naturally satisfied in ordinary settings for nonparametric intensity estimation.

\vspace{0.4cm}

\begin{theorem}\label{admissible}~~~~

\begin{enumerate}
 \item
The Bayesian predictive density based on
$\pi_{\alpha,\gamma}(\dd \mu)
:=p^{\Di}_\alpha(\dd \overline{\mu}) \, p^\mathrm{Ga}_{|\alpha|-\gamma,\infty}(\masslambda) \dd \masslambda$
is admissible under the Kullback--Leibler loss if $|\alpha| - 1 \leq \gamma < |\alpha|$.
\vspace{0.3cm}

\item
The generalized Bayes estimator $\lambda_{\alpha,\gamma,\xx}$ based on
$\pi_{\alpha,\gamma}(\dd \mu)
:=p^{\Di}_\alpha(\dd \overline{\mu}) \, p^\mathrm{Ga}_{|\alpha|-\gamma,\infty}(\masslambda) \dd \masslambda$
is admissible under the Kullback--Leibler loss if $|\alpha| - 1 \leq \gamma < |\alpha|$.

\end{enumerate}
\end{theorem}

\vspace{0.4cm}

The proof is given in Appendix A.

In particular,
if $|\alpha| > 1$, then the Bayesian predictive density based on $\pi_{\alpha,\gamma=|\alpha|-1}$
dominating that based on $\pi_{\alpha} = \pi_{\alpha,\gamma=0}$ is admissible.

\section{Numerical evaluation of predictive densities and estimators}\label{numerical}

In this section, we explore numerical methods by which to evaluate Bayesian predictive densities and Bayes estimates.

In the posterior \eqref{posterior} with respect to the prior
$\pi_{\alpha,\beta,\gamma}(\dd \mu) :=  p^{\Di}_\alpha(\dd \overline{\mu}) \, p^\mathrm{Ga}_{|\alpha|-\gamma,\beta}(\masslambda) \dd w$,
$w$ and $\overline{\mu}$ are independently distributed.
The posterior distribution of $w$ is $p^\mathrm{Ga}_{|\alpha|-\gamma+\xnum,\beta/(s\beta+1)}(\masslambda) \, \dd \masslambda$.
The posterior distribution of $\overline{\mu}$ is proportional to a Dirichlet process mixture

\begin{align}
\int & \biggl\{ \prod_{l=1}^\xnum \int k(x_l,u_l) \overline{\mu}(\dd u_l) \biggr\} p_\alpha^{\Di} (\dd \overline{\mu})
= \idotsint \biggl\{ \prod_{l=1}^\xnum k(x_l,u_l) \biggr\}
p^\mathrm{CR}_{\alpha}(\dd u_1,\ldots, \dd u_\xnum),
\label{CRrepresentation}
\end{align}

where

\begin{align*}
p^\mathrm{CR}_{\alpha}(\dd u_1,\ldots, \dd u_\xnum)
= \int \Bigl\{ \prod_{l=1}^\xnum \overline{\mu}(\dd u_l) \Bigr\} p_\alpha^{\Di} (\dd \overline{\mu})
\end{align*}

is the distribution of the Chinese restaurant process with measure $\alpha$
\citep[e.g.][p.~193]{Phadia16}.
If random variables $u_1,u_2,\ldots,u_\xnum$ are
distributed according to
a Chinese restaurant process $P^{\mathrm{CR}}_\alpha(\dd u_1, \ldots, \dd u_\xnum)$,
then they are sequentially distributed according to

\begin{align*}
P^{\mathrm{CR}}_\alpha(\dd u_1)
= \frac{1}{|\alpha|} \alpha(\dd u_1)
\end{align*}

and

\begin{align*}
P^{\mathrm{CR}}_\alpha(\dd u_{k+1} \mid u_1,\ldots,u_k)
= \frac{1}{|\alpha| + k} \Bigl\{ \alpha(\dd u_{k+1})
+ \sum_{i=1}^k \delta_{u_i} (\dd u_{k+1})
\Bigr\} \hspace{0.5cm} (k=1,2,\ldots,\xnum-1),
\end{align*}

see e.g.\ \citet{Phadia16}.

We can numerically evaluate quantities concerning the posterior density of $\lambda$ such as
the Bayes estimate of $\lambda$ and Bayesian predictive density of $\yy$ given $\xx$
using MCMC based on the representation \eqref{CRrepresentation}.
Various MCMC methods for density estimation
\citep[e.g.][]{Phadia16} based on nonparametric Bayes with Dirichlet priors can be applied
to our problem.

From \eqref{posteriormean}, the posterior mean $\overline{\lambda}$ with respect to
$\pi_{\alpha,\beta,\gamma}$ and observation $\xx$ is

\begin{align}
\overline{\lambda}_{\alpha,\xx}
&=
\frac{\int k(y,\overline{\mu}) \{ \prod_{l=1}^\xnum k(x_l,\overline{\mu}) \} p_\alpha^{\Di} (\dd \overline{\mu})}
{\int \{ \prod_{l=1}^\xnum k(x_l,\overline{\mu}) \} p_\alpha^{\Di} (\dd \overline{\mu})} \notag \\
&=
\frac{\idotsint k(y,u_{\xnum+1}) \{\prod_{l=1}^\xnum k(x_l,u_l)\}
p^\mathrm{CR}_{\alpha}(\dd u_1,\ldots, \dd u_\xnum, \dd u_{\xnum+1})}
{\idotsint \{ \prod_{l=1}^\xnum k(x_l,u_l)\}
p^\mathrm{CR}_{\alpha}(\dd u_1,\ldots,\dd u_\xnum)} \notag \\
&=
\frac{\idotsint k(y,u_{\xnum+1}) p^\mathrm{CR}_{\alpha}(\dd u_{\xnum+1} \mid u_1,\ldots,u_\xnum) \{\prod_{l=1}^\xnum k(x_l,u_l)\}
p^\mathrm{CR}_{\alpha}(\dd u_1,\ldots, \dd u_\xnum)}
{\idotsint \{ \prod_{l=1}^\xnum k(x_l,u_l)\}
p^\mathrm{CR}_{\alpha}(\dd u_1,\ldots, \dd u_\xnum)},
\label{posmeanlambdabar}
\end{align}

which does not depend on $\beta$ or $\gamma$.
Thus, the posterior mean of $\overline{\lambda}$ with respect to the prior
$\pi_{\alpha,\beta,\gamma}$ is $\overline{\lambda}_{\alpha,\xx}$,
which does not depend on $\beta$ or $\gamma$.

