Hypothesis Testing of Mixture Distributions using Compressed Data

Hypothesis testing with communication constraints is a classic problem in information theory. The problem was initiated by Ahswede and Csiszár in [1] as well as by Berger in [2]. It is assumed that a pair of data sequences $(x^{n},y^{n})$ is observed at separate locations. The sequence $x^{n}$ is compressed and sent to the location of $y^{n}$ via a noiseless channel. The decision center there decides whether $(x^{n},y^{n})$ is iid generated from the null hypothesis with a distribution $P_{XY}$ or from the alternative hypothesis with a distribution $Q_{XY}$. The Neyman-Pearson framework was used to study the trade-off between the probabilities of errors. The main interest was to establish the maximum error exponent of the second type of error when the probability of the first type of error is bounded as the number of samples tends to infinity. A single letter formulation was given in [1] for the testing against independence scenario and strong converse was proven for the general setting under a special condition. Various lower bounds for the general setting have been proposed in [3] and [4]. The work [3] also established the optimal error exponent in the zero-compression rate regime when $\epsilon$ is small enough. Using the same condition as in [1], the work [5] established the optimal $\epsilon$-error exponent also in the zero-compression rate regime. The optimality of the random binning scheme proposed in [4] was shown for the conditional independence setting in [6]. In [7] the authors extended the testing against independence study to the successive refinement setting. The work [8] studied the non-asymptotic regime of the general setting under the two-sided zero-compression rate constraint. Yet, in all of the above studies, the distributions in both hypotheses are assumed to be iid.

Mixture distributions are prevalent in practice, cf. [9] for a comprehensive list of applications. However, hypothesis testing for mixture distributions is an under-examined direction in information theory. Notable studies are given in [10, 11, 12]. Communication constraints are not included in the above works. In this work we study the following binary hypothesis problem in which the hypotheses are given by

Mixture of iid components is also known as mixture of repeated measurements or mixture of grouped observations in the literature. By no means exhaustive, we list a few works with applications in topic modeling [13], in cognitive psychology [14], and in developmental psychology [15], cf. also [16]. In machine learning applications, to allow flexible modeling it is often assumed that within each component the joint distribution has a product form without the requirement of having the same marginal distribution, cf. [17] and [18]. We keep the iid condition in the null hypothesis for tractable analysis. A motivating example is provided in the following.

Assume that two statisticians observe two bags of words $x^{n}$ and $y^{n}$ taken as excerpts from some documents where $n$ denote the number of words in each bag. For simplicity we assume that in each bag words have an identical marginal distribution and also the order of words is not important. Then it is likely that the bag of words $x^{n}$ ($y^{n}$) is not generated iid, since for example knowing the first word $x_{1}$ ($y_{1}$) to be in Latin (German) gives us a guess that the whole $x^{n}$ ($y^{n}$) is in Latin (German) [19]. Since the order of words does not matter, $x^{n}$ and $y^{n}$ are sequences of exchangeable observations. It is natural to approximate the distributions of $x^{n}$ and $y^{n}$ by finite mixtures of iid distributions, due to de Finetti’s theorem [20], in which each underlying state represents a topic. The two statisticians then form a hypothesis testing problem. In the null hypothesis, they assume that the two bags of words are generated jointly iid according to $P_{Y_{i}X_{i}}^{\otimes n}$ from an unknown topic $i\in[1:m]$ with probability $\nu_{i}$, for example $x_{l}$ is a direct translation or a synonym of $y_{l}$ for all $l\in[1:n]$. In the alternative hypothesis they assume that the two bags of words are generated independently and iid according to $Q_{Y_{j}}^{\otimes n}\times Q_{X_{t}}^{\otimes n}$ from an unknown topic $(j,t)$ with probability $\tau_{jt}$.

We similarly are interested in the optimal error exponent of the second type of error when the probability of the first type is restricted by some $\epsilon\in[0,1)$. The iid assumption of each component in the null hypothesis as well as the factorization assumption in the alternative hypothesis are used to facilitate the derivation. In contrast to the classic strong converse result in [1], the maximum $\epsilon$-achievable error exponent generally depends on not only the prior distribution $(\nu_{i})$ but also relations between different constituent components in the mixture distributions. When no compression is involved, the weak law of large numbers can be used to establish the maximum $\epsilon$-achievable error exponent, cf. [11, 12]. However, this argument is no longer applicable in the presence of compression. We need to use an argument established through proving exponential strong converses of constituent components.

We summarize contents of our work in the following.

• •

  We use code transformation arguments to unveil a new property of the classic Ahlswede-Csiszár problem in a general setting, cf. Theorem 7 in Appendix A. The arguments are also useful in simplifying proofs of different results in our work.

