130 research outputs found
Estimating Ratios of Normalizing Constants Using Linked Importance Sampling
Ratios of normalizing constants for two distributions are needed in both
Bayesian statistics, where they are used to compare models, and in statistical
physics, where they correspond to differences in free energy. Two approaches
have long been used to estimate ratios of normalizing constants. The `simple
importance sampling' (SIS) or `free energy perturbation' method uses a sample
drawn from just one of the two distributions. The `bridge sampling' or
`acceptance ratio' estimate can be viewed as the ratio of two SIS estimates
involving a bridge distribution. For both methods, difficult problems must be
handled by introducing a sequence of intermediate distributions linking the two
distributions of interest, with the final ratio of normalizing constants being
estimated by the product of estimates of ratios for adjacent distributions in
this sequence. Recently, work by Jarzynski, and independently by Neal, has
shown how one can view such a product of estimates, each based on simple
importance sampling using a single point, as an SIS estimate on an extended
state space. This `Annealed Importance Sampling' (AIS) method produces an
exactly unbiased estimate for the ratio of normalizing constants even when the
Markov transitions used do not reach equilibrium. In this paper, I show how a
corresponding `Linked Importance Sampling' (LIS) method can be constructed in
which the estimates for individual ratios are similar to bridge sampling
estimates. I show empirically that for some problems, LIS estimates are much
more accurate than AIS estimates found using the same computation time,
although for other problems the two methods have similar performance. Linked
sampling methods similar to LIS are useful for other purposes as well
Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification
Gaussian processes are a natural way of defining prior distributions over
functions of one or more input variables. In a simple nonparametric regression
problem, where such a function gives the mean of a Gaussian distribution for an
observed response, a Gaussian process model can easily be implemented using
matrix computations that are feasible for datasets of up to about a thousand
cases. Hyperparameters that define the covariance function of the Gaussian
process can be sampled using Markov chain methods. Regression models where the
noise has a t distribution and logistic or probit models for classification
applications can be implemented by sampling as well for latent values
underlying the observations. Software is now available that implements these
methods using covariance functions with hierarchical parameterizations. Models
defined in this way can discover high-level properties of the data, such as
which inputs are relevant to predicting the response
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