We can obtain the Bayes estimates of $\lambda$ based on $\pi_{\alpha,\beta,\gamma}$
and $\pi_{\alpha,\gamma} := \pi_{\alpha,\beta=\infty,\gamma}$
using \eqref{pomeanw}, \eqref{pomeanlambda}, \eqref{pomeanw2}, \eqref{pomeanlambda2}, and
\eqref{posmeanlambdabar}.

\vspace{0.4cm}

\noindent
Example.
We consider intensity functions on a unit circle $[0,2\pi)$
for simplicity.
Let the true intensity function be

\[
 \lambda(u) = \sin(u) + 2.
\]

We obtain Bayes estimates of $\lambda$ based on priors $\pi_{\alpha=1,\beta=\infty,\gamma=0}$
and $\pi_{\alpha=1,\beta=\infty,\gamma=|\alpha|-1}$.
Then, $|\alpha| := \int_0^{2\pi} \alpha(u) \dd u = 2 \pi > 1$.
We adopt the von Mises kernel

\begin{align*}
 k_\kappa(x;\mu) = \frac{\exp \{\kappa \cos (x-\mu) \}}{2 \pi I_0(\kappa)}
\end{align*}

with $\kappa = 5$.
The observation time length $s$ is set to be $1$.

In Figure 1, Bayesian estimates of $\lambda$ based on non-shrinkage prior $\pi_{\alpha=1,\beta=\infty,\gamma=0}$
and shrinkage prior $\pi_{\alpha=1,\beta=\infty,\gamma=|\alpha|-1}$
with respect to observation $
(0.29, 1.55, 2.06, 2.85, 2.87, 3.60, 5.55, 5.61, 5.65, 6.01)$ are shown.

The Bayesian estimates based on the non-shrinkage prior and the shrinkage prior are
$\lambda_{\alpha = 1,\gamma = 0, \boldsymbol{x}}
= (\xnum + 2 \pi) \blambda_{\alpha, \boldsymbol{x}}$,
and
$\lambda_{\alpha = 1,\gamma = \absalpha-1, \boldsymbol{x}}
= (\xnum + 1) \blambda_{\alpha, \boldsymbol{x}}$,
respectively,
where $\blambda_{\alpha, \boldsymbol{x}}$ is numerically evaluated by MCMC.

From Theorem \ref{domination},
the estimator based on the shrinkage prior dominates that based on the non-shrinkage prior.
Specifically, the Kullback--Leibler risk of the estimator based on the shrinkage prior is smaller
than that of the estimator based on the non-shrinkage prior for all $\lambda$.
This example illustrates how shrinkage priors improve Bayes estimators based on non-shrinkage priors.

\begin{figure}[!htbp]
\begin{center}
\includegraphics[height=8cm]{allfigure.eps}
\end{center}
\caption{Bayesian estimates of $\lambda$ based on non-shrinkage prior $\pi_{\alpha=1,\beta=\infty,\gamma=0}$
and shrinkage prior $\pi_{\alpha=1,\beta=\infty,\gamma=|\alpha|-1}$.
The (sorted) observed points indicated by the vertical ticks at the bottom
are located at $0.29, 1.55, 2.06, 2.85, 2.87, 3.60, 5.55, 5.61, 5.65$, and $6.01$.}
\end{figure}

\section{Discussion and Conclusion}

A class of improper shrinkage priors for nonparametric Bayesian inference
of the nonhomogeneous Poisson processes with kernel mixture is investigated.
The Bayesian predictive densities and the Bayes estimators based on the priors are admissible
under the Kullback--Leibler loss.
The class of improper  priors could be useful as objective priors for nonhomogeneous Poisson models.

The information theoretic properties of the Kullback--Leibler loss play essential roles in our theory.
Our results can be easily generalized to various methods for intensity estimation
including models based on kernels with parameters.

Although few studies on admissibility of Bayesian prediction and inference based on improper priors
for infinite-dimensional models have been carried out,
our approach could be useful for various infinite-dimensional problems.

One important direction of future research is the generalization of the present results
for nonhomogeneous Poisson process models
to those for other stochastic process models such as nonhomogeneous negative binomial process models.
For finite dimensional models, \citet{HK:MMS2019,HK:JMVA2020} extended the theory for finite dimensional Poisson models
to finite dimensional negative binomial models and negative multinomial models.
These generalizations require techniques not used in the theory for the Poisson models.

Hazard rate models closely related to nonhomogeneous Poisson models are widely used in applications and kernel mixtures with general random
measures have been investigated \citep{AS:James2005, AS:DBPP2009}.
The use of various random measures other than the gamma random measure has been studied.
The relation between theories on hazard rate models with general random measures
and the present study is also an important topic for future study.

Nonhomogeneous Poisson process models have a variety of applications.
For example, they have recently been used in neural information processing \citep{SK17,SK20}.
The development of data analyzing methods for such applications based on the theory presented here is also a future challenge.
In applications such as survival analysis, weighted gamma processes are often used as priors.
By extending the theorems in \citet{K:JMVA15} to those for infinite-dimensional models,
we could generalize our results to improve the limits of weighted gamma process priors.

Furthermore, estimation and prediction with infinite dimensional statistical models based on Brownian motions
could be studied in line with the approach adopted in the present paper.

\vspace{0.4cm}

\noindent
{\bf Acknowledgments.}
The author is grateful to two reviewers for their helpful and constructive comments.
This research was supported by MEXT KAKENHI Grant Number 16H06533,
JST CREST Grant Number JPMJCR1763,
and AMED Grant Numbers JP20dm0207001 and JP20dm0307009.