• •

  We provide the optimal achievable error exponent in Theorem 3 and show in Corollary 2 that for $\epsilon>0$ small enough the obtained error exponent is also $\epsilon$-optimal. Our proof is based on studying a compound hypothesis testing problem of differentiating between $\{P_{Y_{i}X_{i}}^{\otimes n}\}$ and $\{Q_{Y_{j}^{n}}\times Q_{X_{t}^{n}}\}$ under communication constraints. The results are established when $\{(Q_{Y_{j}^{n}})\}$ and $\{(Q_{X_{t}^{n}})\}$ are stationary memoryless processes or finite Markov processes with stationary transition probabilities.

• •

  Under a further sufficient condition on the sets of distributions $\{P_{Y_{i}X_{i}}^{\otimes n}\}$ and $\{Q_{Y_{j}^{n}}\}$, Assumption 2, we provide a complete characterization of the maximum $\epsilon$-error exponent in Theorem 4. It is shown that even if the strong converse is available for the compound problem of testing $\{P_{Y_{i}X_{i}}^{\otimes n}\}$ against $\{Q_{Y_{j}^{n}}\times Q_{X_{t}^{n}}\}$, it is not sufficient for establishing the maximum $\epsilon$-error exponent in our mixture setting. Our derivation is based on the exponentially strong converse of the false alarm probabilities under the assumed condition.

• •

  We refine a recently established connection in [21] between the Wyner-Ahlswede-Körner (WAK) problem [22, 23] and the hypothesis testing against independence problem. While the previous result holds only under the stationary ergodic assumption, our new connection in Theorem 5 is valid under a more general assumption. We then use the refined connection to establish the corresponding minimum achievable compression rate and the minimum $\epsilon$-achievable compression rate for the WAK problem with mixture distributions.

Our paper is organized as follows. In Section II we review previous results and define quantities which are essential to characterize optimal error exponents in our study. We then establish various results for the compound setting, some might be of independent interest in Section III. In Section IV, we provide the maximum achievable error exponent in the mixture setting. We state the sufficient condition and use it to establish the maximum $\epsilon$-achievable error exponent in the mixture setting in Section V. Then the refined connection between the WAK problem and the hypothesis testing against independence problem is given in Section VI. Consequences of the new connection are also given therein.

We focus on finite alphabets in this paper. Before we begin we make the following conventions. Given a probability measure $\mu$, $\mu^{\otimes n}$ denotes its $n$-fold product measure extension. $\log(\textperiodcentered)$ denotes the natural logarithm. For any two distributions $P$ and $Q$ on an alphabet $\mathcal{U}$, assume that $Q(u)=0$, if $P$ is absolutely continuous w.r.t $Q$, denoted by $P\ll Q$, holds then we define $\iota_{P\|Q}(u)=0$, otherwise we define $\iota_{P\|Q}(u)=+\infty$, irrespective of whether $P(u)$ is equal to 0 or not. We also define¹¹1For simplicity we use the convention $\log 0=-\infty$ in the following. $\iota_{P\|Q}(u)=\log P_{Y}(u)/Q_{Y}(u)$ when $Q(u)>0$. The relative entropy between two distributions $P$ and $Q$ is defined as $D(P\|Q)=\mathbb{E}[\iota_{P\|Q}(U)]$ where $U\sim P$. For a joint distribution $P_{UV}$ on $\mathcal{U}\times\mathcal{V}$ if $P_{U}(u)=0$ or $P_{V}(v)=0$ holds then we define $\iota_{P_{UV}}(u;v)=0$ as $P_{UV}\ll P_{U}\times P_{V}$ is valid. Otherwise we define $\iota_{P_{UV}}(u;v)=\log P_{UV}(u,v)/(P_{U}(u)\times P_{V}(v))$. The mutual information between $U$ and $V$ that is jointly distributed according to $P_{UV}$ is defined as $I(U;V)=\mathbb{E}[\iota_{P_{UV}}(U;V)]$. For a finite set $\mathcal{A}$, we use $|\mathcal{A}|$ and $\mathcal{A}^{c}$ to denote its cardinality and its complement. For a mapping $\phi\colon\mathcal{X}\to\mathcal{M}$ we define $|\phi|$ to be the cardinality of its range, $|\phi|\triangleq|\mathcal{M}|$. For a given distribution $P$ on $\mathcal{X}$ and a stochastic mapping $f\colon\mathcal{X}\to\{0,1\}$ with the corresponding transition kernel $W$ we define

The following result characterizes the optimal achievable error exponent in the compound hypothesis testing problem.

