Concept

Marginal likelihood — where it appears

The probability of the data with the parameters integrated out against the prior, which is what a Bayes factor is a ratio of. It depends on the prior in a way the posterior largely does not, so a prior chosen to be uninformative can decide the comparison.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

What eight groups say about τ, when the truth is 1. The posterior density for the population spread after eight groups whose standard errors run from 0.5 to 2.1. The shaded band is the central 95% interval, from 1.02 to 4.44; the posterior median is 1.99 and the mean 2.18. The vertical mark at 1.07 is the moment estimate that empirical Bayes substitutes and then treats as known.

What the plug-in forgets

The shrinkage weight needs a population spread, and the population spread has to be estimated from eight numbers. Empirical Bayes estimates it, substitutes it, and proceeds as though it were known — and the interval that comes out covers 79% rather than the 95% it claims.

fullbayes · Shrinkage

Three priors on the spread, at a true τ of 0.5. The posterior for τ under a flat prior (mean 1.66), a half-Cauchy of scale 1 (1.32) and one of scale 0.25 (1.15). The three answers differ by 30% of the widest. The prior does visible work when eight groups cannot separate a small spread from none, and almost none when they can.

A prior on the spread

Integrating over the population spread means putting a prior on it, which sounds like the objection rather than the repair. The prior's effect is measurable, it is invisible where the groups are clearly different, and the reflex choice for a scale parameter turns out not to have a posterior at all.

fullbayes · Prior

16 groups shrunk towards a fitted line, at γ = 1.2. Hollow circles are the groups' own values, filled ones the estimates after pooling, and the diagonal is the line fitted through them with each group weighted by how well it is measured — slope 1.21, intercept 0.20. The horizontal rule is where the same 16 groups would have been shrunk to with no covariate. The spread left to borrow against is 0.64 with the covariate against 1.28 without, so every group is pulled further in than it would otherwise have been.

Borrowing towards a line

A group shrunk towards the average of all groups is being compared with groups it has nothing in common with. Fit a group-level predictor and it is shrunk towards what the predictor says a group like it should be — which halves the spread left to borrow against and takes a quarter off the squared error.

multilevel · Levels

Named alongside it

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

Hierarchical model Empirical Bayes Flat prior Partial pooling Posterior Posterior mean Prior Shrinkage Confounding Coverage The half-Cauchy prior Improper posterior

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