\vspace{0.4cm}

\bibliographystyle{sjs}


text:
In \cite{DFK08}, we introduced a family of weighted graphs $\Gamma_\bm$
constructed from a Motzkin path $\bm$. Here we recall the definition of these graphs, as well as the sequence of bijections which brings them to form of the weighted graphs $G_\bm$ which we use in the main body of the paper. It is important to understand this connection, because the most basic proof of positivity relies on the fact that the graphs $\Gamma_\bm$ are positively weighted with weights which are monomials in the initial data $R(\bm)$.

In fact, we describe a sequence of simple bijections
$$
\bm \leftrightarrow \Gamma_\bm \leftrightarrow \Gamma'_\bm \leftrightarrow G_\bm.
$$
The graph $\Gamma_\bm'$ is the ``compactified" graph of \cite{DFK09}. Note that in general, only the graph $\Gamma_\bm$ has manifestly positive weights.

Below, let $\bm$ be a fixed Motzkin path of length $r$.
\subsection{The graphs $\Gamma_\bm$}
\begin{enumerate}
\item Decompose the Motzkin path $\bm$ into a disjoint union of maximal ``descending" segments, including those of length one, such that $\bm = \underset{i}{\sqcup} (m_i,m_{i+1}, ..., m_{i+k_i-1})$. Here,
 $m_{i+a} = m_{i}-a$, $0\leq a < k$ in the segment beginning with index $i$ and having length $k$.
 \begin{example}\label{samplemot}
 If $\bm = (2,1,2,2,3,2,1,1)$ then
 $\bm = (2,1)\sqcup (2)\sqcup(2) \sqcup (3,2,1) \sqcup (1)$.
 \end{example}
 For notational purposes, we call the maximal descending segment starting with $\bm_i$ $\bm^{(i)}$, so that $\bm = \sqcup_i \bm^{(i)}$.
\item To each maximal descending segment $\bm^{(i)}$ of length, say, $k$, is associated a graph
$\Gamma_{\bm^{(i)}}$. It has $2(k+1)$ vertices, labeled $(i,i+1,...,i+k)$ and
$(i',(i+1)',...,(i+k)')$. It has the following list of
oriented edges: $(a,a+1), (a+1,a), (a,a'), (a',a)$ (for all
$a$), and $(b,a)$ for all $k+i\geq b>a+1 \geq 2$. Below we give a picture of the graph $\Gamma_{\bm^{(1)}}$ for the example of the descending subpath $\bm^{(1)}=(3,2,1,0)$.
\begin{center}
\begin{tabular}{ccc}
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(15,22)
\psline(0,17.5)(15,2.5)
\multips(0,17.5)(5,-5){4}{\pscircle*[linecolor=blue](0,0){.7}}
\rput(2,17.5){$m_k$}
\rput(8,12.5){$m_{k-1}$}
\rput(17,2.5){$m_1$}
\rput(7.5,-2){$\mathbf m^{(1)}$}
\end{pspicture}
&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(5,22)
\psline[arrowsize=1.5]{->}(0,10)(5,10)
\end{pspicture}&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(12,22)
\psline(5,0)(5,20)
\multips(0,0)(0,5){5}{\psline(5,0)}
\multips(0,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}
\rput(0,10){\psarc[linecolor=blue,arrowsize=1.5]{-}{11.5}{-60}{0}}
\rput(0,10){\psarc[linecolor=blue,arrowsize=1.5]{<-}{11.5}{0}{60}}
\rput(1,7.5){\psarc[linecolor=blue]{-}{8.75}{-60}{0}}
\rput(1,7.5){\psarc[linecolor=blue,arrowsize=1.5]{<-}{8.75}{0}{60}}
\rput(2.5,5){\psarc[linecolor=blue]{-}{6}{-60}{0}}
\rput(2.5,5){\psarc[linecolor=blue,arrowsize=1.5]{<-}{6}{0}{60}}
\rput(1,12.5){\psarc[linecolor=blue]{-}{8.75}{-60}{0}}
\rput(1,12.5){\psarc[linecolor=blue,arrowsize=1.5]{<-}{8.75}{0}{60}}
\rput(2.5,10){\psarc[linecolor=blue]{-}{6}{-60}{0}}
\rput(2.5,10){\psarc[linecolor=blue,arrowsize=1.5]{<-}{6}{0}{60}}
\rput(2.5,15){\psarc[linecolor=blue]{-}{6}{-60}{0}}
\rput(2.5,15){\psarc[linecolor=blue,arrowsize=1.5]{<-}{6}{0}{60}}
\multips(5,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}

\rput(7,0){1}
\rput(-2,0){1'}
\rput(7,5){2}
\rput(-2,5){2'}
\rput(9,20){$k+1$}
\rput(-5,20){$(k+1)'$}
\rput(2,-3){$\Gamma_{\mathbf m^{(1)}}$}
\end{pspicture}
\end{tabular}
\end{center}
\vskip.1in
(In this picture, an unoriented edge on the graph $\Gamma$ is considered to be doubly-oriented, for ease of reading.)
\item Consecutive maximal descending segments
 are separated by either a ``flat'' step of the form $m_a=m_{a+1}$, or an
``ascending'' step, $m_a+1=m_{a+1}$.
\begin{example}
For the Motzkin path in Example \ref{samplemot}, the step separating $\bm^{(1)}=(2,1)$ and $\bm^{(3)}=(2)$ is ascending, and the step separating $\bm^{(3)}=(2)$ from $\bm^{(4)}=(2)$ is flat.
\end{example}

We ``glue" consecutive graphs $\Gamma_1=\Gamma_{\bm^{(i)}}$ and $\Gamma_2=\Gamma_{\bm^{(j)}}$ where $m^{(i)}$ and $m^{(j)}$ are neighboring descending segments in $\bm$ with $j>i$. The gluing procedure is dictated by whether the segments are separated by an ascending or flat step. We assume $\bm^{(i)}$ has length $k_i$ and $\bm^{(j)}$ has length $k_j$.

\begin{enumerate}
\item If the separation between the segments is flat, we identify the edge and vertices $(j',j)\in \Gamma_2$ with $((k_i+i)',k_i+i)\in \Gamma_1$.

