Posterior predictive simulation I

You can create a posterior predictive distribution of the mean number of chocolate chips per cookie for each location as well as their variability estimates (i.e. standard deviation) to simulate the posterior predictive distribution of these parameters for a new location. For instance, you might want to know what the lambda parameter would look like should you open a new location. Further, you might want to know what the predicted probability is that a new location produced an average number of chocolate chips per cookie (lambda parameter) greater than 15. You would discover that the probability of this happening is pretty low, because there is less than 2% chance that it occurs.

mean(lam_pred > 15)

[1] 0.01766667

Posterior predictive simulation II

Using these lambda draws obtained earlier, you can go to the observation level and simulate the actual number of chips per cookie taking into account the uncertainty in that parameter. The histogram represents the posterior predictive distribution of the number of chocolate chips per cookie from a new location, which averages over the uncertainty of that new location lambda parameter. You could then answer questions such as: what is the posterior probability that the number of chocolate chips produced from a cookie at a new location would be greater that 15? At this level, the chance of observing such event would be about 6%.

mean(y_pred > 15)

[1] 0.0596

Posterior predictive simulation III

In order to make predictions for a specific location, you draw from the combined simulations for the location you’re interested in to get an idea of what it looks like. Then, you can use the estimates to answer questions such as: what is the (posterior) probability that the next cookie produced at location 1 will have fewer than 7 chips? You quickly discover there would be about 19% chance that that happens.

mean(y_pred1 < 7)

[1] 0.1910667