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I received a script from a friend and at the end, he's keeping the values that agree to a certain threshold for the Posterior inclusion probability (PIP).

(This is coming from a GEMMA analysis calculating the effect of SNPs to explain the variance in phenotypes)

pip01       <- para.mean[which(para.mean$gamma >= 0.01),] # snps with gamma (i.e. PIP) > 0.01
pip10       <- para.mean[which(para.mean$gamma >= 0.10),] # gamma > 0.10
pip50       <- para.mean[which(para.mean$gamma >= 0.50),] # gamma > 0.50

I'm wondering what does this mean. How can we put a posterior inclusion probability in a sentence?

Is it like, I'm 99% confident that the values that I extracted (with a pip greater than 0.01) are "significantly" having an effect?

Here is a definition of PIP:

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

Also:

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.

So is it similar to model averaging? Why talking about "individual models" in the above commentary?

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    $\begingroup$ I don't think the model is using inclusion the way I do in the book, Doing Bayesian Data Analysis. When I talk about inclusion probability, I'm referring to a discrete parameter, with values 0 or 1, that marks whether or not another parameter (such as a regression coefficient) is included in a model. There is no thresholding of a continuous parameter! $\endgroup$ Commented Jan 23, 2017 at 19:17

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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 included in the true model.

It is not model averaging. In model averaging you want to compute some summary measure and treat the models as a sort of nuisance 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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  • $\begingroup$ And I guess that it is more likely when the pip is smaller? $\endgroup$ Commented Jan 18, 2017 at 16:25
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    $\begingroup$ no...bigger. Posterior INCLUSION probability. As in probability this variable is IN the model. $\endgroup$
    – bdeonovic
    Commented Jan 18, 2017 at 16:29
  • $\begingroup$ So if we put it in a sentence with the exemple I have, it would be "I save the parameters that have a pip (or gamma in my case) that are greater than 0.01, .1 and 0.5 so 1%, 10% and 50% chance of being included in the model. $\endgroup$ Commented Jan 18, 2017 at 16:45
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    $\begingroup$ I would say something like: "Parameters with gamma values 0.01, 0.1, and 0.5 have posterior inclusion probabilities of 1%, 10%, and 50%." $\endgroup$
    – bdeonovic
    Commented Jan 18, 2017 at 17:21

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