\item If the separation between the segments is ``ascending'', we
  identify the vertices and edges $(j,j')\in\Gamma_2$ with $((k_i+i)',k_i+i)\in \Gamma_1$ (i.e., the identification is in the opposite sense).

\end{enumerate}

\end{enumerate}
We define $\Gamma_\bm$ to be the result of gluing the graphs
corresponding to consecutive strictly descending pieces of $\bm$
according to the procedure given in (4) above.
\begin{example}\label{exzero}
The purely ascending Motzkin path $\bm=\bm_1=(0,1,...,k)$ maps to the chain with $2k+2$ vertices, connected by doubly-oriented edges. The flat Motzkin path $\bm_0=(0,...,0)$ of length $k$ maps to the graph
\begin{center}

\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(12,22)
\psline(5,0)(5,20)
\multips(0,0)(0,5){5}{\psline(5,0)}
\multips(0,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}
\multips(5,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}

\rput(7,0){1}
\rput(-2,0){1'}
\rput(7,5){2}
\rput(-2,5){2'}
\rput(9,20){$k+1$}
\rput(-5,20){$(k+1)'$}
\rput(-10,10){$\Gamma_{\mathbf m_0}=$}
\end{pspicture}
\end{center}
As an example of the gluing procedure, in the case of $r=3$ and $\bm=(0,1,0)$, we have
\begin{center}

\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(12,22)
\psline(0,0)(5,0)(5,20)(0,20)
\psline(0,15)(5,15)

\rput(2.5,15){\psarc[linecolor=blue]{-}{6}{-60}{0}}
\rput(2.5,15){\psarc[linecolor=blue,arrowsize=1.5]{<-}{6}{0}{60}}
\multips(0,15)(0,5){2}{\pscircle*[linecolor=red](0,0){.7}}
\rput(0,0){\pscircle*[linecolor=red](0,0){.7}}
\multips(5,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}

\rput(-10,10){$\Gamma_{(0,1,0)}=$}
\end{pspicture}
\end{center}
\end{example}
$\Gamma_\bm$ has a root vertex, which is the vertex $1'$ in the picture of $\Gamma_{\bm_0}$, or the bottom-most vertex in $\Gamma_\bm$ in general.

The weights attached to the graph $\Gamma_\bm$ are as follows.
Each edge connecting neighboring vertices and pointing towards the root vertex $1'$ carries weight
$t y_a$, with the edges numbered $a=1,2,...,2r+1$ from the root vertex on
(with edge $(i',i)$ preceding $(i+1,i)$ in cases where it exists).
The weights $y_a$ are called the {\it skeleton weights}
of $\Gamma_\bm$.

Let $w_{a,b}$ be the weight corresponding to the edge connecting vertex $a$ to $b$. Then $w_{a,a+1}=w_{a,a'}=1$, $w_{a+1,a}=ty_{2a} $ and $w_{a',a}=t y_{2a-1}$.

The weights $w_{a,b}$ of the edges $(a,b)$ with $(a-b>1)$ are called {\it redundant weights},
and are expressed solely in terms of skeleton weights between the vertices $a,b$ as:
\begin{equation}\label{redunco}
w_{a,b}= t\prod_{i=b+1}^{a}y_{2i-2}
\prod_{i=b+1}^{a-1}y_{2i-1}^{-1}, (a-b\geq 1).
\end{equation}

These graphs have manifestly positive weights.  Let $\bm$ be a Motzkin path with first component $m_1$. Then:
\begin{thm}\cite{DFK08}
The ratio $R_{1,n+m_1}/R_{1,m_1}$
is the coefficient of $t^n$ in the partition function of paths on
$\Gamma_\bm$ from the vertex $1'$ to itself.
\end{thm}

This theorem can be stated in terms of the resolvent of the transfer matrix
associated with the graph in the following way. Let $T_{\Gamma_\bm}$ be the matrix with entries
$(T_{\Gamma_\bm})_{a,b} = w_{a,b}$.
Then the generating function for paths from vertex $1'$ to itself on $\Gamma_\bm$ is the resolvent
\begin{equation}\label{resolvent}
F_\bm(t) = \left ((\mathbb I - T_{\Gamma_\bm})^{-1}\right)_{1',1'} =\sum_{n\geq 0} t^n R_{1,n+m_1}/R_{1,m_1} .
\end{equation}

\subsection{Compactified graphs}
We now define the graphs $\Gamma'_\bm$, which have the property that
$$
((\mathbb I - T_{\Gamma_\bm})^{-1})_{1,1}=((\mathbb I - T_{\Gamma'_\bm})^{-1})_{1,1}.
$$
Note that $Z_{1',1'}(t) = 1 + t y_{1} ((\mathbb I - T_{\Gamma'_\bm})^{-1})_{1,1}$.

The compactified graphs $\Gamma'_\bm$ are obtained from $\Gamma_\bm$
as explained in Section 6 of \cite{DFK09}. This process involves an identification of pairs of vertices in the graph $\Gamma_\bm$, such that the final number of vertices is reduced to $r+1$. This  produces loops and also, in certain cases, extra directed edges.

\begin{enumerate}
\item For subgraphs arising from descending or flat segments of $\bm$, compactified graphs are obtained simply from identification of the vertices $i$ and $i'$ for each $i$, resulting in a loop at vertex $i$ with weight $w_{i',i}$.  All other edges retain their weights.

