Concept

Bayes factor

The ratio of how probable the observed data are under one hypothesis to how probable they are under another, each averaged over its own unknowns. Multiplying prior odds by it gives posterior odds, and under the hypothesis in the denominator it averages exactly one.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as p value — the same set of essays touches all of them, so they are one junction rather than several.

A fixed p-value favours the null more as the sample grows. Even odds at p=0.05 near n=76, at p=0.01 near n=1202, at p=0.001 near n=83176; p=0.05 at n=10⁶ gives 116.9 to one for the fair coin; PEAR data z=3.614, p=0.00030, BF₀₁=11.90.

A result at five per cent that favours the null

Toss a coin a million times and find a count of heads exactly at the edge of significance, p = 0.05. A test at the usual level rejects fairness. A Bayes factor comparing a fair coin with a coin of unknown bias says the opposite, by 117 to one. Both computations are right, and the sample size that separates them is about seventy-six tosses.

probability · Bayes

Peeking until the answer is yes

Toss a fair coin, test for bias after every toss, and stop the moment the result is significant at 5%. Within a thousand tosses nearly half of fair coins have been declared biased, and with no limit on the tosses every one of them is. Watch a Bayes factor the same way, stopping at twenty to one, and no amount of watching gets more than one fair coin in twenty past the line.

probability · Bayes

Named alongside it

The objects these essays reach for when they reach for this one.

Central limit theorem Likelihood P value Bayes' theorem Martingale Model misspecification Posterior Random walk Stopping time

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