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Sunday, October 7, 2012

2-headed coin - Tom's special coin

http://blog.moertel.com/articles/2010/12/07/on-the-evidence-of-a-single-coin-toss is a problem very similar to the regular 2-headed coin problem. If after careful analysis we decide to use initial estimate of 50/50, then the first Head would sway our estimate to 66.66%.

Many follow-up comments point out our trust of Tom is a deciding factor, which I agree. After seeing 100 heads in a row, we are likely to believe Tom more. Now, that is a very tricky statement.

We need to carefully separate 2 kinds of adjustments on our beliefs.
C) corrections on the initial estimate
U) updates based on new evidence. These won't threaten to discredit the initial estimate.

I would say getting 100 heads is a U not a C.

An example of C would be other people's (we trust) endorsements that Tom is trustworthy. In this case as compared to the "pool distribution" case, the initial estimate is more subject to correction. In the pool scenario, initial prior is largely based on estimate of pool distribution. If and when a correction occurs, we must recompute all updated versions.

The way we update our estimate relies on the initial estimate of 50/50. Seeing 100 heads and updating 100 versions of our estimate is valid precisely because the validity of the initial estimate. The latest estimate of Prob(Fair) incorporates the initial estimate of 50/50 + all subsequent updates.

If you really trust Tom more, then what if it's revealed the 100 heads are an illusion show by a neighbor magician (Remember we are in a pub). Nothing to with Tom's coin. The entire "new information" is recalled. Would you still trust Tom more? If not, then there's no reason to "Correct" the initial estimate. There's no Corrective evidence on the scene.