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_18: In this paper, we propose some algorithms for the simulation of the distribution of certain diffusions conditioned on a terminal point. We prove that the conditional distribution is absolutely continuous with respect to the distribution of another diffusion which is easy for simulation, and the formula for the density is given explicitly. An example of parameter estimation for a Duffing-Van der Pol oscillator is given as an application.
@cite_3: A Monte Carlo method for simulating a multi-dimensional diffusion process conditioned on hitting a fixed point at a fixed future time is developed. Proposals for such diffusion bridges are obtained by superimposing an additional guiding term to the drift of the process under consideration. The guiding term is derived via approximation of the target process by a simpler diffusion processes with known transition densities. Acceptance of a proposal can be determined by computing the likelihood ratio between the proposal and the target bridge, which is derived in closed form.We show under general conditions that the likelihood ratio is well defined and show that a class of proposals with guiding term obtained from linear approximations fall under these conditions.
@cite_19: The problem of formal likelihood-based (either classical or Bayesian) inference for discretely observed multidimensional diffusions is particularly challenging. In principle, this involves data augmentation of the observation data to give representations of the entire diffusion trajectory. Most currently proposed methodology splits broadly into two classes: either through the discretization of idealized approaches for the continuous-time diffusion setup or through the use of standard finite-dimensional methodologies discretization of the diffusion model. The connections between these approaches have not been well studied. This article provides a unified framework that brings together these approaches, demonstrating connections, and in some cases surprising differences. As a result, we provide, for the first time, theoretical justification for the various methods of imputing missing data. The inference problems are particularly challenging for irreducible diffusions, and our framework is correspondingly mo...
@cite_31: In this paper, we adapt recently developed simulation-based sequential algorithms to the problem concerning the Bayesian analysis of discretely observed diffusion processes. The estimation framework involves the introduction of m?1 latent data points between every pair of observations. Sequential MCMC methods are then used to sample the posterior distribution of the latent data and the model parameters on-line. The method is applied to the estimation of parameters in a simple stochastic volatility model (SV) of the U.S. short-term interest rate. We also provide a simulation study to validate our method, using synthetic data generated by the SV model with parameters calibrated to match weekly observations of the U.S. short-term interest rate.
@cite_25: 
In this section we point out similarities and differences of this paper with respect to related approaches in @cite_31 , @cite_30 , @cite_18 , @cite_25 and @cite_19 . @cite_18 introduced proposals with dynamics governed by the SDE where @math . Sufficient conditions for absolute continuity for both cases are given in @cite_18 . The pulling term @math in the SDE forces @math to hit @math at time @math and the case @math takes the drift of the process into account when proposing bridges. However, for both @math and @math the resulting bridges may not resemble true bridges. An illuminating example is given in the introductory section of @cite_3 .