66 research outputs found
Particle Gibbs Split-Merge Sampling for Bayesian Inference in Mixture Models
This paper presents a new Markov chain Monte Carlo method to sample from the
posterior distribution of conjugate mixture models. This algorithm relies on a
flexible split-merge procedure built using the particle Gibbs sampler. Contrary
to available split-merge procedures, the resulting so-called Particle Gibbs
Split-Merge sampler does not require the computation of a complex acceptance
ratio, is simple to implement using existing sequential Monte Carlo libraries
and can be parallelized. We investigate its performance experimentally on
synthetic problems as well as on geolocation and cancer genomics data. In all
these examples, the particle Gibbs split-merge sampler outperforms
state-of-the-art split-merge methods by up to an order of magnitude for a fixed
computational complexity
Exponential Ergodicity of the Bouncy Particle Sampler
Non-reversible Markov chain Monte Carlo schemes based on piecewise
deterministic Markov processes have been recently introduced in applied
probability, automatic control, physics and statistics. Although these
algorithms demonstrate experimentally good performance and are accordingly
increasingly used in a wide range of applications, geometric ergodicity results
for such schemes have only been established so far under very restrictive
assumptions. We give here verifiable conditions on the target distribution
under which the Bouncy Particle Sampler algorithm introduced in \cite{P_dW_12}
is geometrically ergodic. This holds whenever the target satisfies a curvature
condition and has tails decaying at least as fast as an exponential and at most
as fast as a Gaussian distribution. This allows us to provide a central limit
theorem for the associated ergodic averages. When the target has tails thinner
than a Gaussian distribution, we propose an original modification of this
scheme that is geometrically ergodic. For thick-tailed target distributions,
such as -distributions, we extend the idea pioneered in \cite{J_G_12} in a
random walk Metropolis context. We apply a change of variable to obtain a
transformed target satisfying the tail conditions for geometric ergodicity. By
sampling the transformed target using the Bouncy Particle Sampler and mapping
back the Markov process to the original parameterization, we obtain a
geometrically ergodic algorithm.Comment: 30 page
Randomized Hamiltonian Monte Carlo as Scaling Limit of the Bouncy Particle Sampler and Dimension-Free Convergence Rates
The Bouncy Particle Sampler is a Markov chain Monte Carlo method based on a
nonreversible piecewise deterministic Markov process. In this scheme, a
particle explores the state space of interest by evolving according to a linear
dynamics which is altered by bouncing on the hyperplane tangent to the gradient
of the negative log-target density at the arrival times of an inhomogeneous
Poisson Process (PP) and by randomly perturbing its velocity at the arrival
times of an homogeneous PP. Under regularity conditions, we show here that the
process corresponding to the first component of the particle and its
corresponding velocity converges weakly towards a Randomized Hamiltonian Monte
Carlo (RHMC) process as the dimension of the ambient space goes to infinity.
RHMC is another piecewise deterministic non-reversible Markov process where a
Hamiltonian dynamics is altered at the arrival times of a homogeneous PP by
randomly perturbing the momentum component. We then establish dimension-free
convergence rates for RHMC for strongly log-concave targets with bounded
Hessians using coupling ideas and hypocoercivity techniques.Comment: 47 pages, 2 figure
Non-Reversible Parallel Tempering: a Scalable Highly Parallel MCMC Scheme
Parallel tempering (PT) methods are a popular class of Markov chain Monte
Carlo schemes used to sample complex high-dimensional probability
distributions. They rely on a collection of interacting auxiliary chains
targeting tempered versions of the target distribution to improve the
exploration of the state-space. We provide here a new perspective on these
highly parallel algorithms and their tuning by identifying and formalizing a
sharp divide in the behaviour and performance of reversible versus
non-reversible PT schemes. We show theoretically and empirically that a class
of non-reversible PT methods dominates its reversible counterparts and identify
distinct scaling limits for the non-reversible and reversible schemes, the
former being a piecewise-deterministic Markov process and the latter a
diffusion. These results are exploited to identify the optimal annealing
schedule for non-reversible PT and to develop an iterative scheme approximating
this schedule. We provide a wide range of numerical examples supporting our
theoretical and methodological contributions. The proposed methodology is
applicable to sample from a distribution with a density with respect
to a reference distribution and compute the normalizing constant. A
typical use case is when is a prior distribution, a likelihood
function and the corresponding posterior.Comment: 74 pages, 30 figures. The method is implemented in an open source
probabilistic programming available at
https://github.com/UBC-Stat-ML/blangSD
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