We introduce a nonparametric approach for estimating drift and diffusion functions in systems of stochastic differential equations from observations of the state vector. Gaussian processes are used as flexible models for these functions and estimates are calculated directly from dense data sets using Gaussian process regression. We also develop an approximate expectation maximization algorithm to deal with the unobserved, latent dynamics between sparse observations. The posterior over states is approximated by a piecewise linearized process of the Ornstein-Uhlenbeck type and the maximum a posteriori estimation of the drift is facilitated by a sparse Gaussian process approximation.

The process of dynamic state estimation (filtering) based on point process observations is in general intractable. Numerical sampling techniques are often practically useful, but lead to limited conceptual insight about optimal encoding/decoding strategies, which are of significant relevance to Computational Neuroscience. We develop an analytically tractable Bayesian approximation to optimal filtering based on point process observations, which allows us to introduce distributional assumptions about sensor properties, that greatly facilitate the analysis of optimal encoding in situations deviating from common assumptions of uniform coding. Numerical comparison with particle filtering demonstrate the quality of the approximation. The analytic framework leads to insights which are difficult to obtain from numerical algorithms, and is consistent with biological observations about the distribution of sensory cells' tuning curve centers.

We describe and analyze some novel approaches for studying the dynamics of Ising spin glass models. We first briefly consider the variational approach based on minimizing the Kullback-Leibler divergence between independent trajectories and the real ones and note that this approach only coincides with the mean field equations from the saddle point approximation to the generating functional when the dynamics is defined through a logistic link function, which is the case for the kinetic Ising model with parallel update. We then spend the rest of the paper developing two ways of going beyond the saddle point approximation to the generating functional. In the first one, we develop a variational perturbative approximation to the generating functional by expanding the action around a quadratic function of the local fields and conjugate local fields whose parameters are optimized. We derive analytical expressions for the optimal parameters and show that when the optimization is suitably restricted, we recover the mean field equations that are exact for the fully asymmetric random couplings (M\'ezard and Sakellariou, 2011). However, without this restriction the results are different. We also describe an extended Plefka expansion in which in addition to the magnetization, we also fix the correlation and response functions. Finally, we numerically study the performance of these approximations for Sherrington-Kirkpatrick type couplings for various coupling strengths, degrees of coupling symmetry and external fields. We show that the dynamical equations derived from the extended Plefka expansion outperform the others in all regimes, although it is computationally more demanding. The unconstrained variational approach does not perform well in the small coupling regime, while it approaches dynamical TAP equations of (Roudi and Hertz, 2011) for strong couplings.

Most Machine Learning (ML) methods, from clustering to classification, rely on a distance function to describe relationships between datapoints. For complex datasets it is hard to avoid making some arbitrary choices when defining a distance function. To compare images, one must choose a spatial scale, for signals, a temporal scale. The right scale is hard to pin down and it is preferable when results do not depend too tightly on the exact value one picked. Topological data analysis seeks to address this issue by focusing on the notion of neighbourhood instead of distance. It is shown that in some cases a simpler solution is available. It can be checked how strongly distance relationships depend on a hyperparameter using dimensionality reduction. A variant of dynamical multi-dimensional scaling (MDS) is formulated, which embeds datapoints as curves. The resulting algorithm is based on the Concave-Convex Procedure (CCCP) and provides a simple and efficient way of visualizing changes and invariances in distance patterns as a hyperparameter is varied. A variant to analyze the dependence on multiple hyperparameters is also presented. A cMDS algorithm that is straightforward to implement, use and extend is provided. To illustrate the possibilities of cMDS, cMDS is applied to several real-world data sets.

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.

The process of dynamic state estimation (filtering) based on point process observations is in general intractable. Numerical sampling techniques are often practically useful, but lead to limited conceptual insight about optimal encoding/decoding strategies, which are of significant relevance to Computational Neuroscience. We develop an analytically tractable Bayesian approximation to optimal filtering based on point process observations, which allows us to introduce distributional assumptions about sensory cell properties, that greatly facilitates the analysis of optimal encoding in situations deviating from common assumptions of uniform coding. The analytic framework leads to insights which are difficult to obtain from numerical algorithms, and is consistent with experiments about the distribution of tuning curve centers. Interestingly, we find that the information gained from the absence of spikes may be crucial to performance.

The process of dynamic state estimation (filtering) based on point process observations is in general intractable. Numerical sampling techniques are often practically useful, but lead to limited conceptual insight about optimal encoding/decoding strategies, which are of significant relevance to Computational Neuroscience. We develop an analytically tractable Bayesian approximation to optimal filtering based on point process observations, which allows us to introduce distributional assumptions about sensory cell properties, that greatly facilitates the analysis of optimal encoding in situations deviating from common assumptions of uniform coding. The analytic framework leads to insights which are difficult to obtain from numerical algorithms, and is consistent with experiments about the distribution of tuning curve centers. Interestingly, we find that the information gained from the absence of spikes may be crucial to performance.

We introduce a model where the rate of an inhomogeneous Poisson process is modified by a Chinese restaurant process. Applying a MCMC sampler to this model allows us to do posterior Bayesian inference about the number of states in Poisson-like data. Our sampler is shown to get accurate results for synthetic data and we apply it to V1 neuron spike data to find discrete firing rate states depending on the orientation of a stimulus.

Agents acting in the natural world aim at selecting appropriate actions based on noisy and partial sensory observations. Many behaviors leading to decision making and action selection in a closed loop setting are naturally phrased within a control theoretic framework. Within the framework of optimal Control Theory, one is usually given a cost function which is minimized by selecting a control law based on the observations. While in standard control settings the sensors are assumed fixed, biological systems often gain from the extra flexibility of optimizing the sensors themselves. However, this sensory adaptation is geared towards control rather than perception, as is often assumed. In this work we show that sensory adaptation for control differs from sensory adaptation for perception, even for simple control setups. This implies, consistently with recent experimental results, that when studying sensory adaptation, it is essential to account for the task being performed.

Variational inference is a powerful concept that underlies many iterative approximation algorithms; expectation propagation, mean-field methods and belief propagations were all central themes at the school that can be perceived from this unifying framework. The lectures of Manfred Opper introduce the archetypal example of Expectation Propagation, before establishing the connection with the other approximation methods. Corrections by expansion about the expectation propagation are then explained. Finally some advanced inference topics and applications are explored in the final sections.