SummaryFinding 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.

Computing the value of a high-dimensional integral can often be reduced to the problem of finding the ratio between the measures of two sets. Monte Carlo methods are often used to approximate this ratio, but often one set will be exponentially larger than the other, which leads to an exponentially large variance. A standard method of dealing with this problem is to interpolate between the sets with a sequence of nested sets where neighboring sets have relative measures bounded above by a constant. Choosing such a well-balanced sequence can rarely be done without extensive study of a problem. Here a new approach that automatically obtains such sets is presented. These well-balanced sets allow for faster approximation algorithms for integrals and sums using fewer samples, and better tempering and annealing Markov chains for generating random samples. Applications, such as finding the partition function of the Ising model and normalizing constants for posterior distributions in Bayesian methods, are discussed.

Computing the value of a high-dimensional integral can often be reduced to the problem of finding the ratio between the measures of two sets. Monte Carlo methods are often used to approximate this ratio, but often one set will be exponentially larger than the other, which leads to an exponentially large variance. A standard method of dealing with this problem is to interpolate between the sets with a sequence of nested sets where neighboring sets have relative measures bounded above by a constant. Choosing such a well-balanced sequence can rarely be done without extensive study of a problem. Here a new approach that automatically obtains such sets is presented. These well-balanced sets allow for faster approximation algorithms for integrals and sums using fewer samples, and better tempering and annealing Markov chains for generating random samples. Applications, such as finding the partition function of the Ising model and normalizing constants for posterior distributions in Bayesian methods, are discussed.

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