I am given that we have data $s$ and that there are $\textbf{Two}$ statisticians analyzing these data using the same sampling model but with different priors, and they are asked to assess a hypothesis $H_0$.
Both statisticians report a Bayes factor in favor of $H_0$ equal to $100$
$\textbf{Statistician I}$ assigned a prior probability $\frac{1}{2}$ to $H_0$
$\textbf{statistician II}$ assigned prior probability $\frac{1}{4}$ to $H_0$
How do we determine which statistician has the greatest posterior degree of belief in $H_0$ being true?
having the same Bayes Factor, the posterior degree of belief in $H_0$ is greater for Stat I as Stat II gives a bigger prior probability to $H_1$
The decision in favour of $H_0$ is governed by the following rule:
$$\frac{P(\theta=\theta_0)}{P(\theta=\theta_1)}\times \underbrace{\frac{p(\mathbf{x}|\theta_0)}{p(\mathbf{x}|\theta_1)}}_{\text{Bayes Factor}}>1$$
the likelihood ratio is called "Bayes Factor"
Thus for Stat 1 the decision rule is 1:1 while for Stat II the decision rule is 1:3. This means that the degree of belief in $H_0$ is greater for Stat I who beliefs the two hypotheses are equal in probability. Stat II beliefs that $H_1$ is more probabily. Decision could change after seeing the data, but as per the fact that BF is the same for both...