A note on the geometry of the MAP partition in Conjugate Exponential Bayesian Mixture Models

TL;DR

This paper explores the geometric structure of the MAP partition in Bayesian Mixture Models with conjugate exponential families, revealing linear or quadratic cluster separation surfaces depending on the model specifics.

Contribution

It provides a geometric characterization of MAP partitions, showing they are separated by contour lines of linear or quadratic functions, extending understanding of Bayesian clustering geometry.

Findings

Clusters are separated by contour lines of a linear functional of the sufficient statistic.

In Bayesian Mixture of Normals, clusters are separated by quadratic surfaces.

Results connect Bayesian clustering geometry with Fisher Discriminant Analysis.

Abstract

We investigate the geometry of the maximal a posteriori (MAP) partition in the Bayesian Mixture Model where the component and the base distributions are chosen from conjugate exponential families. We prove that in this case the clusters are separated by the contour lines of a linear functional of the sufficient statistic. As a particular example, we describe Bayesian Mixture of Normals with Normal-inverse-Wishart prior on the component mean and covariance, in which the clusters in any MAP partition are separated by a quadratic surface. In connection with results of Rajkowski (2018), where the linear separability of clusters in the Bayesian Mixture Model with a fixed component covariance matrix was proved, it gives a nice Bayesian analogue of the geometric properties of Fisher Discriminant Analysis (LDA and QDA).

Figures

Equations

$$\begin{array}[]{rcl}\bm{p}=(p_{i})_{i=1}^{m}&\sim&\mathcal{P}\\ \bm{\uptheta}=(\bm{\theta}_{i})_{i=1}^{m}&\stackrel{{\scriptstyle\textrm{iid}}}{{\sim}}&\pi\\ \mathbf{x}=(\bm{x}_{1},\ldots,\bm{x}_{n})\,|\,\bm{p},\bm{\uptheta}&\stackrel{{\scriptstyle\textrm{iid}}}{{\sim}}&\sum_{i=1}^{m}p_{i}g_{\bm{\theta}_{i}}.\end{array}$$

$$\begin{array}[]{rcl}\bm{p}=(p_{i})_{i=1}^{m}&\sim&\mathcal{P}\\ \bm{\uptheta}=(\bm{\theta}_{i})_{i=1}^{m}&\stackrel{{\scriptstyle\textrm{iid}}}{{\sim}}&\pi\\ \bm{\upphi}=(\bm{\phi}_{1},\ldots,\bm{\phi}_{n})\,|\,\bm{p},\bm{\uptheta}&\stackrel{{\scriptstyle\textrm{iid}}}{{\sim}}&\sum_{i=1}^{m}p_{i}\delta_{\bm{\theta}_{i}}\\ \bm{x}_{i}\,|\,\bm{p},\bm{\uptheta},\bm{\upphi}&\sim&g_{\bm{\phi}_{i}}\quad\textrm{ independently for all $i\leq n$.}\end{array}$$

$$\mathbb{P}(\bm{\Pi}=\mathcal{I})=\mathbb{P}\Big{(}\bm{\Pi}=\big{\{}\{\sigma(i)\colon i\in I\}\colon I\in\mathcal{I}\big{\}}\Big{)}.$$

$$f_k({\mathbf{u}}_1,\dots ,{\mathbf{u}}_k):={\int }_{\Theta }\pi (\mathbf{\theta })\prod _{i=1}^k{g}_{\mathbf{\theta }}({\mathbf{u}}_i)\text{d}\mathbf{\theta }.$$

$$\begin{array}[]{rcl}\mathcal{I}&\sim&\textrm{ERP}_{n}\\ \mathbf{x}_{I}:=(\bm{x}_{i})_{i\in I}\,|\,\mathcal{I}&\sim&f_{|I|}\quad\textrm{ independently for all $I\in\mathcal{I}$.}\end{array}$$

$$f(\mathbf{x}|\mathcal{I}):=\prod _{I\in \mathcal{I}}{f}_{|I|}({\mathbf{x}}_I).$$

$$\begin{array}[]{rcl}\mathcal{I}&\sim&\textrm{ERP}_{n}\\ \mathbf{x}\,|\,\mathcal{I}&\sim&f(\cdot\,|\,\mathcal{I}).\end{array}$$

$$\begin{array}[]{rcl}G&\sim&DP(\alpha,G_{0})\\ \bm{\upphi}=(\bm{\phi}_{1},\ldots,\bm{\phi}_{n})\,|\,G&\stackrel{{\scriptstyle\textrm{iid}}}{{\sim}}&G\\ \bm{x}_{i}\,|\,G,\bm{\upphi}&\sim&g_{\bm{\phi}_{i}}\quad\textrm{ independently for all $i\leq n$.}\end{array}$$

$$p_n(\mathcal{I})=\frac{{\alpha }^{|\mathcal{I}|}}{{\alpha }^{(n)}}\prod _{I\in \mathcal{I}}(|I|-1)!,$$

$$p_n(\mathcal{I})f({\mathbf{x}}_I|\mathcal{I})\le p_n(\hat{\mathcal{I}})f({\mathbf{x}}_I|\hat{\mathcal{I}}).$$

$$\hat{\mathcal{I}}\in \underset{\text{partitions I of [n]}}{\mathrm{argmax}}{p}_n(\mathcal{I})f({\mathbf{x}}_I|\mathcal{I}).$$

$$\hat{\mathcal{I}}\in\operatorname*{argmax}_{\textrm{partitions $\mathcal{I}$ of $[n]$}}\Big{(}\ln p_{n}(\mathcal{I})+\sum_{I\in\mathcal{I}}\ln f_{|I|}(\mathbf{x}_{I})\Big{)}.$$

$$g_{\mathbf{\theta }}(\mathbf{x})=h(\mathbf{x})\cdot \exp \left\{\mathbf{T}(\mathbf{x}{)}^{\top }\mathbf{\eta }(\mathbf{\theta })-B(\mathbf{\theta })\right\}.$$

