We finished off the discussion of model selection/averaging.

We then went into missing data models. In some cases the fact that data are not observed can have no effect on the inference whatsoever. In other cases, the effects can be profound. Sometimes (as in Example 2) the effect only appears through a prior that connects the probability of selection to the parameter being estimated. Censorship (where you know that an observation has been made, but the answer is only that the observation is not within the range of the measurement process) and truncation (where you are only given the successful measurements and do not know how many of the measurements were unsuccessful, as in a survey of voters where the voters who hung up the phone are not reported) are examples of situations where missing data affects the likelihood function and hence the inference.

We discussed a general approach to problems of this sort, where the likelihood is explicitly written to include both the observed and the missing data. The missing data are considered as latent variables, which thus require a prior and should be marginalized out. We verified that (in Example 4) this approach gives the same results as we got earlier.

In a sampling scheme, we sample on all parameters, including the latent variables that represent the missing data. We started on discussing a sampling scheme for Example 4, and will finish it next time.