The achievability proof of Theorem 1 is given in Appendix B. The converse proof of Theorem 1 follows from the one of Theorem 3. The proof of Theorem 1 uses a combination of techniques from [11], [25] and our new mixing idea. In the following we provide an overview of steps in the proof of Theorem 1.

Assume that the set of marginal distributions on $\mathcal{X}$, $\mathcal{P}_{\mathcal{X}}$, consists of a single element. Assume further that the number of components in the null hypothesis is two, i.e. $m=2$ and $\mathfrak{F}_{1}=\{1,2\}$. First we check whether the sequence $x^{n}$ is “typical” in the sense that

This helps us to perform the change of measure step from $Q_{X_{t}^{n}}$ to $P_{X}^{\otimes n}$ in the analysis of the type-II (or miss detection) probability. The above condition is violated with vanishing probability in the analysis of the false alarm probability. We then select a test channel $P_{U|X}$ and generate a codebook from the marginal distribution $P_{U}^{\otimes n}$. In our proof we do not estimate $i$ from the sequence $y^{n}$ to avoid potential complications in the analysis of the miss detection probability. We artificially create instead the following joint distribution

where $\nu_{i}$ are positive probability weights. $P_{Y^{n}U^{n}}$ shifts the burden from calculating the miss detection probabilities to bounding the false alarm probabilities, which is less complex. We then consider the following score function which is helpful in defining a deterministic decision mapping $\psi_{n}$

In the score function $\zeta(\cdot,\cdot)$ the first term resolves the uncertainty within the set of marginal distributions $\{P_{Y_{i}}^{\otimes n}\}$, while the second term resolves the mismatch between two sets of distributions $\{P_{Y_{i}}^{\otimes n}\}$ and $\{Q_{Y_{t}^{n}}\}$. The second term also indirectly checks whether $y^{n}$ is “typical”.
Given a chosen codeword, which we explain how to obtain later, we decide that the null hypothesis is true if $\zeta(y^{n},u^{n})>n(E-d_{1}^{\mathrm{x}})$ is fulfilled. Given this decision the miss detection probabilities can be deduced based on the following chain of measure changing steps

as well as the fact that $\sum_{y^{n},u^{n}}P_{Y^{n}|U^{n}}(y^{n}|u^{n})P_{\phi(X^{n})}(u^{n})\leq 1$ holds, where the summation is performed over the decision region.
Now to obtain a transmission message index, we search for a codeword such that it yields the lowest maximum false alarm conditional probabilities by looking at

In the analysis of the maximum false alarm probabilities changing measure from $P_{Y^{n}U^{n}}$ to $P_{Y_{i}^{n}U_{i}^{n}}$ in the expression of $\zeta(\cdot,\cdot)$ is relatively standard, cf. [11]. We then can use the standard typical arguments to conclude the existence of a good codebook. In the general case where the marginal set $\mathcal{P}_{\mathcal{X}}$ has multiple elements, we need to estimate $s$. Because of the way that we design $\zeta(\cdot,\cdot)$ this extra step does not affect the exponent of the miss detection probability.

A simple case in which the third statement in Theorem 2 holds, is when $|\mathfrak{F}_{s}|=1$ for all $s\in[1:|\mathcal{P}_{\mathcal{X}}|]$. The proof of Theorem 2 is given in Appendix C. The second part of Theorem 4 will be employed in proving the converse of Theorem 4 for mixture models in Section IV.

Recall that $\mathfrak{F}_{s}$ represents the set of distributions $P_{Y_{i}X_{i}}$ which have the same marginal distribution $P_{\mathcal{X},s}$ on $\mathcal{X}$, cf. Equation (10). In the following we provide an outline of the proofs of the first two points in the statement Theorem 2.

We discuss in this paragraph the first item in the first part of Theorem 2. When $s$ is active, i.e., $\theta_{s}(R_{c})=\min_{i\in\mathfrak{F}_{s}}\xi_{i}(R_{c})$ holds, then it follows from the strong converse bound for testing $P_{Y_{i}X_{i}}^{\otimes n}$ against $Q_{Y_{j_{i}^{\star}}^{n}}\times Q_{X_{t_{s}^{\star}}^{n}}$ for each $i$ inside the class $\mathfrak{F}_{s}$, that $E_{\mathrm{comp},\epsilon}^{\star}(R_{c})\leq\theta_{s}(R_{c})$ holds for all $\epsilon\in[0,1)$, cf. Theorem 7. Therefore we only need to focus on the inactive set $\mathcal{S}$. For simplicity, in this discussion we can assume that $\mathcal{S}=\{1\}$ holds and there are two components inside $\mathfrak{F}_{1}$. We assume further that there is no mismatch, i.e., $\{Q_{Y_{j}^{n}}\}=\{P_{Y_{i}}^{\otimes n}\}$ and $\{Q_{X_{t}^{n}}\}=\{P_{X_{i}}^{\otimes n}\}$ hold. The formulation of $\theta_{1}(R_{c})$ requires the selection of a test channel $P_{U|\bar{X}_{1}}$. To show the strong converse bound, the general idea is hence to identify a common test channel $P_{U|\bar{X}_{1}}$. This can be done by considering relevant sets $\mathcal{V}_{i}$, $i=1,2$, of $x^{n}$ such that for each $i=1,2,$

