Comparing two methods for bootstrapping posterior distributions ∗

Two related methods exist for sampling from posterior distributions of the MLE with a known prior Newton & Raftery (1994) [8], Efron (2011) [4]. We compare them by examining asymptotic Edgeworth expansions of their pivotal distributions. The result is that Newton & Raftery (1994) is 2nd order consistent to the posterior distribution with prior proportional to the Fisher Information, under some conditions on the underlying distribution. When the underlying distribution is an exponential family, this reduces to sampling with a flat prior. On the other hand, Efron (2011) is exponentially convergent in number of bootstrap samples to the posterior. 1 The two bootstrap methods: WLB and paraBayes The WLB is a method for sampling from a class of posterior distributions corresponding to a user specified likelihood, and described in Newton & Raftery (1994) [8]. Specifically, starting with data X = (x1, . . . , xn), and a likelihood f(x, θ), the WLB solves