Using TPA for Bayesian inference

Finding the integrated likelihood of a model given the data requires the integration of a nonnegative function over the parameter space. Classical Monte Carlo methods for numerical integration require a bound or estimate of the variance in order to determine the quality of the output. The method called the product estimator does not require knowledge of the variance in order to produce a result of guaranteed quality, but requires a cooling schedule that must have certain strict properties. Finding a cooling schedule can be difficult, and finding an optimal cooling schedule is usually computationally out of reach. TPA is a method that solves this difficulty, creating an optimal cooling schedule automatically as it is run. This method has its own set of requirements; here it is shown how to meet these requirements for problems arising in Bayesian inference. This gives guaranteed accuracy for integrated likelihoods and posterior means of nonnegative parameters.