holds, where $\eta\in(0,1-(\epsilon+\gamma))$ and $\gamma\in(0,1-\epsilon)$ are arbitrary. By setting $\eta=1/2-(\epsilon+\gamma)$ and using the reverse Markov inequality, we obtain the following inequalities

We require that the intersection $\mathcal{V}=\cap_{i=1}^{2}\mathcal{V}_{i}$ should be non-empty. This allows us to define a joint distribution $P_{\tilde{Y}_{1}^{n}\tilde{Y}_{2}^{n}\tilde{X}^{n}}$ where $P_{\tilde{X}^{n}}$ is supported on $\mathcal{V}$. Note also that $\eta$-restriction in the definition of $\mathcal{V}_{i}$ allows us to obtain the following inequality

With this we can identify $\tilde{U}_{l}=(\phi_{n}(\tilde{X}^{n}),(\tilde{Y}_{i}^{l-1})_{i=1}^{2})$ for all $l\in[1:n]$. The variational arguments in [26] can be used to indeed show that $E_{\mathrm{comp},\epsilon}^{\star}(R_{c})\leq\theta_{1}(R_{c})$ also holds. To make $\mathcal{V}$ non-empty, we must have $\epsilon<1/2=\frac{1}{|\mathfrak{F}_{1}|}$. When the inactive set $\mathcal{S}$ contains more than one element we obtain the corresponding threshold $\min_{s\in\mathcal{S}}\frac{1}{|\mathfrak{F}_{s}|}$.

Now we discuss about the second item in the first part of Theorem 2. Since $E_{\mathrm{comp},\epsilon}^{\star}(R_{c})\leq\xi_{i}(R_{c})$ holds for all $i\in[1:m]$ and $\epsilon\in[0,1)$, cf. Theorem 7, we only explain the achievability direction of the second item. For simplicity we assume that the set of marginal distributions $\mathcal{P}_{\mathcal{X}}$ has a single element, the element is inactive $\mathcal{S}=\{1\}$, and $\mathfrak{F}_{1}=\{1,2\}$. Our achievability idea is to build two sequences of testing schemes separately and then mix them together. For this we need to divide the space $\mathcal{X}^{n}$ into $|\mathfrak{F}_{1}|=2$ partitions $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ such that $P_{\bar{X}_{1}}^{\otimes n}(\mathcal{C}_{l})>1-\epsilon$ for all $l=1,2$ for all sufficiently large $n$. Since $\epsilon>\frac{|\mathfrak{F}_{1}|-1}{|\mathfrak{F}_{1}|}=1/2$, such partitioning can be done. For each $i\in\{1,2\}$, we design a sequences of testing schemes $(\phi_{n}^{1i},\psi_{n}^{1i})$ to differentiate between $P_{Y_{i}X_{i}}^{\otimes n}$ and $Q_{Y_{j_{i}^{\star}}^{n}}\times Q_{X_{t_{1}^{\star}}^{n}}$ such that $\xi_{i}(R_{c})-\gamma$ is achievable. As in the proof of Theorem 1 we also define an auxiliary mixture distribution

The mixture distribution helps to alleviate the estimation of the distribution of $y^{n}$. Once the preparation is complete, we perform the compression as follows.
We first check if $x^{n}$ is a typical sequence in the sense that whether the following condition is fulfilled or not

Again the above inequality also helps resolving the mismatch between two sets of distributions $\{P_{X_{i}}^{\otimes n}\}$ and $\{Q_{X_{t}^{n}}\}$. Suppose that $x^{n}$ is typical. If $x^{n}\in\mathcal{C}_{1}$ then we use $\phi_{n}^{11}$ to compress it, and similarly when $x^{n}\in\mathcal{C}_{2}$ we use $\phi_{n}^{12}$ to compress it. The joint compression mapping is then

The joint compression mapping induces the following distribution from $P_{Y^{n}X^{n}}$

We also define the following score function

We say $H_{0}$ is true if $\zeta(y^{n},\phi_{n}^{1}(x^{n}))>n(E-d_{1}^{\mathrm{x}})$ holds. Let us look at $\alpha_{n}^{(1)}$ which can be upper-bounded as