Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case. We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods.
@cite_30: Computational systems biology unifies the mechanistic approach of systems biology with the data-driven approach of computational biology. Computational systems biology aims to develop algorithms that uncover the structure and parameterization of the underlying mechanistic model--in other words, to answer specific questions about the underlying mechanisms of a biological system--in a process that can be thought of as learning or inference. This volume offers state-of-the-art perspectives from computational biology, statistics, modeling, and machine learning on new methodologies for learning and inference in biological networks.The chapters offer practical approaches to biological inference problems ranging from genome-wide inference of genetic regulation to pathway-specific studies. Both deterministic models (based on ordinary differential equations) and stochastic models (which anticipate the increasing availability of data from small populations of cells) are considered. Several chapters emphasize Bayesian inference, so the editors have included an introduction to the philosophy of the Bayesian approach and an overview of current work on Bayesian inference. Taken together, the methods discussed by the experts in Learning and Inference in Computational Systems Biology provide a foundation upon which the next decade of research in systems biology can be built. Florence d'Alch e-Buc, John Angus, Matthew J. Beal, Nicholas Brunel, Ben Calderhead, Pei Gao, Mark Girolami, Andrew Golightly, Dirk Husmeier, Johannes Jaeger, Neil D. Lawrence, Juan Li, Kuang Lin, Pedro Mendes, Nicholas A. M. Monk, Eric Mjolsness, Manfred Opper, Claudia Rangel, Magnus Rattray, Andreas Ruttor, Guido Sanguinetti, Michalis Titsias, Vladislav Vyshemirsky, David L. Wild, Darren Wilkinson, Guy Yosiphon
@cite_9: 
@cite_23: This paper provides methods for carrying out likelihood based inference for diffusion driven models, for example discretely observed multivariate diffusions, continuous time stochastic volatility models and counting process models. The diffusions can potentially be non-stationary. Although our methods are sampling based, making use of Markov chain Monte Carlo methods to sample the posterior distribution of the relevant unknowns, our general strategies and details are different from previous work along these lines. The methods we develop are simple to implement and simulation efficient. Importantly, unlike previous methods, the performance of our technique is not worsened, in fact it improves, as the degree of latent augmentation is increased to reduce the bias of the Euler approximation. In addition, our method is not subject to a degeneracy that afflicts previous techniques when the degree of latent augmentation is increased. We also discuss issues of model choice, model checking and filtering. The techniques and ideas are applied to both simulated and real data.
Without loss of generality we may assume we have have one observation @math . If @math in , a slight adjustment of the Euler discretisation of the bridge (introduced by @cite_9 ) sets where @math and @math (the adjustment being the addition of the term @math in front of @math ). If @math is small, the right hand side defines a discrete time Markov chain approximation @math to @math termed the Modified Brownian Bridge. We can define a mapping @math by the relation [ ( X^ , , X^ ) =g( , W_1, , W_N), ] where @math and @math . This implies that parameters in the diffusion coefficient can be updated conditional on @math , instead of @math ; thereby preventing degeneracy of the data-aug -men -ta -tion algorithm. This is the approach of @cite_30 . @cite_23 use the same algorithm using Euler discretisation of @math in case @math (so no correction in the diffusion coefficient appears in the discretisation). Note that there is no formal proof that these methods work as @math .