| > Bayes' formula has no place to put for things that could have been observed had things turned out differently, but which didn't actually happen. Sure it does: you have to calculate the probability of your data given the hypothesis. Doing that requires considering all possible outcomes of the hypothesis and their relative likelihood, not just the one you actually observed. > If your prior was that a fraction p of the children would be boys, the odds of the observed outcome would be p^6 (1-p).* The prior would not actually be a single value for p; it would be a distribution for p over the range (0, 1). The distribution I described was a narrowly peaked Gaussian around p = 0.5, though, as you point out, that might not be the correct value for the peak (see below). However, for illustration purposes, it is much easier to talk about the (idealized, unrealistic) case where your prior is in fact a single point value for p. However, in order to calculate the odds of the observed outcome, as I said above, you don't just need to know the prior for p. You need to know the process by which the outcomes are generated, according to the hypothesis. The odds you give assume that that process is "bear seven children, regardless of their gender". But that is not the correct process for the actual decision procedure you describe your aunt and uncle as using. That process won't necessarily result in seven children, and the odds of the actually observed outcome will change accordingly. > a Bayesian with access to actual population data would be aware, as you aren't, that globally we average 1.07 boys to each girl at birth Depends on whose data you look at and over what time period. But I agree that the best prior to use in a given case would be whatever distribution you get from the data you already have, and yes, that might not be peaked exactly at 50-50. |