$$\Omega :=\left\{(\mathbf{\chi },\mathbf{\tau })\in {\mathbb{R}}^p\times {\mathbb{R}}^q:{\int }_{\Theta }H(\mathbf{\theta })\cdot \exp \left\{[\mathbf{\eta }(\mathbf{\theta }{)}^{\top },\mathbf{B}(\mathbf{\theta }{)}^{\top }]\left[\begin{array}{c}\mathbf{\chi }\\ \mathbf{\tau }\end{array}\right]\right\}\text{d}\mathbf{\theta }<\infty \right\}.$$

$$A(\mathbf{\chi },\mathbf{\tau }):=\ln \left({\int }_{\Theta }H(\mathbf{\theta })\cdot \exp \left\{[\mathbf{\eta }(\mathbf{\theta }{)}^{\top },\mathbf{B}(\mathbf{\theta }{)}^{\top }]\left[\begin{array}{c}\mathbf{\chi }\\ \mathbf{\tau }\end{array}\right]\right\}\text{d}\mathbf{\theta }\right)$$

$$\pi (\mathbf{\theta })={\pi }_{\mathbf{\chi },\mathbf{\tau }}(\mathbf{\theta })=H(\mathbf{\theta })\cdot \exp \left\{[\mathbf{\eta }(\mathbf{\theta }{)}^{\top },-\mathbf{B}(\mathbf{\theta }{)}^{\top }]\left[\begin{array}{c}\mathbf{\chi }\\ \mathbf{\tau }\end{array}\right]-A(\mathbf{\chi },\mathbf{\tau })\right\}$$

$$\mathbf{u}\sim f_k(\mathbf{u}):=\prod _{i=1}^{k}h({\mathbf{u}}_i)\cdot \exp \left\{A({\mathbf{\chi }}_{\mathbf{u}},{\mathbf{\tau }}_k)-A(\mathbf{\chi },\mathbf{\tau })\right\}$$

$$\begin{array}[]{rcl}\bm{\Lambda}&\sim&\mathcal{W}^{-1}(\nu_{0}+d+1,\nu_{0}\bm{\Sigma}_{0})\\ \bm{\mu}\,|\,\bm{\Lambda}&\sim&\mathcal{N}(\bm{\mu}_{0},\bm{\Lambda}/\kappa_{0}),\end{array}$$

$$\begin{array}[]{rl}\mathbb{E}\,(\mathbf{V}(\bm{x}\,|\,\bm{\mu},\bm{\Lambda}))&=\mathbb{E}\,\bm{\Lambda}=\bm{\Sigma}_{0},\\ \mathbf{V}(\mathbb{E}\,(\bm{x}\,|\,\bm{\mu},\bm{\Lambda}))&=\mathbf{V}(\bm{\mu})=\mathbb{E}\,\mathbf{V}(\bm{\mu}\,|\,\bm{\Lambda})+\mathbf{V}\mathbb{E}\,(\bm{\mu}\,|\,\bm{\Lambda})=\mathbb{E}\,\bm{\Lambda}/\kappa_{0}+\mathbf{V}(\bm{\mu}_{0})=\bm{\Sigma}_{0}/\kappa_{0},\end{array}$$

$$\begin{array}[]{rcl}\bm{x}\,|\,\bm{\mu},\bm{\Lambda}&\sim&(2\pi)^{-d/2}|\bm{\Lambda}|^{-1/2}\exp\left\{-\frac{1}{2}(\bm{x}-\bm{\mu})^{\top}\bm{\Lambda}^{-1}(\bm{x}-\bm{\mu})\right\}\\ \bm{\mu}\,|\,\bm{\Lambda}&\sim&(2\pi)^{-d/2}\kappa_{0}^{d/2}|\bm{\Lambda}|^{-1/2}\exp\left\{-\frac{1}{2}\kappa_{0}(\bm{\mu}-\bm{\mu}_{0})^{\top}\bm{\Lambda}^{-1}(\bm{\mu}-\bm{\mu}_{0})\right\}\\ \bm{\Lambda}&\sim&\left(\frac{|\nu_{0}\bm{\Sigma}_{2}|}{2^{d}}\right)^{\frac{\nu_{0}+d+1}{2}}\Gamma_{d}\left(\frac{\nu_{0}+d+1}{2}\right)^{-1}|\bm{\Lambda}|^{-\frac{\nu_{0}+2d+2}{2}}\exp\left\{-\frac{1}{2}\nu_{0}\textnormal{tr}(\bm{\Sigma}_{0}\bm{\Lambda}^{-1})\right\}\end{array}$$

$$\begin{array}{rl}h(\mathbf{x})=(2\pi {)}^{-d/2},\mathbf{T}(\mathbf{x})& =\left[\begin{array}{c}-\frac{1}{2}\text{diag}(\mathbf{x}{\mathbf{x}}^{\top })\\ -\text{low}(\mathbf{x}{\mathbf{x}}^{\top })\\ \mathbf{x}\end{array}\right],\mathbf{\eta }(\mathbf{\theta })=\left[\begin{array}{c}\text{diag}({\mathbf{\Lambda }}^{-1})\\ \text{low}({\mathbf{\Lambda }}^{-1})\\ {\mathbf{\Lambda }}^{-1}\mathbf{\mu }\end{array}\right],\\ B(\mathbf{\theta })& =\frac{1}{2}\ln |\mathbf{\Lambda }|+\frac{1}{2}{\mathbf{\mu }}^t{\mathbf{\Lambda }}^{-1}\mathbf{\mu }\end{array}$$

$$\mathbf{B}(\mathbf{\theta })=\left[\begin{array}{c}\frac{1}{2}\ln |\mathbf{\Lambda }|\\ \frac{1}{2}{\mathbf{\mu }}^t{\mathbf{\Lambda }}^{-1}\mathbf{\mu }\end{array}\right],\mathbf{\tau }=\left[\begin{array}{c}{\nu }_0+2d+3\\ {\kappa }_0\end{array}\right],\mathbf{\chi }=\left[\begin{array}{c}-\frac{1}{2}\text{diag}({\nu }_0{\mathbf{\Sigma }}_0+{\mathbf{\mu }}_0{\mathbf{\mu }}_0^{\top })\\ -\text{low}({\nu }_0{\mathbf{\Sigma }}_0+{\mathbf{\mu }}_0{\mathbf{\mu }}_0^{\top })\\ {\kappa }_0{\mathbf{\mu }}_0\end{array}\right]$$

