What is the connection between credible regions and Bayesian hypothesis tests?

Question

In frequentist statistics, there is a close connection between confidence intervals and tests. Using inference about $\mu$ in the $\rm N(\mu,\sigma^2)$ distribution as an example, the $1-\alpha$ confidence interval $$\bar{x}\pm t_{\alpha/2}(n-1)\cdot s/\sqrt{n}$$ contains all values of $\mu$ that aren't rejected by the $t$-test at the significance level $\alpha$.

Frequentist confidence intervals are in this sense inverted tests. (Incidentally, this means that we can interpret the $p$-value as the smallest value of $\alpha$ for which the null value of the parameter would be included in the $1-\alpha$ confidence interval. I find that this can be a useful way to explain what $p$-values really are to people who know a bit of statistics.)

Reading about the decision-theoretic foundation of Bayesian credible regions, I started to wonder whether there is a similar connection/equivalence between credible regions and Bayesian tests.

Is there a general connection? If there is no general connection, are there examples where there is a connection? If there is no general connection, how can we see this?

Answer 1

I managed to come up with an example where a connection exists. It seems to depend heavily on my choice of loss function and the use of composite hypotheses though.

I start with a general example, which is then followed by a simple special case involving the normal distribution.

General example

For an unknown parameter $\theta $, let $\Theta$ be the parameter space and consider the hypothesis $\theta\in\Theta_0$ versus the alternative $\theta\in\Theta_1=\Theta\backslash\Theta_0$.

Let $\varphi$ be a test function, using the notation in Xi'an's The Bayesian Choice (which is sort of backwards to what I at least am used to), so that we reject $\Theta_0$ if $\varphi=0$ and accept $\Theta_0$ if $\varphi=1$. Consider the loss function $$ L(\theta,\varphi) = \begin{cases} 0, & \mbox{if } \varphi=\mathbb{I}_{\Theta_0}(\theta) \\ a_0, & \mbox{if } \theta\in\Theta_0 \mbox{ and }\varphi=0\\ a_1, & \mbox{if } \theta\in\Theta_1 \mbox{ and }\varphi=1. \end{cases} $$ The Bayes test is then $$\varphi^\pi(x)=1\quad \rm if\quad P(\theta\in\Theta_0|x)\geq a_1(a_0+a_1)^{-1}.$$

Take $a_0=\alpha\leq 0.5$ and $a_1=1-\alpha$. The null hypothesis $\Theta_0$ is accepted if $\rm P(\theta\in\Theta_0|x)\geq 1-\alpha$.

Now, a credible region $\Theta_c$ is a region such that $\rm P(\Theta_c|x)\geq 1-\alpha$. Thus, by definition, if $\Theta_0$ is such that $\rm P(\theta\in\Theta_0|x)\geq 1-\alpha$, $\Theta_c$ can be a credible region only if $\rm P(\Theta_0\cap\Theta_c|x)>0$.

We accept the null hypothesis if an only if each $1-\alpha$-credible region contains a non-null subset of $\Theta_0$.

A simpler special case

To better illustrate what kind of test we have in the above example, consider the following special case.

Let $x\sim \rm N(\theta,1)$ with $\theta\sim \rm N(0,1)$. Set $\Theta=\mathbb{R}$, $\Theta_0=(-\infty,0]$ and $\Theta_1=(0,\infty)$, so that we wish to test whether $\theta\leq 0$.

Standard calculations give $$\rm P(\theta\leq 0|x)=\Phi(-x/\sqrt{2}),$$ where $\Phi(\cdot)$ is the standard normal cdf.

Let $z_{1-\alpha}$ be such that $\Phi(z_{1-\alpha})=1-\alpha$. $\Theta_0$ is accepted when $-x/\sqrt{2}>z_{1-\alpha}$.

This is equivalent to accepting when $x\leq \sqrt{2}z_{\alpha}.$ For $\alpha=0.05$, $\Theta_0$ is therefore rejected when $x>-2.33$.

If instead we use the prior $\theta\sim \rm N(\nu,1)$, $\Theta_0$ is rejected when $x>-2.33-\nu$.

Comments

The above loss function, where we think that falsely accepting the null hypothesis is worse than falsely rejecting it, may at first glance seem like a slightly artifical one. It can however be of considerable use in situations where "false negatives" can be costly, for instance when screening for dangerous contagious diseases or terrorists.

The condition that all credible regions must contain a part of $\Theta_0$ is actually a bit stronger than what I was hoping for: in the frequentist case the correspondence is between a single test and a single $1-\alpha$ confidence interval and not between a single test and all $1-\alpha$ intervals.

Answer 2

Michael and Fraijo suggested that simply checking whether the parameter value of interested was contained in some credible region was the Bayesian equivalent of inverting confidence intervals. I was a bit skeptical about this at first, since it wasn't obvious to me that this procedure really resulted in a Bayesian test (in the usual sense).

As it turns out, it does - at least if you're willing to accept a certain type of loss functions. Many thanks to Zen, who provided references to two papers that establish a connection between HPD regions and hypothesis testing:

• Pereira & Stern (1999), Evidence and credibility: full Bayesian significance test for precise hypotheses, Entropy
• Madruga, Esteves & Wechsler (2001), On the Bayesianity of Pereira-Stern tests, Test

I'll try to summarize them here, for future reference. In analogue with the example in the original question, I'll treat the special case where the hypotheses are $$H_0: \theta\in\Theta_0=\{\theta_0\}\qquad \mbox{and}\qquad H_1: \theta\in\Theta_1=\Theta\backslash \Theta_0,$$ where $\Theta$ is the parameter space.

