# What's the meaning of a posterior inclusion probability (PIP) in Bayesian?

### Definition

First we calculate the posterior inclusion probability, which is the sum of all posterior probabilities of all the regressions including the specific variable (regressor). The posterior inclusion probability is a ranking measure to see how much the data favors the inclusion of a variable in the regression.

### In Kruschke's book:

It is the proportion of steps in the overall MCMC chain that include the predictor

While the overall inclusion probabilities provide a di erent perspective on the predictors than individual models, be careful not to think that the marginal inclusion probabilities can be multiplied to derive the model probabilities.

### Extended explaination

Think of the inclusion/exclusion of a variable in your model as a random variable. The posterior distribution of this variable is obtained (probably) by some MCMC sampling scheme. The PIP is the mean of the posterior. You can think of it as a measure of how likely it is that this variable is in the true model.

Its not model averaging. In model averaging you want to compute some summary measure and treat the models as a sort of nuissance parameter that you want to integrate out.

So BMA gives you summary measures of interest that take all models into account. PIP is a value for each variable that indicates how likely it is to be included in the true model.

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