$$H(\mathbf{\theta })=(2\pi {)}^{-d/2},A(\mathbf{\chi },\mathbf{\tau })=-\frac{d}{2}\ln {\kappa }_0-\frac{{\nu }_0+d+1}{2}\ln \frac{|{\nu }_0{\mathbf{\Sigma }}_0|}{2^d}+\ln {\Gamma }_d\left(\frac{{\nu }_0+d+1}{2}\right).$$

$$\begin{array}[]{rcl}\bm{\mu}&\sim&\mathcal{N}(\bm{\mu}_{0},\bm{\Psi}_{0})\end{array}$$

$$\mathbb{E}(\mathbf{V}(\mathbf{x}|\mathbf{\mu }))={\mathbf{\Sigma }}_0,\mathbf{V}(\mathbb{E}(\mathbf{x}|\mathbf{\mu }))=\mathbf{V}(\mathbf{\mu })={\mathbf{\Psi }}_0.$$

$$\begin{array}[]{rcl}\bm{x}\,|\,\bm{\mu}&\sim&(2\pi)^{-d/2}|\bm{\Sigma}_{0}|^{-1/2}\exp\left\{-\frac{1}{2}(\bm{x}-\bm{\mu})^{\top}\bm{\Sigma}_{0}^{-1}(\bm{x}-\bm{\mu})\right\}\\ \bm{\mu}&\sim&(2\pi)^{-d/2}|\bm{\Psi}_{0}|^{-1/2}\exp\left\{-\frac{1}{2}(\bm{\mu}-\bm{\mu}_{0})^{\top}\bm{\Psi}_{0}^{-1}(\bm{\mu}-\bm{\mu}_{0})\right\}\end{array}$$

$$\begin{array}{rl}h(\mathbf{x})=(2\pi {)}^{-d/2}|{\mathbf{\Sigma }}_0{|}^{-1/2}\exp \left\{-\frac{1}{2}{\mathbf{x}}^{\top }{\mathbf{\Sigma }}_0^{-1}\mathbf{x}\right\},\mathbf{T}(\mathbf{x})& ={\mathbf{\Sigma }}_0^{-1}\mathbf{x},\mathbf{\eta }(\mathbf{\theta })=\mathbf{\mu },\\ B(\mathbf{\theta })& =\frac{1}{2}{\mathbf{\mu }}^{\top }{\mathbf{\Sigma }}_0^{-1}\mathbf{\mu }\end{array}$$

$$\mathbf{B}(\mathbf{\theta })=\left[\begin{array}{c}\frac{1}{2}\text{diag}(\mathbf{\mu }{\mathbf{\mu }}^{\top })\\ \text{low}(\mathbf{\mu }{\mathbf{\mu }}^{\top })\end{array}\right],\mathbf{\tau }=\left[\begin{array}{c}\text{diag}({\mathbf{\Psi }}_0^{-1})\\ \text{low}({\mathbf{\Psi }}_0^{-1})\end{array}\right],\mathbf{\chi }={\mathbf{\Psi }}_0^{-1}{\mathbf{\mu }}_0$$

$$H(\mathbf{\theta })=(2\pi {)}^{-d/2},A(\mathbf{\chi },\mathbf{\tau })=\frac{1}{2}\ln |{\mathbf{\Psi }}_0|+\frac{1}{2}{\mathbf{\mu }}_0^{\top }{\mathbf{\Psi }}_0{\mathbf{\mu }}_0.$$

$$\begin{array}[]{rcl}\lambda&\sim&\mathcal{G}^{-1}(\beta_{0}+1,\beta_{0})\\ \bm{\mu}\,|\,\lambda&\sim&\mathcal{N}(\bm{\mu}_{0},\lambda\bm{\Psi}_{0})\\ x\,|\,\bm{\mu},\lambda&\sim&\mathcal{N}(\bm{\mu},\lambda\bm{\Sigma}_{0})\end{array}$$

Full text

Keywords: Bayesian Mixture Models, Maximal a Posteriori Partition

1 Introduction

In the standard setting of Bayesian Mixture Models we assume that the target distribution is a random mixture of distributions from some parametrized family. We assume that the probabilities of components are sampled from a (perhaps infinitely dimensional) simplex and the parameters of the component distribution are sampled independently for each component. A prominent example is the Dirichlet Process Mixture Model (Antoniak (1974)), where the prior distribution on the probability weights is Sethuraman’s stick breaking process (Sethuraman (1994)).

A popular choice of the component distribution is multivariate Normal, which gives Normal Bayesian Mixture Model. There are two standard conjugate prior distributions on the component mean and covariance matrix (Gelman et al. (2013), Chapter 3.6): Normal distribution on the mean with fixed component covariance matrix or Normal-inverse-Wishart distribution (here the terminology is adopted from Murphy (2007)), where the component covariance matrix follows the inverse-Wishart distribution and the component mean (conditioned on the component covariance matrix) is Normal. The exact specification of these priors is given in Section 2.4.

Bayesian Mixture Models give a basis for cluster analysis. Indeed, one can translate the distribution on component probabilities into a discrete prior distribution on the possible data partitions – just like Sethuraman’s construction translates in to the Generalised Pólya Urn Scheme (Blackwell et al. (1973)), also known as the Chinese Restaurant Process (Aldous (1985)). The inference about clusters is based on the posterior distribution on the space of partitions. Analysing partition which maximises the posterior probability (the MAP partition) seems to be a natural choice.

In Rajkowski (2018) it is proved that in the Normal Bayesian Mixture Model, when the component covariance matrix is fixed and the prior on the component mean is Normal, the MAP partition is convex, i.e. the convex hulls of clusters are disjoint. An equivalent formulation is: for every two clusters in the MAP there exists a hyperplane that separates them.

