4,182 research outputs found
A population Monte Carlo scheme with transformed weights and its application to stochastic kinetic models
This paper addresses the problem of Monte Carlo approximation of posterior
probability distributions. In particular, we have considered a recently
proposed technique known as population Monte Carlo (PMC), which is based on an
iterative importance sampling approach. An important drawback of this
methodology is the degeneracy of the importance weights when the dimension of
either the observations or the variables of interest is high. To alleviate this
difficulty, we propose a novel method that performs a nonlinear transformation
on the importance weights. This operation reduces the weight variation, hence
it avoids their degeneracy and increases the efficiency of the importance
sampling scheme, specially when drawing from a proposal functions which are
poorly adapted to the true posterior.
For the sake of illustration, we have applied the proposed algorithm to the
estimation of the parameters of a Gaussian mixture model. This is a very simple
problem that enables us to clearly show and discuss the main features of the
proposed technique. As a practical application, we have also considered the
popular (and challenging) problem of estimating the rate parameters of
stochastic kinetic models (SKM). SKMs are highly multivariate systems that
model molecular interactions in biological and chemical problems. We introduce
a particularization of the proposed algorithm to SKMs and present numerical
results.Comment: 35 pages, 8 figure
Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems
Approximate Bayesian computation methods can be used to evaluate posterior
distributions without having to calculate likelihoods. In this paper we discuss
and apply an approximate Bayesian computation (ABC) method based on sequential
Monte Carlo (SMC) to estimate parameters of dynamical models. We show that ABC
SMC gives information about the inferability of parameters and model
sensitivity to changes in parameters, and tends to perform better than other
ABC approaches. The algorithm is applied to several well known biological
systems, for which parameters and their credible intervals are inferred.
Moreover, we develop ABC SMC as a tool for model selection; given a range of
different mathematical descriptions, ABC SMC is able to choose the best model
using the standard Bayesian model selection apparatus.Comment: 26 pages, 9 figure
Robust Covariance Adaptation in Adaptive Importance Sampling
Importance sampling (IS) is a Monte Carlo methodology that allows for
approximation of a target distribution using weighted samples generated from
another proposal distribution. Adaptive importance sampling (AIS) implements an
iterative version of IS which adapts the parameters of the proposal
distribution in order to improve estimation of the target. While the adaptation
of the location (mean) of the proposals has been largely studied, an important
challenge of AIS relates to the difficulty of adapting the scale parameter
(covariance matrix). In the case of weight degeneracy, adapting the covariance
matrix using the empirical covariance results in a singular matrix, which leads
to poor performance in subsequent iterations of the algorithm. In this paper,
we propose a novel scheme which exploits recent advances in the IS literature
to prevent the so-called weight degeneracy. The method efficiently adapts the
covariance matrix of a population of proposal distributions and achieves a
significant performance improvement in high-dimensional scenarios. We validate
the new method through computer simulations
Continuum variational and diffusion quantum Monte Carlo calculations
This topical review describes the methodology of continuum variational and
diffusion quantum Monte Carlo calculations. These stochastic methods are based
on many-body wave functions and are capable of achieving very high accuracy.
The algorithms are intrinsically parallel and well-suited to petascale
computers, and the computational cost scales as a polynomial of the number of
particles. A guide to the systems and topics which have been investigated using
these methods is given. The bulk of the article is devoted to an overview of
the basic quantum Monte Carlo methods, the forms and optimisation of wave
functions, performing calculations within periodic boundary conditions, using
pseudopotentials, excited-state calculations, sources of calculational
inaccuracy, and calculating energy differences and forces
Boosting Bayesian Parameter Inference of Nonlinear Stochastic Differential Equation Models by Hamiltonian Scale Separation
Parameter inference is a fundamental problem in data-driven modeling. Given
observed data that is believed to be a realization of some parameterized model,
the aim is to find parameter values that are able to explain the observed data.
In many situations, the dominant sources of uncertainty must be included into
the model, for making reliable predictions. This naturally leads to stochastic
models. Stochastic models render parameter inference much harder, as the aim
then is to find a distribution of likely parameter values. In Bayesian
statistics, which is a consistent framework for data-driven learning, this
so-called posterior distribution can be used to make probabilistic predictions.