\item Consider the vertices and edges arising from each ascending segment of length $k$ in $\bm$. This is a chain of $2k+2$ vertices.
\begin{enumerate}
\item Name the vertices from bottom to top, $1,...,2k+2$ and identify the vertices $2a-1,2a$ ($a = 1,...,k+1$). This leaves $k+1$ vertices.
\item Rename the vertices from bottom to top again, $1,...,k+1$.
\item Identification of vertices results in a loop at each vertex $a$ with weight inherited from the weight associated with the edge connecting $2a$ to $2a-1$, $w_{2a,2a-1}$. The spine retains weights $w_{2a+1,2a}$ along each edge $(a+1,a)$.
\item In addition we adjoin ascending directed edges, connecting vertex $a$ to $b$ for all $1\leq a<b\leq k+1$, with weight $(-1)^{b-a+1}$.
\end{enumerate}
\end{enumerate}

\begin{remark}\label{appremark}
In fact, the construction of the compactified graph directly from the Motzkin path is easy to describe. The Motzkin path is decomposed into maximal ascending, descending and flat segments, the corresponding compactified graphs and their weights are constructed as above, and the graphs are glued: The top two vertices corresponding to a segment are identified with the bottom two vertices corresponding to the next segment. Incident edges between these vertices are identified. (See pictures below.)
\end{remark}

\begin{lemma}\cite{DFK09} We have the equality of partition functions
$$
\left((\mathbb I - T_{\Gamma_\bm})^{-1}\right)_{1,1}=\left((\mathbb I - T_{\Gamma'_\bm})^{-1}\right)_{1,1}
$$
\end{lemma}

In light of remark \ref{appremark}, we can focus on graph segments corresponding to segments of $\bm$ which do not change direction.

\subsection{The graph $G_\bm$}
We now turn to the equivalence of resolvents of the graph $G_\bm$, defined in Section 2,  to those on the graph $\Gamma'_\bm$. The difference is the presence of long directed edges in $\Gamma'_\bm$ (connecting vertices along the spine of the graph with indices which differ by 2 or more).

On the level of the transfer matrix $T_{\Gamma'_\bm}$, these edges are represented by non-zero elements above or below the second diagonals.
These entries can be eliminated using row (column) elimination for descending (ascending) edges, respectively. Conveniently,
\begin{lemma}
The matrices $E$ and $F$ such that $E(\mathbb I-T_{\Gamma'_\bm})F$ has no non-trivial entries away from the second diagonals are such that $E$ is upper unitriangular and $F$  is lower unitriangular.
\end{lemma}
The $(1,1)$-resolvent
$$
((\mathbb I-T_{\Gamma'_\bm})^{-1})_{1,1}
$$
is obviously unchanged under such transformations.
\begin{proof}
It is sufficient to show this for each block of $T'=T_{\Gamma'_\bm}$ corresponding to maximal descending and maximal ascending  pieces of $\bm$. The case of flat pieces is trivial as it has no entries off the second diagonals.

\begin{enumerate}
\item {\bf Descending pieces.} The matrix $\mathbb I - T'$ related  to the vertices in the graph $\Gamma'_{\bm}$ corresponding to a descending path $(k-1,k-2,...,0)$ is the $(k+1)\times (k+1)$-matrix
$$\mathbb I - T' =
\left( \begin{array}{cccccc}
1-x_{1} & -1 &  0 & \cdots &  & 0\\
-x_2 & 1-x_3 & -1 & 0 & \cdots & \\
-\frac{x_2 x_4}{x_3} & -x_4 & 1-x_5 & -1 & 0 & \\
-\frac{x_2 x_4 x_6}{x_3 x_5} & -\frac{x_4 x_6}{x_5}& -x_6 &1-x_7 & -1 & \vdots\\
\vdots & & \ddots & \ddots & \ddots &\\
 & \cdots & & -x_{2k-2} & 1-x_{2k-1} & -1\\
\vdots & \cdots & &-\frac{x_{2k-2}x_{2k}}{x_{2k-1}} & -x_{2k} & 1-x_{2k+1}
\end{array}
\right).
$$
Here, $x_{2a-1} =w_{a',a}$ and $x_{2a} = w_{a+1,a}$. Clearly, there is an lower triangular matrix which eliminates all of the entries below the second diagonal when multiplied on the right:
$$
F_{i,i} = 1, (1\leq i \leq k+1), \quad F_{i+1,i} = -\frac{x_{2i}}{x_{2i+1}} (1\leq k-1),\quad F_{i,j}=0 \hbox{ otherwise}.
$$
The resulting matrix $M=(\mathbb I- T')F$ has non-vanishing entries $M_{i,i+1}=-1$, and for $i\leq k-1$,
\begin{equation}\label{equals}
M_{ii} = 1- (x_{2i-1} - \frac{x_{2i}}{x_{2i+1}}),\quad
M_{i+1,i} = -\frac{x_{2i}}{x_{2i+1}}.
\end{equation}
In addition, the last two columns of the matrix $M$ are those of $\mathbb I-T'$.

