Estimating Ratios of Normalizing Constants Using Linked Importance Sampling

By Radford M. Neal. 2005

65 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Statistics and Probability, Probabilistic & Bayesian Methods

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Citations in the corpus

On the quantitative analysis of deep belief networks (2008, methodology): “We also have to rely on the empirical estimate of AIS accuracy, which could potentially be very misleading (Neal, 2001; Neal, 2005).”
Distilling Intractable Generative Models (2015, methodology): “Monte Carlo methods [3, 4, 5] typically avoid the curse of dimensionality, but are often inaccurate or slow.”
On Tracking The Partition Function (2011, methodology): “Σ ∆ β is once again a diagonal covariance matrix, updated online from the variance of the log-importance weights u and v [19].”
A survey on approaches for reliability-based optimization (2010, methodology): “…Simulation (Au and Beck, 2001), Domain Decomposition Method (Katafygiotis and Cheung, 2006), Auxiliary Domain Method (Katafygio- tis et al, 2007), Linked Importance Sampling (Katafygiotis and Zuev, 2007; Neal, 2005), Horseracing Simulation method (Katafygiotis and Zuev, 2009; Zuev, 2009), etc.”
Training Restricted Boltzmann Machines (2015, methodology): “We focused on Parallel Tempering (PT) and path sampling for generating samples, but Linked Importance Sampling (Neal, 2005) also fits into this scheme.”
Fitting the Phenomenological MSSM (2009, methodology): “osed [43] as an exact, and potentially faster, means of evaluating evidences, but is restricted to the special case of nested hypotheses and a separable prior on the model parameters. Bridge sampling [44, 45, 46] allows the evaluation of the ratio of Bayesian evidence of two models and was implement”
MCMC for Doubly-intractable Distributions (2006, background): “Linked importance sampling (Neal, 2005) could also be used as a drop-in replacement.”
Learning and Evaluating Boltzmann Machines (2008, methodology): “It is shown in [15, 16 ] that for sufficiently large number of in termediate distributions K, the variance of ˆ rAIS will be proportional to 1/MK.”
Sequential importance sampling for structural reliability analysis (2016, methodology): “Therein, estimation of the intermediate distributions was performed following an algorithm proposed in [16].”