This paper describes a general model that subsumes many parametric models for continuous data. The model comprises hidden layers of state-space or dynamic causal models, arranged so that the output of one provides input to another. The ensuing hierarchy furnishes a model for many types of data, of arbitrary complexity. Special cases range from the general linear model for static data to generalised convolution models, with system noise, for nonlinear time-series analysis. Crucially, all of these models can be inverted using exactly the same scheme, namely, dynamic expectation maximization. This means that a single model and optimisation scheme can be used to invert a wide range of models. We present the model and a brief review of its inversion to disclose the relationships among, apparently, diverse generative models of empirical data. We then show that this inversion can be formulated as a simple neural network and may provide a useful metaphor for inference and learning in the brain.

We apply the techniques of stochastic integration with respect to the frac-tional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by fractional Brownian motion with any level of Holder-regularity (any Hurst parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We also prove that a version of the MLE using only discrete observations is still a strongly consistent estimator.

This article focuses on two methods to approximate the loglikelihood function for univariate diffusions: 1) the simulation approach using a modified Brownian bridge as the importance sampler; and 2) the recent closed-form approach. For the case of constant volatility, we give a theoretical justification of the modified Brownian bridge sampler by showing that it is exactly a Brownian bridge. We also discuss computational issues in the simulation approach such as accelerating numerical variance stabilizing transformation, computing derivatives of the simu-lated loglikelihood, and choosing initial values of parameter estimates. The two approaches are compared in the context of financial applications with annualized parameter values, where the diffusion model has an unknown transition density and has no analytical variance stabilizing transformation. The closed-form expansion, particularly the second-order closed-form, is found to be computationally efficient and very accurate when the observation frequency is monthly or higher. It is more accurate in the center than in the tail of the transition density. The simulation approach combined with the variance stabilizing transformation is found to be more reliable than the closed-form approach when the observation frequency is lower. Both methods performs better when the volatility level is lower, but the simulation method is more robust to the volatility nature of the diffusion model. When applied to two well known datasets of daily observations, the two methods yield similar parameter estimates in both datasets but slightly different loglikelihood in the case of higher volatility.

transformations

Abstract not found

This chapter considers stochastic differential equations for Systems Biology models derived from the Chemical Langevin Equation (CLE). After outlining the derivation of such models, Bayesian inference for the parameters is considered, based on state-of-the-art Markov chain Monte Carlo algorithms. Starting with a basic scheme for models observed perfectly, but discretely in time, problems with standard schemes and their solutions are discussed. Extensions of these schemes to partial observation and observations subject to measurement error are also considered. Finally, the techniques are demonstrated in the context of a simple stochastic kinetic model of a genetic regulatory network. 1

This paper presents a variational treatment of dynamic models that furnishes time-dependent conditional densities on the path or trajectory of a system's states and the time-independent densities of its parameters. These are obtained by maximising a variational action with respect to conditional densities, under a fixed-form assumption about their form. The action or path-integral of free-energy represents a lower bound on the model's log-evidence or marginal likelihood required for model selection and averaging. This approach rests on formulating the optimisation dynamically, in generalised coordinates of motion. The resulting scheme can be used for online Bayesian inversion of nonlinear dynamic causal models and is shown to outperform existing approaches, such as Kalman and particle filtering. Furthermore, it provides for dual and triple inferences on a system's states, parameters and hyperparameters using exactly the same principles. We refer to this approach as dynamic expectation maximisation (DEM).

anonymous referees, Niels Haldrup, Montserrat Guillén and from participants at the International Symposium

estimation for ergodic diffusions sampled at

I would like to thank a number of people who have accompanied me during the writing of this thesis. First and foremost, my sincere gratitude goes to my supervisors Ludwig Fahrmeir and Gareth Roberts, who enriched my work through their advice, ideas and encouragement. I also thank Leonhard Held for his directions during the first stage of my thesis. My research has financially been supported by the German Research Foundation (DFG), the German Academic Exchange Service (DAAD) and the LMU Mentoring programme, in which Francesca Biagini has been a dedicated mentor to me. I deeply appreciate the careful proof-reading and helpful comments by Michael Höhle. Furthermore, I am grateful to my former and present colleagues for their interest in my research and their friendship, in particular to the members of the Semwiso group, my FRAP collaborators, the advocates of good teaching, my fellow women’s representatives, the Cozi Family and my office mates. My family has been a constant source of support, and I greatly acknowledge their personal way of understanding my work. I owe my heartful gratitude to Florian Fuchs, who has been a strong and close partner during all stages of my thesis and who stayed awake until the last sentence was written.