Placing an inverse-Wishart prior on the cluster covariance structure, with covariances for different clusters, independent of each other, gives better modelling possibilities, since it is unusual for the covariance to be known a priori and the same for different clusters. It would be of interest to characterise the boundaries of the MAP partition in this case. Since cluster covariance structures are no longer fixed, we might expect quadratic boundaries, analogously to the Fisher’s Quadratic Discriminant Analysis (Friedman et al. (2001)).

A natural generalisation of this hypothesis is to establish the separability result in the case when the base and the component distributions in the Bayesian Mixture Model form a conjugate exponential family. One may expect that the separability can be described in terms of the sufficient statistics. This is indeed what happens and it is the goal of the article to prove this.

2 Formal statement of the result

Naming conventions and notation. In order to facilitate the readership, we introduce the following naming convention. To make a distinction between real numbers and vectors or matrices, we denote the latter in bold, i.e. $\bm{x},\bm{\theta},\bm{\bm{\Sigma}}$. We do the same to distinguish between real-valued, and vector-valued functions. We use an upright bold font do denote a sequence of vectors, e.g. $\mathbf{x}=(\bm{x}_{1},\ldots,\bm{x}_{n})$. In such case, when $I$ is a subset of $[n]:=\{1,\ldots,n\}$, we define $\mathbf{x}_{I}:=(\bm{x}_{i})_{i\in I}$.

2.1 Bayesian Mixture Models and the MAP partition

Let $\Theta\subset\mathbb{R}^{p}$ be the parameter space and $\{g_{\bm{\theta}}\colon\bm{\theta}\in\Theta\}$ be a family of probability densities on the observation space $\mathbb{R}^{d}$. Consider a prior distribution on $\Theta$ given by its density $\pi$. Let $\mathcal{P}$ be a probability distribution on the $m$-dimensional simplex \Delta^{m}=\{\bm{p}=(p_{i})_{i=1}^{m}\colon\textrm{\sum_{i=1}^{m}p_{i}=1$and$p_{i}\geq 0$for$i\leq m}\} (where $m\in\mathbb{N}\cup\{\infty\}$). Let

[TABLE]

This is a Bayesian Mixture Model. It can model possible clusters within data; they are defined by deciding which $g_{\bm{\theta}_{i}}$ generated a given data point. In order to formally define the clusters, we need to rewrite (2.1) as

[TABLE]

Then the clusters are the classes of abstraction of the equivalence relation $i\sim j\equiv\bm{\phi}_{i}=\bm{\phi}_{j}$. In this way the distribution on $m$ dimensional simplex generates a probability distribution on the partitions of set $[n]$ into at most $m$ subsets. According to Pitman (2002) this leads to an exchangeable partition, i.e. a random partition whose probability function is invariant with respect to permutations of indices.

Definition 2.1**.**

We say that $\bm{\Pi}$ is an exchangeable random partition of $[n]$ if for every partition $\mathcal{I}$ of $[n]$ and permutation $\sigma\colon[n]\to[n]$,

[TABLE]

In order to indicate that $\bm{\Pi}$ is an exchangeable random partition of $[n]$ we use a generic notation $\bm{\Pi}\sim\textrm{ERP}_{n}$. Moreover we use the notation $p_{n}(\mathcal{I}):=\mathbb{P}(\bm{\Pi}=\mathcal{I})$.

For $\bm{\theta}\sim\pi$, $k\in\mathbb{N}$ and $\mathbf{u}=(\bm{u}_{1},\ldots,\bm{u}_{k})\,|\,\bm{\theta}\stackrel{{\scriptstyle\textrm{iid}}}{{\sim}}g_{\bm{\theta}}$ let $f_{k}$ be the resulting marginal distribution on $\mathbf{u}$, i.e.

[TABLE]

Let $\textrm{ERP}_{n}$ be the exchangeable probability distribution on the space of partitions generated by $\mathcal{P}$. We see that (2.1) is equivalent to

[TABLE]

We stress the fact that the independent sampling on the ‘lower’ level of (2.5) relates to the independence between clusters (conditioned on the random partition); within one cluster the observations are (marginally) dependent. To make the notation more concise we define

[TABLE]

Then (2.5) becomes

[TABLE]

Example 2.2**.**

Consider the Dirichlet Process Mixture Model (Antoniak (1974)). Let $\alpha>0$, $G_{0}$ be a probability measure on $\Theta$ with density $\pi$ and $DP(\alpha,G_{0})$ be the Dirichlet Process on $\Theta$ (Ferguson (1973)). The Dirichlet Process Mixture Model is defined by

[TABLE]

Let $V_{1},V_{2},\ldots\stackrel{{\scriptstyle\textrm{iid}}}{{\sim}}\textnormal{Beta}(1,\alpha)$, $p_{1}=V_{1}$, $p_{k}=V_{k}\prod_{i=1}^{k-1}(1-V_{i})$ for $k>1$. By Sethuraman (1994) by setting $\mathcal{P}$ to be the distribution of $\bm{p}=(p_{1},p_{2},\ldots)$ we get that (2.8) is equivalent to (2.1). The exchangeable random partition that $\mathcal{P}$ generates is the Generalized Polya Urn Scheme (Blackwell et al. (1973)) or the Chinese Restaurant Process (Aldous (1985)) with the probability weight given by

[TABLE]

where $\alpha^{(n)}=\alpha(\alpha+1)\ldots(\alpha+n-1)$. Again, setting $\textrm{ERP}_{n}$ to be the Chinese Restaurant Process with parameter $\alpha$ we get the equivalence between (2.8) and (2.7).