Pereira & Stern proposed a method for testing said hypotheses without having to put prior probabilities on $\Theta_0$ and $\Theta_1$.

Let $\pi(\cdot)$ denote the density function of $\theta$ and define $$T(x)=\{ \theta:\pi(\theta|x)>\pi(\theta_0|x)\}.$$

This means that $T(x)$ is a HPD region, with credibility $P(\theta\in T(x)|x)$.

The Pereira-Stern test rejects $\Theta_0$ when $P(\theta\notin T(x)|x)$ is "small" ($<0.05$, say). For a unimodal posterior, this means that $\theta_0$ is far out in the tails of the posterior, making this criterion somewhat similar to using p-values. In other words, $\Theta_0$ is rejected at the $5~\%$ level if and only if it is not contained the in $95~\%$ HPD region.

Let the test function $\varphi$ be $1$ if $\Theta_0$ is accepted and $0$ if $\Theta_0$ is rejected. Madruga et al. proposed the loss function $$ L(\theta,\varphi,x) = \begin{cases} a(1-\mathbb{I}(\theta\in T(x)), & \mbox{if } \varphi(x)=0 \\ b+c\mathbb{I}(\theta\in(T(x)), & \mbox{if } \varphi(x)=1, \end{cases} $$ with $a,b,c>0$.

Minimization of the expected loss leads to the Pereira-Stern test where $\Theta_0$ is rejected if $P(\theta\notin T(x)|x)<(b+c)/(a+c).$

So far, all is well. The Pereira-Stern test is equivalent to checking whether $\theta_0$ is in an HPD region and there is a loss function that generates this test, meaning that it is founded in decision theory.

The controversial part though is that the loss function depends on $x$. While such loss functions have appeared in the literature a few times, they don't seem to be generally accepted as being very reasonable.

For further reading on this topic, see a list of papers that cite the Madruga et al. article.

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Update October 2012:

I wasn't completely satisfied with the above loss function, as its dependence on $x$ makes the decision-making more subjective than I would like. I spent some more time thinking about this problem and ended up writing a short note about it, posted on arXiv earlier today.

Let $q_{\alpha}(\theta|x)$ denote the posterior quantile function of $\theta$, such that $P(\theta\leq q_{\alpha}(\theta|x))=\alpha$. Instead of HPD sets we consider the central (equal-tailed) interval $(q_{\alpha/2}(\theta|x),q_{1-\alpha/2}(\theta|x))$. To test $\Theta_0$ using this interval can be justified in the decision-theoretic framework without a loss function that depends on $x$.

The trick is to reformulate the problem of testing the point-null hypothesis $\Theta_0=\{\theta_0\}$ as a three-decision problem with directional conclusions. $\Theta_0$ is then tested against both $\Theta_{-1}=\{\theta:\theta<\theta_0\}$ and $\Theta_{1}=\{\theta:\theta>\theta_0\}$.

Let the test function $\varphi=i$ if we accept $\Theta_i$ (note that this notation is the opposite of that used above!). It turns out that under the weighted $0-1$ loss function $$L_2(\theta,\varphi) = \begin{cases} 0, & \mbox{if } \theta\in\Theta_i\mbox{ and }\varphi=i, \quad i\in \{-1,0,1\}, \\ \alpha/2, & \mbox{if } \theta\notin\Theta_0 \mbox{ and }\varphi=0,\\ 1, & \mbox{if } \theta\in\Theta_{i}\cup\Theta_0 \mbox{ and }\varphi=-i,\quad i\in\{-1,1\},\end{cases}$$ the Bayes test is to reject $\Theta_0$ if $\theta_0$ is not in the central interval.

This seems like a quite reasonable loss function to me. I discuss this loss, the Madruga-Esteves-Wechsler loss and testing using credible sets further in the manuscript on arXiv.

Answer 3

I believe the paper http://ba.stat.cmu.edu/vol03is01.php can clarify most of our discussion here.

Answer 4

I coincidentally read your arXiv paper prior to coming to this question and already wrote a blog entry on it (scheduled to appear on October, 08). To sum up, I find your construction of theoretical interest, but also think it is too contrived to be recommended, esp. as it does not seem to solve the point-null hypothesis Bayesian testing problem, which traditionally requires to put some prior mass on the point-null parameter value.

To wit, the solution you propose above (in the October update) and as Theorem 2 in your arXiv paper is not a valid test procedure in that $\varphi$ takes three values, rather than the two values that correspond to accept/reject. Similarly, the loss function you use in Theorem 3 (not reproduced here) amounts to testing a one-sided hypothesis, $H_0: \theta\le\theta_0$, rather than a point-null hypothesis $H_0: \theta= \theta_0$.

My major issue however is that it seems to me that both Theorem 3 and Theorem 4 in your arXiv paper are not valid when $H_0$ is a point-null hypothesis, i.e. when $\Theta_0=\{\theta_0\}$, with no prior mass.

Answer 5

You can use a credible interval (or HPD region) for Bayesian hypothesis testing. I don't think it is common; though, to be fair I do not see much nor do I use formal Bayesian Hypothesis testing in practice. Bayes factors are occasionally used (and in Robert's "Bayesian Core" somewhat lauded) in hypothesis testing set up.

Answer 6

A credible region is just a region where the integral of the posterior density over the region is a specified probability e.g. 0.95. One way to form a Bayesian hypothesis test is to see whether or not the null hypothesized value(s) of the parameter(s) fall in the credible region. In this way we can have a similar 1-1 correspondence between hypothesis tests and credible regions just like the frequentists do with confidence intervals and hypothesis tests. But this is not the only way to do hypothesis testing.