Sub-Gaussian Error Bounds for Hypothesis Testing
^††thanks: This research was supported in part by the US National Science Foundation under grant HDR: TRIPODS 19-34884.

Hypothesis testing is one central task in statistics. One of its simplest forms is the binary case: given $n$ independent and identically distributed (i.i.d.) random variables $X_{1}^{n}\equiv(X_{1},\ldots,X_{n})$, one wants to infer whether the null hypothesis $H_{0}:X_{i}\sim P_{0}$ or the alternative hypothesis $H_{1}:X_{i}\sim P_{1}$ is true. The binary case serves as an important starting point from which further results can be established, in the settings of both classical and quantum hypothesis testing [1, 2]. With $X_{1}^{n}$, one can construct the empirical distribution $\hat{P}_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}$, where $\delta_{X}$ is the Dirac measure that puts unit mass at $X$. Adopting the Kullback-Leibler (KL) divergence as a distance from $\hat{P}_{n}$ to $P_{0}$ or $P_{1}$, one can construct a test function as

where $I\{\cdot\}$ is the indicator function, $c\geq 0$ serves as a threshold beyond which the decision that $\hat{P}_{n}$ is closer to $P_{1}$ than to $P_{0}$ is made, and $D_{\mathrm{KL}}(P\rVert Q)=\int\ln(dP/dQ)dP$ is the KL divergence from probability $P$ to probability $Q$ if $P\ll Q$. Conventionally, if $P$ is not absolutely continuous with respect to $Q$, then $D_{\mathrm{KL}}(P\rVert Q)\equiv\infty$. Note $\hat{P}_{n}$ is discrete; hence if both $P_{0}$ and $P_{1}$ are discrete with the same support, (1) is well defined. Denote the densities of $P_{0}$ and $P_{1}$ with respect to the counting measure as $p_{0}$ and $p_{1}$, respectively, and we have

In fact, in this case, (1) is equivalent to the test function for the likelihood ratio test [4]

where $c^{\prime}=e^{cn}$. In the case that both $P_{0}$ and $P_{1}$ are continuous, the KL divergence difference $D_{\mathrm{KL}}(\hat{P}_{n}\rVert P_{0})-D_{\mathrm{KL}}(\hat{P}_{n}\rVert P_{1})$ is not well defined. Nonetheless, the technically tricky part is the term “$\int\hat{p}_{n}\ln(\hat{p}_{n})d\mu$,” where we use $\hat{p}_{n}$ to denote the density of $\hat{P}_{n}$ with respect to the Lebesgue measure $\mu$ as if it had one. But it appears twice and is cancelled out formally. We might conveniently define the KL divergence difference in this case as (2), and still find the equivalence between (1) and (3). Using the KL divergence in the context of hypothesis testing can be beneficial. Firstly, it provides a clear geometric meaning to the likelihood ratio test, as well as to the general idea underlying hypothesis testing. Secondly, it also offers a geometric, or even physical, interpretation of the lower bound for the resulting statistical errors, as shown below.

Under the null hypothesis $H_{0}$, the type I error rate (or the significance level) $\alpha$ that is incurred by applying (1) for a fixed $c$ is

where $P_{0}^{\otimes n}$ is the product probability measure for $X_{1}^{n}$ under $H_{0}$. In practice, by prescribing the significance level, for example, letting $\alpha=0.05$, one can derive the corresponding $c$ and determine the desired test function. However, in this work, our focus is $not$ to find a test function at given $\alpha$, we mainly deal with the case that $c$ is fixed, and $\alpha$ is obtained in a somewhat passive way. Thanks to the Neyman-Pearson lemma [3], the likelihood ratio test is known to be optimal in the sense of statistical power. Hence, given the incurred $\alpha$, test function (1) has the minimal type II error rate $\beta$ among all possible test functions with the corresponding type I error rate no gretaer than $\alpha$:

where $P_{1}^{\otimes n}$ is the product probability measure for $X_{1}^{n}$ under the alternative hypothesis $H_{1}$.

Controlling statistical errors is of practical importance; however, typically one cannot suppress both types of error simultaneously. Under our i.i.d. setting, a classical result, based on Pinsker’s inequality, concerning the error bound for any (measurable) test function is that [4]

This result is striking in that without going into the details of calculating $\alpha$ and $\beta$, one can have a nontrivial lower bound of their sum in terms of the KL divergence between two candidate probabilities, as long as the right-hand side of (6) is greater than 0. For a fixed $n$, this bound is solely determined by $D_{\mathrm{KL}}(P_{1}\rVert P_{0})$, which reflects the “distance” from $P_{1}$ to $P_{0}$. This result also has a significant physical meaning. At a nonequilibrium steady state, if $P_{1}$ denotes the probability associated with observing a stochastic trajectory in the forward process, and $P_{0}$ in the backward process, then the theory of stochastic thermodynamics tells us that $D_{\mathrm{KL}}(P_{1}\rVert P_{0})$ is equivalent to the average entropy production $\Delta S$ in the forward process, which is always nonnegative [5, 6]. Hence, if one wishes to infer the arrow of time based on observations, then Pinsker’s result (6) implies that the chance of making an error is high if $\Delta S$ is small. In fact, we know that $\Delta S=0$ at equilibrium, and one cannot tell the arrow of time at all; hypothesis testing is just random guess in this case.

While (6) is useful, can we have a tighter and thus more informative bound? In this work, we will show that by taking advantage of the sub-Gaussian property of $\Phi(X_{1}^{n})$ [7, 8], one can derive a bound (17) on statistical errors in terms of its sub-Gaussian norm (as well as the KL divergence from $P_{1}$ to $P_{0}$). We name such an error bound as “sub-Gaussian” to highlight this fact. It turns out that it is a tighter bound than (6) in the sense that it provides a greater lower bound for $\alpha+\beta$ (or for $\beta$ at any given $\alpha\neq 0.5$). In practice, a small $\alpha$ is commonly set as the significance level, and our result can hopefully be more relevant. Moreover, in the case of $M$-ary hypothesis testing where $M>2$ hypotheses are present, we also derive a bound (24) for making incorrect decisions, which is complementary to the celebrated Fano’s inequality [9] when the number of hypotheses $M$ or the sample size $n$ is not large. The error bounds presented in this work are universal and easily applicable. We hope these findings can help better quantify errors in various statistical practices involving hypothesis testing.

We will first introduce the sub-Gaussian norm of $\Phi(X_{1}^{n})$. Then error bounds in the binary and $M$-ary cases are established, respectively.

In this work, by using the sub-Gaussian property of test functions, we uncover two universal error bounds in terms of the sub-Gaussian norm and the Kullback-Leibler divergence. In the case of binary hypothesis testing, our bound (17) is always tighter than Pinsker’s bound (6) for any given $\alpha\neq 0.5$. In the case of $M$-ary hypothesis testing, our result (24) is complementary to Fano’s inequality (29) by providing a more informative bound when the number of hypotheses or the sample size is not large.

Given the universality of our results, we hope, with possible generalizations, they can find potential applications in fields ranging from clinical trials to quantum state discrimination. In particular, the quantum extension of these bounds is of special interest. Due to the experimental cost, it may be important to quantify statistical errors in the presence of a limited number of observations, and nonasymptotic rather than asymptotic results are thus more relevant. Both our bounds hold for any finite sample size, and can hopefully be helpful in such cases.

YW gratefully thank Prof. Dan Nettleton for helpful discussions that stimulated this work and Prof. Huaiqing Wu for a careful review of the manuscript and insightful comments.