Concept

Model misspecification

The situation in which the statistical model used to analyse data does not contain the process that actually produced them. Inference still converges, but to the member of the model closest to the truth, and its stated uncertainty can be badly overconfident.

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

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

A fair coin judged by someone who thinks it is biased. Posterior probabilities of three wrong coin biases over two thousand tosses of a fair coin, for five runs; the bias 0.6 wins every run.

Certain of a coin that is not there

Bayes' theorem is exact, and it is only as good as the list of hypotheses it is given. Hand it a list that leaves out the truth and it does not hesitate: it becomes certain of the entry that is least wrong, in a precise sense — the one closest in Kullback–Leibler divergence — and if two entries are equally wrong it never settles at all. Hand it a model that assumes independence where there is none, and its intervals shrink as fast as they would for honest data while covering the truth less and less often.

probability · Bayes

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

Named alongside it

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

Bayes' theorem Posterior Bayes factor Central limit theorem Coverage Credible interval Independence Likelihood P value Random walk Relative entropy

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