We propose a novel, exact and very efficient approach for generating posterior
parameter distributions, for stochastic differential equation models calibrated
to measured time-series. The algorithm is inspired by re-interpreting the
posterior distribution as a statistical mechanics partition function of an
object akin to a polymer, where the measurements are mapped on heavier beads
compared to those of the simulated data. To arrive at distribution samples, we
employ a Hamiltonian Monte Carlo approach combined with a multiple time-scale
integration. A separation of time scales naturally arises if either the number
of measurement points or the number of simulation points becomes large.
Furthermore, at least for 1D problems, we can decouple the harmonic modes
between measurement points and solve the fastest part of their dynamics
analytically. Our approach is applicable to a wide range of inference problems
and is highly parallelizable.Comment: 15 pages, 8 figure
An Exact Auxiliary Variable Gibbs Sampler for a Class of Diffusions
Stochastic differential equations (SDEs) or diffusions are continuous-valued
continuous-time stochastic processes widely used in the applied and
mathematical sciences. Simulating paths from these processes is usually an
intractable problem, and typically involves time-discretization approximations.
We propose an exact Markov chain Monte Carlo sampling algorithm that involves
no such time-discretization error. Our sampler is applicable to the problem of
prior simulation from an SDE, posterior simulation conditioned on noisy
observations, as well as parameter inference given noisy observations. Our work
recasts an existing rejection sampling algorithm for a class of diffusions as a
latent variable model, and then derives an auxiliary variable Gibbs sampling
algorithm that targets the associated joint distribution. At a high level, the
resulting algorithm involves two steps: simulating a random grid of times from
an inhomogeneous Poisson process, and updating the SDE trajectory conditioned
on this grid. Our work allows the vast literature of Monte Carlo sampling
algorithms from the Gaussian process literature to be brought to bear to
applications involving diffusions. We study our method on synthetic and real
datasets, where we demonstrate superior performance over competing methods.Comment: 37 pages, 13 figure
Stochastic weighted particle methods for population balance equations
A class of stochastic algorithms for the numerical treatment of population balance equations is introduced. The algorithms are based on systems of weighted particles, in which coagulation events are modelled by a weight transfer that keeps the number of computational particles constant. The weighting mechanisms are designed in such a way that physical processes changing individual particles (such as growth, or other surface reactions) can be conveniently treated by the algorithms. Numerical experiments are performed for complex laminar premixed flame systems. Two members of the class of stochastic weighted particle methods are compared to each other and to a direct simulation algorithm. One weighted algorithm is shown to be consistently better than the other with respect to the statistical noise generated. Finally, run times to achieve fixed error tolerances for a real flame system are measured and the better weighted algorithm is found to be up to three times faster than the direct simulation algorithm
A comparison of Monte Carlo-based Bayesian parameter estimation methods for stochastic models of genetic networks
We compare three state-of-the-art Bayesian inference methods for the estimation of the unknown parameters in a stochastic model of a genetic network. In particular, we introduce a stochastic version of the paradigmatic synthetic multicellular clock model proposed by Ullner et al., 2007. By introducing dynamical noise in the model and assuming that the partial observations of the system are contaminated by additive noise, we enable a principled mechanism to represent experimental uncertainties in the synthesis of the multicellular system and pave the way for the design of probabilistic methods for the estimation of any unknowns in the model. Within this setup, we tackle the Bayesian estimation of a subset of the model parameters. Specifically, we compare three Monte Carlo based numerical methods for the approximation of the posterior probability density function of the unknown parameters given a set of partial and noisy observations of the system. The schemes we assess are the particle Metropolis-Hastings (PMH) algorithm, the nonlinear population Monte Carlo (NPMC) method and the approximate Bayesian computation sequential Monte Carlo (ABC-SMC) scheme. We present an extensive numerical simulation study, which shows that while the three techniques can effectively solve the problem there are significant differences both in estimation accuracy and computational efficiency
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