So if you think that the prior odds for a coin being unfair are $~$(1 : 10^{100})$~$ against, and then you see the coin flipped 312 times and coming up heads each time\.\.\. you do not say, "Well, my new posterior odds are $~$(1 : 10^6)$~$ against the coin being unfair\." You say, "I guess I was wrong about the prior odds being that low\."

Be wary here.

We see on the next (log probability) that a plethora of small evidences sums to a very large number of bits.

In the bookcase aliens example, if you went to 312 houses and found that every one of them had a new bookcase, then by this approach, it's time to reexamine the aliens hypothesis.

In practice, it's just simply *not*. Aliens are still just as unlikely as they were previously. New bookcases are now more likely.

It's time to reexamine your 50:1 in favor of aliens estimate for a new bookcase. It's time to check whether there's a really good door-to-door bookcase salesman offering ridiculous deals in the area. Or whether there are new tax incentives for people with more bookcases. Or a zillion other far more likely things than the false dichotomy of "either each person bought bookcases independently with odds of 50:1 against, or it's bookcase aliens."

The corollary of Doyle's "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth" is "make damn sure to eliminate all the probable stuff, before gallivanting into the weeds of the infeasible".