How to get posterior probability from Bayes factor

Question

I ran into this question in my class and am not sure how to solve it:

A positive test result gives you a Bayes factor of 71 in favor of being sick. If your prior probability of Being sick was 0.05, what is your posterior probability of being sick.

I understand that a Bayes factor is basically the likelihood ratio and I know that $p(M_2)=.01$. I guess I am looking for $p(M2|y=\text{testing positive for sick})$, but I am missing all the other variables.

Can someone please give me a hint how to proceed?

Answer 1

Given data $D$ and two models $M_1$ and $M_2$, the Bayes factor is defined as:

$$BF_{1,2} \equiv \frac{p(D|M_1)}{p(D|M_2)}.$$

So you can write the posterior probability of $M_1$ as:

$$\begin{equation} \begin{aligned} \mathbb{P}(M_1|D) &= \frac{p(D|M_1) \mathbb{P}(M_1)}{p(D|M_1) \mathbb{P}(M_1) + p(D|M_2) \mathbb{P}(M_2)} \\[6pt] &= \frac{BF_{1,2} \cdot \mathbb{P}(M_1)}{BF_{1,2} \cdot \mathbb{P}(M_1) + \mathbb{P}(M_2)} \\[6pt] &= \frac{BF_{1,2} \cdot \mathbb{P}(M_1)}{BF_{1,2} \cdot \mathbb{P}(M_1) + (1-\mathbb{P}(M_1))}. \\[6pt] \end{aligned} \end{equation}$$

Also note: Bayesian statistics is named after the statistician Thomas Bayes. Since this is a proper noun, it is always expressed as "Bayes", not "bayes" or "Bays".