# P-values and Posterior Probabilities

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## Given a p-value of 0.045 what is the probability that the null model is true? The answer will surprise you!

You might want to argue that the question is based on a false premise; the p-value is related to the probability of falsely rejecting the null hypothesis conditional on the level ($\alpha$) and the null hypothesis being true. The question of whether the null-model valid or not is only assesed from this viewpoint and we certainly cannot assign a probability it.

However in Bayesian statistics such a question is perfectly valid. The theory of Bayesian robustness examines how sensitive a Bayesian inference is to the choice of prior. Thereby adressing one of the main criticisms of Bayesian statistics. In a seminal paper James O. Berger and Thomas Sellke consider a single realization (the results still holds for multiple realizations) of a normal variable with known variance: $X \sim \mathcal{N}(\theta, 1)$ and a test for the hypotesis $M_0: \theta = 0$ against the alternative $M_1: \theta \neq 0.$ Let’s also phrase a Bayesian model; we give each of the two models an equal prior probability $P(M_0)=\frac{1}{2}.$ We also need a prior for $\theta$ under $M_1$, $g(\theta)$. Berger and Sellke shows that if $|X| > 1$ corresponding to a p-value less than $0.32$, the prior favoring $M_1$ the most within the class of symmetric unimodal distributions is a uniform distribution,

We find the $K$ that maximizes the posterior probability of the alternative for different values of $X$ and their corresponding posterior probabilities and p-values.

This answers our question: With the prior that favors the alternative the most the posterior probability that the null hypothesis is true for a p-value of 0.045 is still $0.28$. Not exactly overwhelming evidence for the alternative.

Below we have plotted the posterior probability for $H_0$ against the p-value for the prior that favors the alternative the most.