517 research outputs found
Inference for reaction networks using the Linear Noise Approximation
We consider inference for the reaction rates in discretely observed networks
such as those found in models for systems biology, population ecology and
epidemics. Most such networks are neither slow enough nor small enough for
inference via the true state-dependent Markov jump process to be feasible.
Typically, inference is conducted by approximating the dynamics through an
ordinary differential equation (ODE), or a stochastic differential equation
(SDE). The former ignores the stochasticity in the true model, and can lead to
inaccurate inferences. The latter is more accurate but is harder to implement
as the transition density of the SDE model is generally unknown. The Linear
Noise Approximation (LNA) is a first order Taylor expansion of the
approximating SDE about a deterministic solution and can be viewed as a
compromise between the ODE and SDE models. It is a stochastic model, but
discrete time transition probabilities for the LNA are available through the
solution of a series of ordinary differential equations. We describe how a
restarting LNA can be efficiently used to perform inference for a general class
of reaction networks; evaluate the accuracy of such an approach; and show how
and when this approach is either statistically or computationally more
efficient than ODE or SDE methods. We apply the LNA to analyse Google Flu
Trends data from the North and South Islands of New Zealand, and are able to
obtain more accurate short-term forecasts of new flu cases than another
recently proposed method, although at a greater computational cost
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
Scalable Inference for Markov Processes with Intractable Likelihoods
Bayesian inference for Markov processes has become increasingly relevant in
recent years. Problems of this type often have intractable likelihoods and
prior knowledge about model rate parameters is often poor. Markov Chain Monte
Carlo (MCMC) techniques can lead to exact inference in such models but in
practice can suffer performance issues including long burn-in periods and poor
mixing. On the other hand approximate Bayesian computation techniques can allow
rapid exploration of a large parameter space but yield only approximate
posterior distributions. Here we consider the combined use of approximate
Bayesian computation (ABC) and MCMC techniques for improved computational
efficiency while retaining exact inference on parallel hardware
Bayesian inference of biochemical kinetic parameters using the linear noise approximation
Background
Fluorescent and luminescent gene reporters allow us to dynamically quantify changes in molecular species concentration over time on the single cell level. The mathematical modeling of their interaction through multivariate dynamical models requires the deveopment of effective statistical methods to calibrate such models against available data. Given the prevalence of stochasticity and noise in biochemical systems inference for stochastic models is of special interest. In this paper we present a simple and computationally efficient algorithm for the estimation of biochemical kinetic parameters from gene reporter data.
Results
We use the linear noise approximation to model biochemical reactions through a stochastic dynamic model which essentially approximates a diffusion model by an ordinary differential equation model with an appropriately defined noise process. An explicit formula for the likelihood function can be derived allowing for computationally efficient parameter estimation. The proposed algorithm is embedded in a Bayesian framework and inference is performed using Markov chain Monte Carlo.
Conclusion
The major advantage of the method is that in contrast to the more established diffusion approximation based methods the computationally costly methods of data augmentation are not necessary. Our approach also allows for unobserved variables and measurement error. The application of the method to both simulated and experimental data shows that the proposed methodology provides a useful alternative to diffusion approximation based methods
Simulation based sequential Monte Carlo methods for discretely observed Markov processes
Parameter estimation for discretely observed Markov processes is a
challenging problem. However, simulation of Markov processes is straightforward
using the Gillespie algorithm. We exploit this ease of simulation to develop an
effective sequential Monte Carlo (SMC) algorithm for obtaining samples from the
posterior distribution of the parameters. In particular, we introduce two key
innovations, coupled simulations, which allow us to study multiple parameter
values on the basis of a single simulation, and a simple, yet effective,
importance sampling scheme for steering simulations towards the observed data.
These innovations substantially improve the efficiency of the SMC algorithm
with minimal effect on the speed of the simulation process. The SMC algorithm
is successfully applied to two examples, a Lotka-Volterra model and a
Repressilator model.Comment: 27 pages, 5 figure
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