Note that we have lost the positivity property of the weights in this case, although the generating function of paths is positive.  Note also that the weights attached to the edges connecting the last two vertices have not changed under this transformation.
\item {\bf Ascending pieces.} The matrix $T'$, attached  to the vertices in the graph $\Gamma'_{\bm}$ corresponding to an ascending path $(0,...,k-1)$ is the $(k+1)\times (k+1)$-matrix
$$\mathbb I - T' =
\left( \begin{array}{ccccccc}
1-x_{1} & -1 & 1 & -1 & 1&\cdots & \pm 1 \\
-x_2 & 1-x_3 & -1 & 1& -1 & \cdots &\\
0 & -x_4 & 1-x_5 & -1 & 1 & & \\
\vdots & 0 & -x_6 & 1-x_7 & -1& \cdots & \\
& & \ddots & \ddots & \ddots& \ddots & \vdots\\
\vdots & & & &-x_{2k-2} & 1-x_{2k-1} & -1\\
0 & \cdots & & & 0 & -x_{2k} & 1-x_{2k+1}
\end{array}
\right).
$$
There is obviously an upper triangular matrix $E$, corresponding to row operations on this matrix, or multiplication on the left, such that $M=E(\mathbb I -T')$ has no entries away from the second off-diagonals. The resolvent is equal to the resolvent of the matrix with non-vanishing entries $M_{i,i+1}=-1$, and if $i\leq k-1$,
\begin{equation}\label{crosses}
M_{ii} = 1-(x_i + x_{i+1}) , \quad M_{i+1,i} = -x_{2i} x_{2i+1}.
\end{equation}
The last two rows of $M$ are those of $\mathbb I-T'$. Note that in this case, we retain the positivity property of the weights.
\end{enumerate}
\end{proof}
To summarize, we can draw the graphs corresponding to the sequence of transformations from $\bm$ (an ascending, descending or flat path) to $G_\bm$ in graphical notation:
\begin{enumerate}
\item
For ascending paths,
\begin{center}
\begin{tabular}{ccccccc}
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(15,22)
\psline(15,17.5)(0,2.5)
\multips(15,17.5)(-5,-5){4}{\pscircle*[linecolor=blue](0,0){.7}}
\rput(12,17.5){$m_k$}
\rput(7,12.5){$m_{k-1}$}
\rput(2.5,2.5){$m_1$}
\rput(7.5,-2){$\mathbf m$}
\end{pspicture}
&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(5,22)
\psline[arrowsize=1.5]{->}(0,10)(5,10)
\end{pspicture}&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(12,22)
\psline(5,0)(5,20)
\psline(0,0)(5,0)
\psline(0,20)(5,20)
\multips(5,0)(0,2.5){9}{\pscircle*[linecolor=red](0,0){.5}}
\rput(0,0){\pscircle*[linecolor=red](0,0){.7}}
\rput(0,20){\pscircle*[linecolor=red](0,0){.7}}
\rput(7,0){2}
\rput(-2,0){1}
\rput(7,2.5){3}

\rput(9,20){$2k+1$}
\rput(-4,20){$2k+2$}
\rput(2,-3){$\Gamma_{\mathbf m}$}
\end{pspicture}
& \psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(5,22)
\psline[arrowsize=1.5]{->}(0,10)(5,10)
\end{pspicture}
&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(12,22)
\psline(5,0)(5,20)
\multips(.25,-2.5)(0,5){5}{\pscircle(2.5,2.5){2}}
\rput(0,10){\psarc[linecolor=blue,arrowsize=1.5]{->}{11.5}{-60}{0}}
\rput(0,10){\psarc[linecolor=blue,arrowsize=1.5]{-}{11.5}{0}{60}}
\rput(1,7.5){\psarc[linecolor=blue,arrowsize=1.5]{->}{8.75}{-60}{0}}
\rput(1,7.5){\psarc[linecolor=blue,arrowsize=1.5]{-}{8.75}{0}{60}}
\rput(2.5,5){\psarc[linecolor=blue,arrowsize=1.5]{->}{6}{-60}{0}}
\rput(2.5,5){\psarc[linecolor=blue,arrowsize=1.5]{-}{6}{0}{60}}
\rput(1,12.5){\psarc[linecolor=blue,arrowsize=1.5]{->}{8.75}{-60}{0}}
\rput(1,12.5){\psarc[linecolor=blue,arrowsize=1.5]{-}{8.75}{0}{60}}
\rput(2.5,10){\psarc[linecolor=blue,arrowsize=1.5]{->}{6}{-60}{0}}
\rput(2.5,10){\psarc[linecolor=blue,arrowsize=1.5]{-}{6}{0}{60}}
\rput(2.5,15){\psarc[linecolor=blue,arrowsize=1.5]{->}{6}{-60}{0}}
\rput(2.5,15){\psarc[linecolor=blue,arrowsize=1.5]{-}{6}{0}{60}}
\multips(5,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}

\rput(7,0){1}
\rput(7,5){2}
\rput(9,20){$k+1$}
\rput(4,-4){$\Gamma'_{\mathbf m}$}

\end{pspicture}
&\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(5,22)
\psline[arrowsize=1.5]{->}(0,10)(5,10)
\end{pspicture}&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(10,22)
\psline(5,0)(5,20)
\multips(.25,-2.5)(0,5){5}{\pscircle(2.5,2.5){2}}
\rput(.75,0){$\times$}
\rput(.75,5){$\times $}
\rput(.75,10){$\times $}
\rput(5,2.5){$\times $}
\rput(5,7.5){$\times $}
\rput(5,12.5){$\times $}

\multips(5,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}

\rput(7,0){1}
\rput(7,5){2}
\rput(9,20){$k+1$}
\rput(4,-4){$G_{\mathbf m}$}

\end{pspicture}
\end{tabular}
\vskip.1in
\end{center}
\item
For descending paths,
\begin{center}
\begin{tabular}{ccccccc}
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(15,22)
\psline(0,17.5)(15,2.5)
\multips(0,17.5)(5,-5){4}{\pscircle*[linecolor=blue](0,0){.7}}
\rput(2,17.5){$m_k$}
\rput(8,12.5){$m_{k-1}$}
\rput(17,2.5){$m_1$}
\rput(7.5,-2){$\mathbf m$}
\end{pspicture}
&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(5,22)
\psline[arrowsize=1.5]{->}(0,10)(5,10)
\end{pspicture}&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(12,22)
\psline(5,0)(5,20)
\multips(0,0)(0,5){5}{\psline(5,0)}
\multips(0,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}
\rput(0,10){\psarc[linecolor=blue,arrowsize=1.5]{-}{11.5}{-60}{0}}
\rput(0,10){\psarc[linecolor=blue,arrowsize=1.5]{<-}{11.5}{0}{60}}
\rput(1,7.5){\psarc[linecolor=blue]{-}{8.75}{-60}{0}}
\rput(1,7.5){\psarc[linecolor=blue,arrowsize=1.5]{<-}{8.75}{0}{60}}
\rput(2.5,5){\psarc[linecolor=blue]{-}{6}{-60}{0}}
\rput(2.5,5){\psarc[linecolor=blue,arrowsize=1.5]{<-}{6}{0}{60}}
\rput(1,12.5){\psarc[linecolor=blue]{-}{8.75}{-60}{0}}
\rput(1,12.5){\psarc[linecolor=blue,arrowsize=1.5]{<-}{8.75}{0}{60}}
\rput(2.5,10){\psarc[linecolor=blue]{-}{6}{-60}{0}}
\rput(2.5,10){\psarc[linecolor=blue,arrowsize=1.5]{<-}{6}{0}{60}}
\rput(2.5,15){\psarc[linecolor=blue]{-}{6}{-60}{0}}
\rput(2.5,15){\psarc[linecolor=blue,arrowsize=1.5]{<-}{6}{0}{60}}
\multips(5,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}