Definition 2.3**.**

Let $\mathbf{x}=(\bm{x}_{1},\ldots,\bm{x}_{n})\in(\mathbb{R}^{d})^{n}$. We say that a partition $\hat{\mathcal{I}}$ of $[n]$ is a MAP of $\mathbf{x}$ if for any other partition $\mathcal{I}$ of $[n]$:

[TABLE]

Notation**.**

Here and below, $\operatorname*{argmax}_{a\in A}\bm{\phi}(a)$ is the set of maximisers of function $\bm{\phi}$ on the set $A$ (note that the maximiser may not be unique). Hence the MAP partition of $\mathbf{x}$ in a Bayesian Mixture Model (2.7) can be defined by

[TABLE]

or, equivalently, using (2.6)

[TABLE]

2.2 Specification of the Exponential Family Bayesian Mixture

Models

In the paper we consider Exponential Family Bayesian Mixture Models in which the base and the component distributions come from a conjugate exponential family.

Let $\mathcal{X}\subseteq\mathbb{R}^{d}$ be the observation space and let $\Theta\subseteq\mathbb{R}^{p}$ be the parameter space, which is an open subset of $\mathbb{R}^{p}$. Let $\bm{T}\colon\mathcal{X}\to\mathbb{R}^{p}$ be a statistic, let $h\colon\mathcal{X}\to\mathbb{R}$ be a function with the support of positive Lebesgue measure and let $\bm{\eta}\colon\Theta\to\mathbb{R}^{p}$. Let $B\colon\Theta\to\mathbb{R}$. Suppose that the family of component densities is given by

[TABLE]

Let $B(\bm{\theta})=\bm{a}^{\top}\bm{B}(\bm{\theta})$ where $\bm{a}\in\mathbb{R}^{q}$ and $\bm{B}(\bm{\theta})=[B_{1}(\bm{\theta}),\ldots,B_{q}(\bm{\theta})]^{\top}$. Let $H\colon\Theta\to\mathbb{R}$ and let

[TABLE]

be a nonempty hyperparameter space. We define $A\colon\mathbb{R}^{p}\times\mathbb{R}^{q}\to\mathbb{R}$ as

[TABLE]

Definition**.**

Canonical Exponential Family Bayesian Mixture Model is a Bayesian Mixture Model in which the component density is given by (2.13) and the base density is

[TABLE]

for some $(\bm{\chi},\bm{\tau})\in\Omega$.

Let $\bm{\theta}\sim\pi_{\bm{\chi},\kappa}$ and $\mathbf{u}=(\bm{u}_{1},\ldots,\bm{u}_{k})\,|\,\bm{\theta}\sim g_{\bm{\theta}}$ then it follows from the multiplication rule that $(\bm{\chi}_{\mathbf{u}},\bm{\tau}_{k})\in\Omega$ almost surely and $\bm{\theta}\,|\,\mathbf{u}\sim\pi_{\bm{\chi}_{\mathbf{u}},\bm{\tau}_{k}}$, where $\bm{\chi}_{\mathbf{u}}:=\bm{\chi}+\sum_{i=1}^{k}\bm{T}(\bm{u}_{i})$ and $\bm{\tau}_{k}:=\bm{\tau}+k\bm{a}$. As the marginal density of $\mathbf{u}$ is the quotient of the joint density of $(\bm{\theta},\mathbf{u})$ and the conditional density of $\bm{\theta}\,|\,\mathbf{u}$, we get that

[TABLE]

2.3 Main result

We now define what we mean by $\bm{T}$-linear separation of clusters.

Definition 2.4**.**

Let $\mathcal{X}$ be a family of subsets of $\mathbb{R}^{d}$ and $\mathcal{L}$ a family of real functions on $\mathbb{R}^{d}$. We say that $\mathcal{X}$ is separated by $\mathcal{L}$ if for every $X,Y\in\mathcal{X}$, $X\neq Y$, there exist $L_{X,Y}\in\mathcal{L}$ such that $L_{X,Y}(\bm{x})\geq 0$ and $L_{X,Y}(\bm{y})<0$ for all $\bm{x}\in X,\bm{y}\in Y$. Moreover, if $\mathcal{L}=\{\bm{a}^{\top}\bm{T}(\bm{x})+b\colon\bm{a}\in\mathbb{R}^{s},b\in\mathbb{R}\}$ for some function $\bm{T}\colon\mathbb{R}^{d}\to\mathbb{R}^{s}$, we say that $\mathcal{X}$* is $\bm{T}$-linearly separated from $\mathcal{Y}$*. If $\bm{T}(\bm{x})=\bm{x}$, we use the term linear separability for short.

Notation**.**

For the notational convenience we will use the separability notions also with respect to the sets of sequences in $\mathbb{R}^{d}$. For example, if $\bm{x}_{1},\ldots,\bm{x}_{n}\in\mathbb{R}^{d}$ and $I,J$ are disjoint subsets of $[n]$ then the expression $\mathbf{x}_{I}$ is linearly separated from $\mathbf{x}_{J}$ means that $\{\bm{x}_{i}\colon i\in I\}$ is linearly separated from $\{\bm{x}_{j}\colon j\in J\}$.

In Rajkowski (2018) it is proved that in the Normal Bayesian Mixture Model with Normal distribution on the component mean and fixed covariance matrix, when the prior on the space of partitions is the Chinese Restaurant Process, the convex hulls of the clusters in the MAP partition are disjoint. Equivalently, the MAP is linearly separable. The following theorem is a generalisation of this result to the case of arbitrary Exponential Family Bayesian Mixture Model.

Theorem 2.5**.**

Let $\bm{x}_{1},\ldots,\bm{x}_{n}\in\mathbb{R}^{d}$ be pairwise distinct and let $\hat{\mathcal{I}}$ be the MAP partition of $\bm{x}_{1},\ldots,\bm{x}_{n}$ in the Bayesian Mixture Model where the prior on component parameters is given by (2.16). Then the family $\{\mathbf{x}_{I}\colon I\in\hat{\mathcal{I}}\}$ is $\bm{T}$-linearly separable.

Proof.

The proof is left for Section 4. ∎

2.4 Example: Normal Bayesian Mixture Models

As a commonly used in practice example, we consider Normal Bayesian Mixture Models in which the component distributions are multivariate Normal, so $\mathcal{X}=\mathbb{R}^{d}$. The mean is uknown, but the covariance matrix may be treated as uknown, known or known up to a scaling factor.

