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I have read http://www.sumsar.net/blog/2018/12/visualizing-the-beta-binomial/ this simple explanation of how the posterior is changing while more data are added: in this visualization there are six data (F,T,F,F,F,T), my question is: how can be treated the case where some data is missing or, better, unknown, let’s say (F,NA,F,F,F,T)?

enter image description here

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  • $\begingroup$ Why is it missing? $\endgroup$ – Tim Feb 20 at 20:46
  • $\begingroup$ @Tim it is missing because it was not observed, it did happen but no one knows the value $\endgroup$ – Alessandro Jacopson Feb 20 at 20:48
  • $\begingroup$ Did this happen randomly, or is there any pattern? Why was it not recorded? $\endgroup$ – Tim Feb 20 at 20:51
  • $\begingroup$ @Tim there is a pattern, let’s say I observe $n_1$ consecutive values and skip the following $n_2$ values and so on. $n_2$ can be much greater than $n_1$ $\endgroup$ – Alessandro Jacopson Feb 20 at 20:58
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This is very simple model, so the answer is pretty straightforward.

If the values are missing at random, then given the fact that your data is assumed to be independent and identically distributed, you can simply ignore the missing values. If you thrown a coin ten times, but one result was not recorded because your pencil got broken, then it is the same as if you've thrown the coin nine times (you just have less data). This also applies to your comment that $n$ consecutive values are missing, because since you assume the data to be i.i.d., you could shuffle it in any order and it wouldn't change anything.

On another hand, if the values are not missing at random, not much can be done. For example, if your study was about estimating probability of committing crime among Afro-Americans and the data was gathered by a racist researcher who didn't record any result that he didn't like, then the data is rubbish. Of course, if you had way of estimating and correcting for this bias, this would be a different story, since it goes far beyond beta-binomial model.

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