# Bayesian interpretation of FDR

Suppose we want to apply a rejection rule on N tests. We may write the following table:

       Decision
Actual
| Accept H0 | Reject H0 | Total
-------------------------------
H0     | N0 - a    | a         | N0
Non H0 | N1 - b    | b         | N1
Total  | N - R     | R         | N


The false discovery proportion (Fdp) is defined as $a/R$.

In a Bayesian setting, we can define a threshold $z_0$ such that we reject $H_{0,i}$ if $|z_i|>z_0$. Then the Fdp will be:

$$Fdp = P(H_{0,i} | |z_i|>z_0) = \frac{P(|z_i|>z_0 | H_{0,i}) \pi_0}{P(|z_i|>z_0)},$$

where $\pi_0$ is the prior of $H_{0,i}$. My problem is that I do not understand why we want to compute $P(H_{0,i} | |z_i|>z_0)$ in the first instance. Shouldn't we compute:

$$Fdp = P(\text{reject} \quad H_{0,i} | |z_i|<z_0),$$

or by taking the complement of both we get the first formula?

• I guess your last formula implies the rejection of "true" null hypothesis, doesn't it? Otherwise I do not understand how it is connected with FDR. Commented Dec 18, 2017 at 12:16
• @GermanDemidov The last formula is how I would translate Fdp using Bayes' rule, but I do not understand if it is the same as the one I wrote above. Commented Dec 18, 2017 at 12:18
• if I am correct, your formula is the probability of rejection of hypothesis without any assumption on is this hypothesis true or not. FDP, as written above, is the proportion of false discoveries among all discoveries. There should be some statement about if $H_{0,i}$ is true or not. Commented Dec 18, 2017 at 12:21
• $z_i$ is an observed test statistic (on which you look at whether such an extreme value is likely to have occured under the null hypothesis), not some parameter of the model. So, either your last formula makes no sense. Or to put it in another way in the frequentist setting P(reject $H_0$ | $|Z_i|<z_0$)=0, if your test is being done so that rejection occurs when $|z_i|>z_0$. Commented Dec 18, 2017 at 13:03

The setup is a bit confusing because your first formula uses notation $H_{0,i}$ to mark the event "true hypothesis for test $i$ is null". It says nothing about rejection or test results. These are contained in the event $|z_i| > z_0$, which is the same as saying "null was rejected for test $i$". So top line says:
$$Fdp = P(H_0\ is\ true | H_0\ was\ rejected) = ...$$
Your bottom line doesn't add up to the notation in first line, but it is equivalent to $P(reject\ null | test\ results\ don't\ reject\ null)$, which is 0.