\rput(7,0){1}
\rput(-2,0){1'}
\rput(7,5){2}
\rput(-2,5){2'}
\rput(9,20){$k+1$}
\rput(-5,20){$(k+1)'$}
\rput(2,-3){$\Gamma_{\mathbf m}$}
\end{pspicture}
& \psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(5,22)
\psline[arrowsize=1.5]{->}(0,10)(5,10)
\end{pspicture}
&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(12,22)
\psline(5,0)(5,20)
\multips(.25,-2.5)(0,5){5}{\pscircle(2.5,2.5){2}}
\rput(0,10){\psarc[linecolor=blue,arrowsize=1.5]{-}{11.5}{-60}{0}}
\rput(0,10){\psarc[linecolor=blue,arrowsize=1.5]{<-}{11.5}{0}{60}}
\rput(1,7.5){\psarc[linecolor=blue]{-}{8.75}{-60}{0}}
\rput(1,7.5){\psarc[linecolor=blue,arrowsize=1.5]{<-}{8.75}{0}{60}}
\rput(2.5,5){\psarc[linecolor=blue]{-}{6}{-60}{0}}
\rput(2.5,5){\psarc[linecolor=blue,arrowsize=1.5]{<-}{6}{0}{60}}
\rput(1,12.5){\psarc[linecolor=blue]{-}{8.75}{-60}{0}}
\rput(1,12.5){\psarc[linecolor=blue,arrowsize=1.5]{<-}{8.75}{0}{60}}
\rput(2.5,10){\psarc[linecolor=blue]{-}{6}{-60}{0}}
\rput(2.5,10){\psarc[linecolor=blue,arrowsize=1.5]{<-}{6}{0}{60}}
\rput(2.5,15){\psarc[linecolor=blue]{-}{6}{-60}{0}}
\rput(2.5,15){\psarc[linecolor=blue,arrowsize=1.5]{<-}{6}{0}{60}}
\multips(5,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}

\rput(7,0){1}
\rput(7,5){2}
\rput(9,20){$k+1$}
\rput(4,-4){$\Gamma'_{\mathbf m}$}

\end{pspicture}
&\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(5,22)
\psline[arrowsize=1.5]{->}(0,10)(5,10)
\end{pspicture}&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(10,22)
\psline(5,0)(5,20)
\multips(.25,-2.5)(0,5){5}{\pscircle(2.5,2.5){2}}
\rput(.75,0){$=$}
\rput(.75,5){$=$}
\rput(.75,10){$=$}
\rput(5,2.5){$=$}
\rput(5,7.5){$=$}
\rput(5,12.5){$=$}

\multips(5,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}

\rput(7,0){1}
\rput(7,5){2}
\rput(9,20){$k+1$}
\rput(4,-4){$G_{\mathbf m}$}

\end{pspicture}
\end{tabular}
\vskip.1in
\end{center}
\item
For flat paths,
\begin{center}
\begin{tabular}{ccccccc}
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(7,22)
\psline(0,17.5)(0,2.5)
\multips(0,17.5)(0,-5){4}{\pscircle*[linecolor=blue](0,0){.7}}
\rput(2,17.5){$m_k$}
\rput(3,12.5){$m_{k-1}$}
\rput(2,2.5){$m_1$}
\rput(4,-2){$\mathbf m$}
\end{pspicture}
&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(5,22)
\psline[arrowsize=1.5]{->}(0,10)(5,10)
\end{pspicture}&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(12,12)
\psline(5,0)(5,20)
\multips(0,0)(0,5){5}{\psline(5,0)}
\multips(0,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}
\multips(5,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}

\rput(7,0){1}
\rput(-2,0){1'}
\rput(7,5){2}
\rput(-2,5){2'}
\rput(9,20){$k+1$}
\rput(-5,20){$(k+1)'$}
\rput(2,-3){$\Gamma_{\mathbf m}$}
\end{pspicture}
& \psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(5,12)
\psline[arrowsize=1.5]{->}(0,10)(5,10)
\end{pspicture}
&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(12,22)
\psline(5,0)(5,20)
\multips(.25,-2.5)(0,5){5}{\pscircle(2.5,2.5){2}}
\multips(5,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}

\rput(7,0){1}
\rput(7,5){2}
\rput(9,20){$k+1$}
\rput(4,-4){$\Gamma'_{\mathbf m}$}

\end{pspicture}
&\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(5,12)
\psline[arrowsize=1.5]{->}(0,10)(5,10)
\end{pspicture}&
\psset{unit=2mm,linewidth=.4mm,dimen=middle}
\begin{pspicture}(0,-2)(10,22)
\psline(5,0)(5,20)
\multips(.25,-2.5)(0,5){5}{\pscircle(2.5,2.5){2}}

\multips(5,0)(0,5){5}{\pscircle*[linecolor=red](0,0){.7}}

\rput(7,0){1}
\rput(7,5){2}
\rput(9,20){$k+1$}
\rput(4,-4){$G_{\mathbf m}$}

\end{pspicture}
\end{tabular}
\end{center}
\end{enumerate}
\vskip.1in
In the final graphs $G_{\bm}$ for ascending and descending Motzkin paths,
the symbols $\times$ and $=$ along certain edges indicates that the weights
have been renormalized from the initial values $x_{2i-1}$ and $x_{2i}$ to
the values $x_{2i-1}-x_{2i}/x_{2i+1}$ and $x_{2i}/x_{2i+1}$ for $=$ (see Eq.\eqref{equals})
and to the values $x_{2i-1}+x_{2i}$ and $x_{2i}x_{2i+1}$ for $\times$ (see Eq.\eqref{crosses}).
For $x_i=ty_i$ with the appropriate labeling of skeleton weights,
these are precisely the weights $\widehat y$ of Definition \ref{weightdefco} of Section \ref{sectiontwo}.

