MCMC inference for Markov Jump Processes via the Linear Noise Approximation

Bayesian analysis for Markov jump processes is a non-trivial and challenging problem. Although exact inference is theoretically possible, it is computationally demanding thus its applicability is limited to a small class of problems. In this paper we describe the application of Riemann manifold MCMC methods using an approximation to the likelihood of the Markov jump process which is valid when the system modelled is near its thermodynamic limit. The proposed approach is both statistically and computationally efficient while the convergence rate and mixing of the chains allows for fast MCMC inference. The methodology is evaluated using numerical simulations on two problems from chemical kinetics and one from systems biology

Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterising stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealisations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with MATLAB implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community

Efficient Parameter Inference for Stochastic Chemical Kinetics

Parameter inference for stochastic systems is considered as one of the fundamental classical problems in the domain of computational systems biology. The problem becomes challenging and often analytically intractable with the large number of uncertain parameters. In this scenario, Markov Chain Monte Carlo (MCMC) algorithms have been proved to be highly effective. For a stochastic system, the most accurate description of the kinetics is given by the Chemical Master Equation (CME). Unfortunately, analytical solution of CME is often intractable even for considerably small amount of chemically reacting species due to its super exponential state space complexity. As a solution, Stochastic Simulation Algorithm (SSA) using Monte Carlo approach was introduced to simulate the chemical process defined by the CME. SSA is an exact stochastic method to simulate CME but it also suffers from high time complexity due to simulation of every reaction. Therefore computation of likelihood function (based on exact CME) and hence the rejection step (in an acceptance-rejection based MCMC like Metropolis-Hastings) becomes expensive. In this generic work, we introduce different approximations of CME as a pre-conditioning step to the full MCMC in order to make rejection cheaper. The goal is to avoid expensive computation of exact CME as far as possible. We show that, with effective pre-conditioning scheme, one can save a considerable amount of exact CME computations maintaining similar convergence characteristics. Additionally, we investigate three different sampling techniques (dense sampling of the same process, longer time sampling of the same process and i.i.d sampling of different processes) under which convergence of MCMC using exact CME for parameter inference can be analyzed. We find that under i.i.d sampling, better convergence can be achieved than that of other two techniques (at least for the processes, we have investigated). We verify our theoretical findings for two different fundamental processes: linear birth-death and dimerization. Although, we succeed in saving a considerable amount of CME computations for two simple one-dimesional processes, challenges remain in extending it for higher dimensions which is a non-trivial problem

Bayesian inference of chemical kinetic models from proposed reactions

Bayesian inference provides a natural framework for combining experimental data with prior knowledge to develop chemical kinetic models and quantify the associated uncertainties, not only in parameter values but also in model structure. Most existing applications of Bayesian model selection methods to chemical kinetics have been limited to comparisons among a small set of models, however. The significant computational cost of evaluating posterior model probabilities renders traditional Bayesian methods infeasible when the model space becomes large. We present a new framework for tractable Bayesian model inference and uncertainty quantification using a large number of systematically generated model hypotheses. The approach involves imposing point-mass mixture priors over rate constants and exploring the resulting posterior distribution using an adaptive Markov chain Monte Carlo method. The posterior samples are used to identify plausible models, to quantify rate constant uncertainties, and to extract key diagnostic information about model structure—such as the reactions and operating pathways most strongly supported by the data. We provide numerical demonstrations of the proposed framework by inferring kinetic models for catalytic steam and dry reforming of methane using available experimental data

Metropolis Methods for Quantum Monte Carlo Simulations

Since its first description fifty years ago, the Metropolis Monte Carlo method has been used in a variety of different ways for the simulation of continuum quantum many-body systems. This paper will consider some of the generalizations of the Metropolis algorithm employed in quantum Monte Carlo: Variational Monte Carlo, dynamical methods for projector monte carlo ({\it i.e.} diffusion Monte Carlo with rejection), multilevel sampling in path integral Monte Carlo, the sampling of permutations, cluster methods for lattice models, the penalty method for coupled electron-ionic systems and the Bayesian analysis of imaginary time correlation functions.Comment: Proceedings of "Monte Carlo Methods in the Physical Sciences" Celebrating the 50th Anniversary of the Metropolis Algorith

Integrating discrete stochastic models with single-cell and single-molecule experiments

2019 Summer.Includes bibliographical references.Modern biological experiments can capture the behaviors of single biomolecules within single cells. Much like Robert Brown looking at pollen grains in water, experimentalists have noticed that individual cells that are genetically identical behave seemingly randomly in the way they carry out their most basic functions. The field of stochastic single-cell biology has been focused developing mathematical and computational tools to understand how cells try to buffer or even make use of such fluctuations, and the technologies to measure such fluctuations has vastly improved in recent years. This dissertation is focused on developing new methods to analyze modern single-cell and single-molecule biological data with discrete stochastic models of the underlying processes, such as stochastic gene expression and single-mRNA translation. The methods developed here emphasize a strong link between model and experiment to help understand, design, and eventually control biological systems at the single-cell level