We consider the inverse problem of reconstructing the posterior measure over the trajec- tories of a diffusion process from discrete time observations and continuous time constraints. We cast the problem in a Bayesian framework and derive approximations to the posterior distributions of single time marginals using variational approximate inference. We then show how the approximation can be extended to a wide class of discrete-state Markov jump pro- cesses by making use of the chemical Langevin equation. Our empirical results show that the proposed method is computationally efficient and provides good approximations for these classes of inverse problems.
@cite_2 and @cite_4 propose a continuous time extension of the popular unscented transformation in to obtain Gaussian state space approximations in SDE models with time-only dependent diffusion terms and both non-linear non-Gaussian discrete and continuous time observation. In the authors compare these approaches to the variational method in which they then use to improve on their smoothing estimates. In a recent work @cite_12 present a mean-field variational approximation where they approximate the posterior process with a set of independent univariate Gaussian processes (factorised approximation). The considered model has polynomial drift terms and state-independent diffusion terms and the observations are at discrete time-points. Due to a clever parameterisations (piecewise polynomials) of the mean and the variance function of the variational approximation the dimensionality of the state can scale to thousands.
@cite_4 @cite_12 @cite_2
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This paper describes a generalized axiomatic scale-space theory that makes it possible to derive the notions of linear scale-space, affine Gaussian scale-space and linear spatio-temporal scale-space using a similar set of assumptions (scale-space axioms). The notion of non-enhancement of local extrema is generalized from previous application over discrete and rotationally symmetric kernels to continuous and more general non-isotropic kernels over both spatial and spatio-temporal image domains. It is shown how a complete classification can be given of the linear (Gaussian) scale-space concepts that satisfy these conditions on isotropic spatial, non-isotropic spatial and spatio-temporal domains, which results in a general taxonomy of Gaussian scale-spaces for continuous image data. The resulting theory allows filter shapes to be tuned from specific context information and provides a theoretical foundation for the recently exploited mechanisms of shape adaptation and velocity adaptation, with highly useful applications in computer vision. It is also shown how time-causal spatio-temporal scale-spaces can be derived from similar assumptions. The mathematical structure of these scale-spaces is analyzed in detail concerning transformation properties over space and time, the temporal cascade structure they satisfy over time as well as properties of the resulting multi-scale spatio-temporal derivative operators. It is also shown how temporal derivatives with respect to transformed time can be defined, leading to the formulation of a novel analogue of scale normalized derivatives for time-causal scale-spaces. The kernels generated from these two types of theories have interesting relations to biological vision. We show how filter kernels generated from the Gaussian spatio-temporal scale-space as well as the time-causal spatio-temporal scale-space relate to spatio-temporal receptive field profiles registered from mammalian vision. Specifically, we show that there are close analogies to space-time separable cells in the LGN as well as to both space-time separable and non-separable cells in the striate cortex. We do also present a set of plausible models for complex cells using extended quasi-quadrature measures expressed in terms of scale normalized spatio-temporal derivatives. The theories presented as well as their relations to biological vision show that it is possible to describe a general set of Gaussian and or time-causal scale-spaces using a unified framework, which generalizes and complements previously presented scale-space formulations in this area. This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies. It is well documented that natural images are compressible in wavelet bases and tend to exhibit fractal properties. In this paper, we investigate statistical models that mimic these behaviors. We then use our models to make predictions on the statistics of the wavelet coefficients. Following an innovation modeling approach, we identify a general class of finite-variance self-similar sparse random processes. We first prove that spatially dilated versions of self-similar sparse processes are asymptotically Gaussian as the dilation factor increases. Based on this fundamental result, we show that the coarse-scale wavelet coefficients of these processes are also asymptotically Gaussian, provided the wavelet has enough vanishing moments. Moreover, we quantify the degree of Gaussianity by deriving the theoretical evolution of the kurtosis of the wavelet coefficients across scales. Finally, we apply our analysis to one- and two-dimensional signals, including natural images, and show that the wavelet coefficients ... We present an algorithm that infers the model structure of a mixture of factor analysers using an efficient and deterministic variational approximation to full Bayesian integration over model parameters. This procedure can automatically determine the optimal number of components and the local dimensionality of each component (i.e. the number of factors in each factor analyser). Alternatively it can be used to infer posterior distributions over number of components and dimensionalities. Since all parameters are integrated out the method is not prone to overfitting. Using a stochastic procedure for adding components it is possible to perform the variational optimisation incrementally and to avoid local maxima. Results show that the method works very well in practice and correctly infers the number and dimensionality of nontrivial synthetic examples. By importance sampling from the variational approximation we show how to obtain unbiased estimates of the true evidence, the exact predictive density, and the KL divergence between the variational posterior and the true posterior, not only in this model but for variational approximations in general.