Notation**.**

We use two standard notations to denote the determinant of a square matrix $\bm{\Lambda}$: $\det\bm{\Lambda}$ and $|\bm{\Lambda}|$. The latter may seem ambiguous as we also use the symbol $|\cdot|$ to denote the cardinality of a set. However, the meaning of this symbol is always clear from the context.

Notation**.**

To keep the notation precise, in the following we introduce the following convention: if $\bm{\Sigma}$ is a symmetric $d\times d$ matrix, then $\textrm{diag}(\bm{\Sigma})$ is the diagonal of $\bm{\Sigma}$, treated as $d$-dimensional vector, and $\textrm{low}(\bm{\Sigma})$ is the ‘lower triangular’ part of $\bm{\Sigma}$, treated as a $\frac{d(d-1)}{2}$ dimensional vector, whose $\left(\frac{(i-1)(i-2)}{2}+j\right)$-th coordinate is equal to $(i,j)$-th coefficient of $\bm{\Sigma}$, where $i>j$.

2.4.1 Normal-inverse-Wishart

In this case both the mean and the covariance matrix are unknown. The parameter space is therefore equal to $\Theta=\mathbb{R}^{d}\times\mathcal{S}^{d}_{+}$, where $\mathcal{S}^{d}_{+}$ is the space of all positive definite, $d\times d$ matrices. This can be naturally interpreted as an open subset of $\mathbb{R}^{p}$, where $p=\frac{d(d-1)}{2}+d$. For $\bm{\theta}=(\bm{\mu},\bm{\Lambda})\in\Theta$ the component distribution is $\bm{x}\,|\,\bm{\theta}\sim\mathcal{N}(\bm{\mu},\bm{\Lambda})$ and

[TABLE]

where $\mathcal{W}^{-1}$ is the inverse-Wishart distribution. Here the hyperparameters are $\kappa_{0},\nu_{0}>0$, $\bm{\mu}_{0}\in\mathbb{R}^{d}$ and $\bm{\Sigma}_{0}\in\mathcal{S}^{+}$. This model is listed in Gelman et al. (2013) with a slightly different hyperparameters, but we made this modification to obtain

[TABLE]

which is consistent with the remaining two priors.

The conditional densities are given by

[TABLE]

The density of $\bm{x}\,|\,\bm{\theta}$ can be expressed as (2.13) by placing

[TABLE]

We get $\eqref{eq:expdef}$ by placing $\bm{a}=[1,1]^{\top}$,

[TABLE]

and

[TABLE]

Corollary 2.6**.**

Let $\bm{x}_{1},\ldots,\bm{x}_{n}\in\mathbb{R}^{d}$ be pairwise distinct and let $\hat{\mathcal{I}}$ be the MAP partition of $\bm{x}_{1},\ldots,\bm{x}_{n}$ in the Normal Bayesian Mixture Model where the prior on component parameters is given by (2.18). Then the family $\{\mathbf{x}_{I}\colon I\in\hat{\mathcal{I}}\}$ is quadratically separable, i.e. every two clusters are separated by a quadratic surface.

Proof.

It follows from 2.5 and the formula for the sufficient statistic $\bm{T}$ in this model, (2.21). ∎

2.4.2 Normal (fixed covariance)

Here the component covariance matrix is assumed to be known a priori; the component mean is unknown and this is the parameter on which the prior distribution is set, i.e. $\bm{\theta}=\bm{\mu}$, $\Theta=\mathbb{R}^{d}$ and $\bm{x}\,|\,\bm{\mu}\sim\mathcal{N}(\bm{\mu},\bm{\Sigma}_{0})$. The prior is

[TABLE]

The hyperparameters are $\bm{\mu}_{0}\in\mathbb{R}^{d}$ and $\bm{\Psi}_{0},\bm{\Sigma}_{0}\in\mathcal{S}^{+}$. This prior is also listed in Gelman et al. (2013). Clearly

[TABLE]

The conditional densities are given by

[TABLE]

The density of $\bm{x}\,|\,\bm{\theta}$ can be expressed as (2.13) by placing

[TABLE]

We get $\eqref{eq:expdef}$ by placing $\bm{a}=\begin{bmatrix}\textrm{diag}(\bm{\Sigma}_{0}^{-1})\\ \textrm{low}(\bm{\Sigma}_{0}^{-1})\end{bmatrix}$

[TABLE]

and

[TABLE]

Corollary 2.7**.**

Let $\bm{x}_{1},\ldots,\bm{x}_{n}\in\mathbb{R}^{d}$ be pairwise distinct and let $\hat{\mathcal{I}}$ be the MAP partition of $\bm{x}_{1},\ldots,\bm{x}_{n}$ in the Normal Bayesian Mixture Model where the prior on component parameters is given by (2.18). Then the family $\{\mathbf{x}_{I}\colon I\in\hat{\mathcal{I}}\}$ is linearly separable, i.e. every two clusters are separated by a hyperplane.

Proof.

It follows from 2.5 and the formula for the sufficient statistic $\bm{T}$ in this model, (2.21) and the fact that $\bm{\Sigma}_{0}$ is an invertible matrix. ∎

2.4.3 Normal-inverse-Gamma

In Normal-inverse-Gamma model we assume that the base covariance matrix and the component covariance matrix are known up to some scaling factor $\lambda\sim\mathcal{G}^{-1}(\beta_{0}+1,\beta_{0})$. Hence the parameter is $\bm{\theta}=(\bm{\mu},\lambda)$, the parameter space is $\Theta=\mathbb{R}^{d}\times\mathbb{R}\equiv\mathbb{R}^{d+1}$ and

[TABLE]

Here the hyperparameters are $\beta_{0}>0$, $\bm{\mu}_{0}\in\mathbb{R}^{d}$ and $\bm{\Psi}_{0},\bm{\Sigma}_{0}\in\mathcal{S}^{+}$. With this prior

[TABLE]

The conditional densities are given by

[TABLE]

