Critical Particle Limits at Bayesian Posterior Resolution

Importance sampling with the prior as the proposal represents a posterior by reweighting M samples from the prior. As data accumulate, the posterior concentrates and only a few samples keep appreciable weight. The classical quantity N2 = EP[L^2] / EP[L]^2, for likelihood L and prior P, measures how many samples are needed. We study the case in which M grows in proportion to N2. The ratio M/N2, which we call the coverage, then stays fixed, and the approximation error neither vanishes nor blows up. Under a local condition and a tail condition on the likelihood, the reweighted samples, viewed in coordinates rescaled to the width of the posterior, converge to a random distribution: a Poisson cloud of points, each weighted by the limiting likelihood at its location. In regular models N2 grows like n^(r/2) for n observations and r parameters, and the limit is the same for all of them. For the singular model yi = theta^k + epsiloni it grows like n^(1/(2k)), and the limit depends on k. Exact calculations confirm both growth rates, and simulations of the limit agree with finite samples. Equal coverage does not give equal concentration of the weights: at coverage one, a single sample holds over 90% of the weight about 12 percentage points more often for k = 2 than for k = 1.