1,892 research outputs found
Exact sampling for intractable probability distributions via a Bernoulli factory
Many applications in the field of statistics require Markov chain Monte Carlo
methods. Determining appropriate starting values and run lengths can be both
analytically and empirically challenging. A desire to overcome these problems
has led to the development of exact, or perfect, sampling algorithms which
convert a Markov chain into an algorithm that produces i.i.d. samples from the
stationary distribution. Unfortunately, very few of these algorithms have been
developed for the distributions that arise in statistical applications, which
typically have uncountable support. Here we study an exact sampling algorithm
using a geometrically ergodic Markov chain on a general state space. Our work
provides a significant reduction to the number of input draws necessary for the
Bernoulli factory, which enables exact sampling via a rejection sampling
approach. We illustrate the algorithm on a univariate Metropolis-Hastings
sampler and a bivariate Gibbs sampler, which provide a proof of concept and
insight into hyper-parameter selection. Finally, we illustrate the algorithm on
a Bayesian version of the one-way random effects model with data from a styrene
exposure study.Comment: 28 pages, 2 figure
Coupling Control Variates for Markov Chain Monte Carlo
We show that Markov couplings can be used to improve the accuracy of Markov
chain Monte Carlo calculations in some situations where the steady-state
probability distribution is not explicitly known. The technique generalizes the
notion of control variates from classical Monte Carlo integration. We
illustrate it using two models of nonequilibrium transport
Adaptive Threshold Sampling and Estimation
Sampling is a fundamental problem in both computer science and statistics. A
number of issues arise when designing a method based on sampling. These include
statistical considerations such as constructing a good sampling design and
ensuring there are good, tractable estimators for the quantities of interest as
well as computational considerations such as designing fast algorithms for
streaming data and ensuring the sample fits within memory constraints.
Unfortunately, existing sampling methods are only able to address all of these
issues in limited scenarios.
We develop a framework that can be used to address these issues in a broad
range of scenarios. In particular, it addresses the problem of drawing and
using samples under some memory budget constraint. This problem can be
challenging since the memory budget forces samples to be drawn
non-independently and consequently, makes computation of resulting estimators
difficult.
At the core of the framework is the notion of a data adaptive thresholding
scheme where the threshold effectively allows one to treat the non-independent
sample as if it were drawn independently. We provide sufficient conditions for
a thresholding scheme to allow this and provide ways to build and compose such
schemes.
Furthermore, we provide fast algorithms to efficiently sample under these
thresholding schemes
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