The density of $\bm{x}\,|\,\bm{\theta}$ can be expressed as (2.13) by placing

[TABLE]

We get $\eqref{eq:expdef}$ by placing $\bm{a}=\begin{bmatrix}d/2\\ \textrm{diag}(\bm{\Sigma}_{0}^{-1})\\ \textrm{low}(\bm{\Sigma}_{0}^{-1})\end{bmatrix}$

[TABLE]

and

[TABLE]

Corollary 2.8**.**

Let $\bm{x}_{1},\ldots,\bm{x}_{n}\in\mathbb{R}^{d}$ be pairwise distinct and let $\hat{\mathcal{I}}$ be the MAP partition of $\bm{x}_{1},\ldots,\bm{x}_{n}$ in the Normal Bayesian Mixture Model where the prior on component parameters is given by (2.18). Then the family $\{\mathbf{x}_{I}\colon I\in\hat{\mathcal{I}}\}$ is elliptically separable, i.e. every two clusters are separated by a multidimensional ellipse.

Proof.

It follows from 2.5 and the formula for the sufficient statistic $\bm{T}$ in this model, (2.21). ∎

2.4.4 Comparison of the models

If we assume that the component parameters in the Normal Bayesian Mixture Model are distributed by (2.24) then we assume that the covariance matrix in each component is equal to $\bm{\Sigma}_{0}$ which is known to us. The results of Rajkowski (2018) imply that the misspecification of this hyperparameter may lead to serious inference issues regarding the number of clusters, at least as far as the MAP partition is concerned. In the light of these findings, (2.18) seems to be a safer choice of the prior for the component parameters. In this case the covariance matrix is chosen independently for each component according to the inverse-Wishart distribution. Note that although the Normal-inverse-Wishart prior gives more flexibility in terms of the component covariances, it imposes some modelling restriction, namely the expected within and between group covariance matrices are proportional, as is shown by (2.19). This does not affect the fixed-covariance model, cf. (2.25).

This is the reason for which we propose the Normal-inverse-Gamma prior. It is not listed in Gelman et al. (2013) and we were not able to find any reference to it in the literature. It only allows a 1-parameter variation of the covariance function, but no restrictions are imposed on the within-group means, unlike the Normal-inverse-Wishart prior. At the same time, by allowing the component covariance matrix to scale between clusters can be a remedy to the drawbacks of fixed covariance prior that were pointed out in Rajkowski (2018).

As a final point we note that Normal-inverse-Gamma prior is a generalisation of the Normal prior in the sense that (2.30) becomes (2.24) as $\beta_{0}\to\infty$. Analogously, Normal-inverse-Wishart prior is a quasi-extension of the Normal prior, since as $\eta_{0}\to\infty$, (2.18) converges to (2.24), but with $\bm{\Psi}_{0}=\bm{\Sigma}_{0}/\kappa_{0}$.

3 Discussion of potential applications

We proved a separability result concerning the MAP partition in the Exponential Family Bayesian Mixture Models. In particular, we proved linear or quadratic separability of the MAP partition in most popular Normal Bayesian Mixture Models. Apart from an aesthetic analogy to the properties of Fisher Discriminant Analysis, the benefits of such result may be twofold.

In Rajkowski (2018) the linear separability of the MAP partition is crucial for establishing the existence of ‘limits’ of the MAP partitions when the prior on partitions is the Chinese Restaurant Process and the data is independently and identically distributed with some ‘input distribution’. The limit is related to the partitions of observation space which maximises a given functional $\Delta$ (which depends only on the hypeerparameter $\bm{\Sigma}_{0}$ and the input distribution). The linear separability is important for two reasons: firstly, it is possible to consider the limits of sequences of convex sets and secondly: it is possible to apply the Uniform Law of Large Numbers for the family of convex sets. 2.5 should enable an analogous approach for the Normal-inverse-Wishart and Normal-inverse-Gamma priors on the component parameters.

The other kind of application is more practical; 2.5 shows that the search for an MAP partition may be restricted to situations where clusters are quadratically separated. The space of such partitions is still far too large for an exhaustive search, but may help in finding a partition whose score approximates the MAP score.

4 Proofs

4.1 Proof of 2.5

Lemma 4.1**.**

Let $\mathcal{L}$ be a family of real functions on $\mathbb{R}^{d}$. Let $\bm{x}_{1},\ldots,\bm{x}_{n}\in\mathbb{R}^{d}$ and let $\hat{I}$ be the MAP partition for $\bm{x}_{1},\ldots,\bm{x}_{n}$ in some Bayesian Mixture Model, given by (2.5). If for any $\mathcal{U}\subset[n]$, $k,l\in\mathbb{N}$ such that $k+l=|\mathcal{U}|$ and {\hat{I}}_{{\color[rgb]{0,0,0}k},\mathcal{U}} such that

[TABLE]

observations \mathbf{x}_{{\hat{I}}_{{\color[rgb]{0,0,0}k},\mathcal{U}}} and \mathbf{x}_{\mathcal{U}\setminus{\hat{I}}_{{\color[rgb]{0,0,0}k},\mathcal{U}}} are separated by $\mathcal{L}$ then $\{\mathbf{x}_{I}\colon I\in\hat{\mathcal{I}}\}$ is separated by $\mathcal{L}$.

Proof.

