What should be the Bayes factor while testing a simple null against a composite alternative?

Question

Suppose $X_1,X_2,\ldots,X_n$ are i.i.d $\text{Poisson}(\theta)$ where $\theta$ has a Gamma prior. The testing problem is $H_0:\theta=\theta_0$ against $H_1:\theta\ne \theta_0$. I am trying to find the Bayes factor in favour of $H_1$.

Let $\Theta_i$ be the parameter space under $H_i$ and $\Pi(\Theta_i)=\int_{\Theta_i}\pi(\theta)\,d\theta$ where $\pi$ is the prior density, $i=0,1$. Similarly, $\Pi(\Theta_i\mid \boldsymbol x)=\int_{\Theta_i}\pi(\theta\mid \boldsymbol x)\,d\theta$ where $\pi(\theta\mid \boldsymbol x)$ is the posterior density.

My definition of Bayes factor in favour of $H_1$ is $$B=\frac{\text{Posterior odds}}{\text{Prior odds}}=\frac{\Pi(\Theta_1\mid \boldsymbol x)/\Pi(\Theta_0\mid \boldsymbol x)}{\Pi(\Theta_1)/\Pi(\Theta_0)}=\frac{\Pi(\Theta_1\mid \boldsymbol x)/\Pi(\Theta_1)}{\Pi(\Theta_0\mid \boldsymbol x)/\Pi(\Theta_0)}$$

For simple null against simple alternative, say $H_0:\theta=\theta_0$ vs $H_1':\theta=\theta_1(\ne \theta_0)$, Bayes factor is the likelihood ratio:

$$B=\frac{\pi(\theta_1\mid \boldsymbol x)/\pi(\theta_1)}{\pi(\theta_0\mid \boldsymbol x)/\pi(\theta_0)}=\frac{f(\boldsymbol x\mid \theta_1)}{f(\boldsymbol x\mid \theta_0)}$$

But how to use this definition for simple versus composite alternative as I have in the problem above?

Do I have to use a mixture prior for $\theta$ like $$\pi(\theta)=p\delta_{\theta_0}+(1-p)g(\theta)\,,$$ where $\delta_{\theta_0}$ denotes a degenerate distribution at $\theta_0$ and $g(\cdot)$ is the pdf of a Gamma distribution on $\Theta-\{\theta_0\}$? If this is the case, should the Bayes factor be modified as

$$B=\frac{\Pi(\Theta_1\mid \boldsymbol x)/\Pi(\Theta_1)}{\pi(\theta_0\mid \boldsymbol x)/\pi(\theta_0)}=\frac{\Pi(\Theta_1\mid \boldsymbol x)/\Pi(\Theta_1)}{f(\boldsymbol x\mid \theta_0)}\,\,?$$

I guess the posterior would also be a mixed distribution if I go down this road.

Answer 1

As we discussed in our reassessment of his 1939 Theory of Probability, this issue is central to Jeffreys' introduction of the Bayes factor (although an earlier version appears in Haldane, 1932, and Wrinch and Jeffreys, 1921, see Etz & Wagenmakers, 2017).

When considering a point null hypothesis such as $\text{H}_0: \theta=\theta_0$, the prior $\pi$ (if at all available before considering this null hypothesis) need be modified to incorporate a point mass at $\theta_0$. Otherwise, there is no need for data to reach a decision: $\text{H}_0$ is a priori and a posteriori impossible. This was made quite clear in Jeffreys' 1939 Theory of Probability.

Using the (mixture) prior with the point mass at $\theta_0$ $$\pi^\prime(\theta)=p\delta_{\theta_0}+(1-p)\pi(\theta)$$ means that the null hypothesis is a priori possible, with a prior probability of $$\Pi^\prime(\Theta_0) = \Pi^\prime(\{\theta_0\}) = p$$ The Bayes factor then writes as $$\dfrac{\Pi^\prime(\Theta_1\mid \boldsymbol x)/\Pi^\prime(\Theta_1)}{\Pi^\prime(\Theta_0\mid \boldsymbol x)/\Pi^\prime(\Theta_0)} = \dfrac{\Pi^\prime(\Theta_1\mid \boldsymbol x)/(1-p)} {\Pi^\prime(\Theta_0\mid \boldsymbol x)/p} = \dfrac{\int_0^\infty f(x|\theta)\pi(\theta)\,\text d\theta}{f(x|\theta_0)} $$ and the posterior is indeed a mixture of a Dirac mass at $\theta_0$ and of a Gamma (continuous) distribution.