5,605 research outputs found
Automatic Markov Chain Monte Carlo Procedures for Sampling from Multivariate Distributions
Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that only the mode (or an approximation of it) and an oracle is required, i.e., a subroutine that returns the value of the density function at any point x. We show that the number of evaluations of the density increases slowly with dimension. (author's abstract)Series: Preprint Series / Department of Applied Statistics and Data Processin
Simulation of multivariate diffusion bridge
We propose simple methods for multivariate diffusion bridge simulation, which
plays a fundamental role in simulation-based likelihood and Bayesian inference
for stochastic differential equations. By a novel application of classical
coupling methods, the new approach generalizes a previously proposed simulation
method for one-dimensional bridges to the multi-variate setting. First a method
of simulating approximate, but often very accurate, diffusion bridges is
proposed. These approximate bridges are used as proposal for easily
implementable MCMC algorithms that produce exact diffusion bridges. The new
method is much more generally applicable than previous methods. Another
advantage is that the new method works well for diffusion bridges in long
intervals because the computational complexity of the method is linear in the
length of the interval. In a simulation study the new method performs well, and
its usefulness is illustrated by an application to Bayesian estimation for the
multivariate hyperbolic diffusion model.Comment: arXiv admin note: text overlap with arXiv:1403.176
An analytic approximation of the feasible space of metabolic networks
Assuming a steady-state condition within a cell, metabolic fluxes satisfy an
under-determined linear system of stoichiometric equations. Characterizing the
space of fluxes that satisfy such equations along with given bounds (and
possibly additional relevant constraints) is considered of utmost importance
for the understanding of cellular metabolism. Extreme values for each
individual flux can be computed with Linear Programming (as Flux Balance
Analysis), and their marginal distributions can be approximately computed with
Monte-Carlo sampling. Here we present an approximate analytic method for the
latter task based on Expectation Propagation equations that does not involve
sampling and can achieve much better predictions than other existing analytic
methods. The method is iterative, and its computation time is dominated by one
matrix inversion per iteration. With respect to sampling, we show through
extensive simulation that it has some advantages including computation time,
and the ability to efficiently fix empirically estimated distributions of
fluxes
Fast MCMC sampling algorithms on polytopes
We propose and analyze two new MCMC sampling algorithms, the Vaidya walk and
the John walk, for generating samples from the uniform distribution over a
polytope. Both random walks are sampling algorithms derived from interior point
methods. The former is based on volumetric-logarithmic barrier introduced by
Vaidya whereas the latter uses John's ellipsoids. We show that the Vaidya walk
mixes in significantly fewer steps than the logarithmic-barrier based Dikin
walk studied in past work. For a polytope in defined by
linear constraints, we show that the mixing time from a warm start is bounded
as , compared to the mixing time
bound for the Dikin walk. The cost of each step of the Vaidya walk is of the
same order as the Dikin walk, and at most twice as large in terms of constant
pre-factors. For the John walk, we prove an
bound on its mixing time and conjecture
that an improved variant of it could achieve a mixing time of
. Additionally, we propose variants
of the Vaidya and John walks that mix in polynomial time from a deterministic
starting point. The speed-up of the Vaidya walk over the Dikin walk are
illustrated in numerical examples.Comment: 86 pages, 9 figures, First two authors contributed equall
Approximate Bayesian Computation by Modelling Summary Statistics in a Quasi-likelihood Framework
Approximate Bayesian Computation (ABC) is a useful class of methods for
Bayesian inference when the likelihood function is computationally intractable.
In practice, the basic ABC algorithm may be inefficient in the presence of
discrepancy between prior and posterior. Therefore, more elaborate methods,
such as ABC with the Markov chain Monte Carlo algorithm (ABC-MCMC), should be
used. However, the elaboration of a proposal density for MCMC is a sensitive
issue and very difficult in the ABC setting, where the likelihood is
intractable. We discuss an automatic proposal distribution useful for ABC-MCMC
algorithms. This proposal is inspired by the theory of quasi-likelihood (QL)
functions and is obtained by modelling the distribution of the summary
statistics as a function of the parameters. Essentially, given a real-valued
vector of summary statistics, we reparametrize the model by means of a
regression function of the statistics on parameters, obtained by sampling from
the original model in a pilot-run simulation study. The QL theory is well
established for a scalar parameter, and it is shown that when the conditional
variance of the summary statistic is assumed constant, the QL has a closed-form
normal density. This idea of constructing proposal distributions is extended to
non constant variance and to real-valued parameter vectors. The method is
illustrated by several examples and by an application to a real problem in
population genetics.Comment: Published at http://dx.doi.org/10.1214/14-BA921 in the Bayesian
Analysis (http://projecteuclid.org/euclid.ba) by the International Society of
Bayesian Analysis (http://bayesian.org/
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