Firstly note that by (2.11)

[TABLE]

Assume that the assumptions of 4.1 hold. Suppose that $\hat{\mathcal{I}}$ is not separated by $\mathcal{L}$. Then there exist ${\hat{I}},{\hat{J}}\in\hat{\mathcal{I}}$ such that $\mathbf{x}_{\hat{I}}$ and $\mathbf{x}_{\hat{J}}$ are not separated by $\mathcal{L}$. Let $\mathcal{U}={\hat{I}}\cup{\hat{J}}$ and {\color[rgb]{0,0,0}k}=|{\hat{I}}|. Let \tilde{I}=I_{{\color[rgb]{0,0,0}k},\mathcal{U}} and $\tilde{J}=\mathcal{U}\setminus\tilde{I}$. Moreover let $\tilde{\mathcal{I}}$ be a partition of $[n]$ obtained by replacing ${\hat{I}},{\hat{J}}$ by $\tilde{I},\tilde{J}$, i.e. $\tilde{\mathcal{I}}=\hat{\mathcal{I}}\setminus\{{\hat{I}},{\hat{J}}\}\cup\{\tilde{I},\tilde{J}\}$. Note that $p_{n}(\hat{\mathcal{I}})=p_{n}(\tilde{\mathcal{I}})$ (we have $|\hat{I}|=|\tilde{I}|$ and $|\hat{J}|=|\tilde{J}|$, so we use the exchangeability of $\bm{\Pi}$). Moreover $\mathbf{x}_{{\hat{I}}}$ and $\mathbf{x}_{{\hat{J}}}$ are not separated by $\mathcal{L}$ so by the assumptions of 4.1

[TABLE]

and hence

[TABLE]

This means that

[TABLE]

which contradicts the definition of $\hat{\mathcal{I}}$ and the proof follows. ∎

Lemma 4.2**.**

Let $V\subseteq\mathbb{R}^{D}$ be a convex set. Let $\alpha\colon V\to\mathbb{R}$ be a strictly concave function, $z_{1},\ldots,z_{k+l}\in\mathbb{R}^{D}$ are pairwise distinct. If $\sum_{i\in I}z_{i}\in V$ for every $I\subseteq[k+l]$ such that $|I|=k$ and

[TABLE]

then $\bm{z}_{{\hat{J}}}$ and $\bm{z}_{[k+l]\setminus{\hat{J}}}$ are linearly separable.

Proof.

Consider the set of all possible sums of {\color[rgb]{0,0,0}k} distinct vectors $z_{i}$, i.e. \mathcal{S}_{\color[rgb]{0,0,0}k}=\{\sum_{i\in I}z_{i}\colon I\subset[n],|I|={\color[rgb]{0,0,0}k}\} and let \hat{s}_{\color[rgb]{0,0,0}k}\in\operatorname*{argmin}_{s\in\mathcal{S}_{\color[rgb]{0,0,0}k}}\alpha(s). Since $\alpha$ is strictly concave, then \hat{s}_{\color[rgb]{0,0,0}k} is a vertex of \textnormal{conv}\ {\mathcal{S}_{\color[rgb]{0,0,0}k}}. This means that there exist a vector $v_{0}\in\mathbb{R}^{d}$ such that \hat{s}_{\color[rgb]{0,0,0}k}\in\operatorname*{argmax}_{s\in\mathcal{S}_{\color[rgb]{0,0,0}k}}\langle s,v_{0}\rangle (cf. Moszynska (2006), Corollary 3.3.6), where $\langle\cdot,\cdot\rangle$ is a standard Euclidean scalar product. I can also choose $v_{0}$ so that $\langle z_{i},v_{0}\rangle$ are are all different (because we are dealing with a discrete set). Let $z_{(1)},\ldots,z_{(k+l)}$ be a decreasing ordering of vectors $z_{i}$ ‘in the direction $v_{0}$’, i.e. $\{z_{(1)},\ldots,z_{(k+l)}\}=\{z_{1},\ldots,z_{k+l}\}$ and $\langle z_{(i)},v_{0}\rangle>\langle z_{(j)},v_{0}\rangle$ if $i<j$. Note that

[TABLE]

and therefore ${\hat{I}}=\{z_{(1)},\ldots,z_{(k)}\}$. Thus the sets $\{z_{i}\colon i\in{\hat{I}}\}$ and $\{z_{i}\colon i\notin{\hat{I}}\}$ are linearly separated by the hyperplane $\{u\in\mathbb{R}^{D}\colon\langle u,v_{0}\rangle=\langle z_{(k)}+z_{(k+1)},v_{0}\rangle/2\}$.

Remark 4.3**.**

Let us assume the notation of Section 2.2. Then the set $\Omega$ is a convex set and the function $A\colon\Omega\to\mathbb{R}$ is strictly convex.

Proof.

It is a well known property of exponential families in canonical form (cf. Diaconis and Ylvisaker (1979)). ∎

Proof of 2.5. Let $\mathcal{U}\subseteq[n]$, {\color[rgb]{0,0,0}k},l\in\mathbb{N} and {\hat{I}}_{{\color[rgb]{0,0,0}k},\mathcal{U}} be as in 4.1. Plugging the formula (2.17) into (4.1) gives:

[TABLE]

Let $\bm{t}_{i}=\begin{bmatrix}\bm{T}(\bm{x}_{i}),\bm{a}\end{bmatrix}$ and let $\bm{t}_{0}=\begin{bmatrix}\bm{\chi},\bm{\tau}\end{bmatrix}$ and $\bm{t}_{\mathcal{U}}=\sum_{i\in\mathcal{U}}\bm{t}_{i}$. By 4.3, $A$ is a strictly convex function. Hence the functions $\underline{\alpha}(\bm{t})=A(\bm{t}_{0}+\bm{t})$ and $\overline{\alpha}(\bm{t})=A(\bm{t}_{0}+\bm{t}_{\mathcal{U}}-\bm{t})$ are also strictly convex and so is their sum, $\alpha(\bm{t})=\underline{\alpha}(\bm{t})+\overline{\alpha}(\bm{t})$. By (4.8) we get

[TABLE]

Therefore by 4.2 we obtain that \bm{t}_{{\hat{I}}_{{\color[rgb]{0,0,0}k},\mathcal{U}}} and \mathbf{t}_{\mathcal{U}\setminus{\hat{I}}_{{\color[rgb]{0,0,0}k},\mathcal{U}}} are linearly separable (i.e. in terms of the base functions). Obviously this yields $\bm{T}$-linear separability of \mathbf{x}_{{\hat{I}}_{{\color[rgb]{0,0,0}k},\mathcal{U}}} and \mathbf{x}_{\mathcal{U}\setminus{\hat{I}}_{{\color[rgb]{0,0,0}k},\mathcal{U}}} and the proof follows.∎∎