Gluing two path pieces, $\bm$ followed by $\bm'$ into a longer Motzkin path  corresponds to identifying the two top vertices (and incident edges) of $G_\bm$, the lower graph, with the bottom two of $G_{\bm'}$, the top graph. In the cases where the weights have been renormalized in the lower part of the upper path, the renormalized weights appear in the glued graph.


TITLE: Complex Integration using Cauchy's Theorem: what do I set my $f(z)$ to?
QUESTION [0 upvotes]: I think that I have to use Cauchy's integration formula but I haven't been able to figure out how to use it in this case.
$$\int\frac{e^z+\sin z}{z^5}dz,$$
integrated along the path $r:[0,1]\mapsto\mathbb{C}$ where $r(t) = 3 + e^{2\pi i t}$.

REPLY [1 votes]: You can use Cauchy's theorem.  Since the integrand is analytic within the circle, the integral is zero.


TITLE: Minimum value of $p=3x+\frac{1}{15x}+5y+\frac{25}{y}+z+\frac{1}{36z},$ where $x,y,z\in \mathbb{R}^+$.
QUESTION [1 upvotes]: Find the minimum value of $$p=3x+\frac{1}{15x}+5y+\frac{25}{y}+z+\frac{1}{36z},$$ where $x,y,z\in \mathbb{R}^+.$

Applying the AM-GM inequality,
$$
\begin{aligned}\frac{p}{6} & \geqslant\left(3x\cdot\frac{1}{15x}\cdot 5y\cdot\frac{25}{y}\cdot z\cdot \frac{1}{36z}\right)^{1/6} \\
\frac{p}{6} & \geqslant \left(\frac{5}{6}\right)^{1/3}\\
p & \geqslant 6\left(\frac{5}{6}\right)^{1/3}
\end{aligned}$$
$$\implies \text{The minimum value of the expression is } 6\left(\frac{5}{6}\right)^{1/3}
$$
Now, consider $f(x)=3x+\dfrac{1}{15x},\ g(y) = 5y+\dfrac{25}{y}$ and $h(z)=z+\dfrac{1}{36z}.$
$$\begin{aligned}f'(x) &= \frac{\mathrm d}{\mathrm dx}\left(3x+\frac{1}{15x}\right) = 3 - \frac{1}{15x^2}\\f''(x)&=\frac{2}{15x^3}\end{aligned}$$
At the critical points, $f'(x) = 0 \implies x = \dfrac{\pm1}{3\sqrt{5}}.$
$f''\left(\dfrac{1}{3\sqrt5}\right) > 0\implies f(x)$ has a local minima at $x = \dfrac{1}{3\sqrt5}.$
Similarly, the local minima of $g(y)$ is at $y = \sqrt5$ and the local minima of $h(z)$ is at $z=1/6.$
Substituting these values into the original expression, the minimum value of the expression comes out to be
$$
\begin{aligned}
p & = f\left(\dfrac{1}{3\sqrt5}\right)+g\left(\sqrt5\right)+h\left(\frac{1}{6}\right) \\
& = \frac{1}{3} + \frac{52}{\sqrt5}.
\end{aligned}
$$
The answer according to the AM-GM inequality is $\approx5.646$ and according to calculus is $\approx23.588$, which are way off. Also, the functions have only two points of inflection, one is the maxima (for values less than $0$) and the other is the minima (for values greater than $0$). As the question clearly states "for $x,y,z \in \mathbb{R}^+$", the "correct" answer should be $\approx23.588$, shouldn't it?
Why are the answers different? Also, is it possible to figure out the individual $x, y$ and $z$ values for which the expression has the minimum value (in case of AM-GM inequality)?

Note: This question is from a Test and the correct answer according to the "test creators" is $$6\left(\dfrac{5}{6}\right)^{1/3}.$$

$\text{Graph made using Desmos.}$

REPLY [0 votes]: Use Binary AM-GM as
$$3x+\frac{1}{15x}\ge \frac{1}{\sqrt{5}}, ~\text{equality when}~ x=\frac{1}{\sqrt{45}}\tag1$$
$$5y+\frac{25}{y} \ge 2\sqrt{125}, ~\text{equality when}~ y=\sqrt{5} \tag2$$
$$z+\frac{1}{36z} \ge \frac{1}{3} ~ \text{equality when} z=\frac{1}{6} \tag3$$
Adding them, we get
$$3x+\frac{1}{15x}+5y+\frac{25}{y}+z+\frac{1}{36z} \ge \frac{1}{\sqrt{5}}+ 10 \sqrt{5}+\frac{1}{3}.$$
The equality exists for afore mentioned values of $x,y,z$


TITLE: How do I determine linear independence when I don't know the values of the vectors?
QUESTION [1 upvotes]: I am studying for a test. The textbook problem is:
The vectors $v_1, v_2, v_3, v_4$ are linearly independent. Determine if the following vectors are also independent.
$v_1-v_2, 2(v_2-v_3), 3(v_3-v_4)$
I have no idea how to determine this. I would greatly appreciate any help. Thank you!

REPLY [0 votes]: Since ${\mathbf{v_1},\mathbf{v_2},\mathbf{v_3},\mathbf{v_4}}$ is a set of linearly dependent vectors in $\mathbb{R}^n$, that means that there exist a set of numbers of cardinality 4 different that zero, such that $a_1\mathbf{v_1}+a_2\mathbf{v_2}+a_3\mathbf{v_3}+a_4\mathbf{v_4}=0$. Thats the key, now try to rearrange the terms of the equality that you need to prove if the last set were independent.