Suppose I want to find what the bias of a coin is. To do this, I flip the coin N times and record the result. How can I update my best guess of what the bias of the coin is given that my initial belief is that the coin is fair (p=0.5)? Without prior belief, I would use the MLE to find this answer. How would I incorporate this prior?
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Prior beliefs are usually not of the form "the coin is fair", but rather "the heads probability, $p$, is distributed as
". – Ami Tavory Aug 15 '17 at 20:35 -
1If your initial belief is that the coin is fair, no amount of experimenting can change it. The prior is a point mass at 1/2 and hence is the posterior. – Xi'an Aug 16 '17 at 14:53
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If you initially have some belief that the coin is perfectly fair but also possibly unfair you could represent this with a mixed spike-and-slab prior with a point mass in $p=1/2$ and some continuous distribution (for example a uniform or beta distribution) representing your belief about $p$ given that the coin is unfair.
You then update this using Bayes theorem. Several outcomes are then possible. For example, if the data indicate that $p$ is close $1/2$, the posterior point mass in 1/2 would initially increase in size but if $p$ in reality is slightly different from 1/2 the point mass in 1/2 would eventually go to zero.
If the coin in reality is perfectly fair, this will be reflected in the point mass in $p=1/2$ increasing in size towards one as more and more data accumulates. So in this restricted sense, this approach would allow you to "prove" a point null hypothesis. But most people may not agree with this interpretation, see related thread.
Jarle Tufto
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