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Learning accurate predictive models of real-world dynamic phenomena (e.g., climate, biological) remains a challenging task. One key issue is that the data generated by both natural and artificial processes often comprise time series that are irregularly sampled and/or contain missing observations. In this work, we propose the Neural Continuous-Discrete State Space Model (NCDSSM) for continuous-time modeling of time series through discrete-time observations. NCDSSM employs auxiliary variables to disentangle recognition from dynamics, thus requiring amortized inference only for the auxiliary variables. Leveraging techniques from continuous-discrete filtering theory, we demonstrate how to perform accurate Bayesian inference for the dynamic states. We propose three flexible parameterizations of the latent dynamics and an efficient training objective that marginalizes the dynamic states during inference. Empirical results on multiple benchmark datasets across various domains show improved imputation and forecasting performance of NCDSSM over existing models.

Computational models have become a powerful tool in the quantitative sciences to understand the behaviour of complex systems that evolve in time. However, they often contain a potentially large number of free parameters whose values cannot be obtained from theory but need to be inferred from data. This is especially the case for models in the social sciences, economics, or computational epidemiology. Yet many current parameter estimation methods are mathematically involved and computationally slow to run. In this paper we present a computationally simple and fast method to retrieve accurate probability densities for model parameters using neural differential equations. We present a pipeline comprising multi-agent models acting as forward solvers for systems of ordinary or stochastic differential equations, and a neural network to then extract parameters from the data generated by the model. The two combined create a powerful tool that can quickly estimate densities on model parameters, even for very large systems. We demonstrate the method on synthetic time series data of the SIR model of the spread of infection, and perform an in-depth analysis of the Harris-Wilson model of economic activity on a network, representing a non-convex problem. For the latter, we apply our method both to synthetic data and to data of economic activity across Greater London. We find that our method calibrates the model orders of magnitude more accurately than a previous study of the same dataset using classical techniques, while running between 195 and 390 times faster.

Self organizing complex systems can be modeled using cellular automaton models. However, the parametrization of these models is crucial and significantly determines the resulting structural pattern. In this research, we introduce and successfully apply a sound statistical method to estimate these parameters. The method is based on constructing Gaussian likelihoods using characteristics of the structures such as the mean particle size. We show that our approach is robust with respect to the method parameters, domain size of patterns, or CA iterations.

Probabilistic graphical models (PGMs) provide a compact and flexible framework to model very complex real-life phenomena. They combine the probability theory which deals with uncertainty and logical structure represented by a graph which allows one to cope with the computational complexity and also interpret and communicate the obtained knowledge. In the thesis, we consider two different types of PGMs: Bayesian networks (BNs) which are static, and continuous time Bayesian networks which, as the name suggests, have a temporal component. We are interested in recovering their true structure, which is the first step in learning any PGM. This is a challenging task, which is interesting in itself from the causal point of view, for the purposes of interpretation of the model and the decision-making process. All approaches for structure learning in the thesis are united by the same idea of maximum likelihood estimation with the LASSO penalty. The problem of structure learning is reduced to the problem of finding non-zero coefficients in the LASSO estimator for a generalized linear model. In the case of CTBNs, we consider the problem both for complete and incomplete data. We support the theoretical results with experiments.

Image reconstruction based on indirect, noisy, or incomplete data remains an important yet challenging task. While methods such as compressive sensing have demonstrated high-resolution image recovery in various settings, there remain issues of robustness due to parameter tuning. Moreover, since the recovery is limited to a point estimate, it is impossible to quantify the uncertainty, which is often desirable. Due to these inherent limitations, a sparse Bayesian learning approach is sometimes adopted to recover a posterior distribution of the unknown. Sparse Bayesian learning assumes that some linear transformation of the unknown is sparse. However, most of the methods developed are tailored to specific problems, with particular forward models and priors. Here, we present a generalized approach to sparse Bayesian learning. It has the advantage that it can be used for various types of data acquisitions and prior information. Some preliminary results on image reconstruction/recovery indicate its potential use for denoising, deblurring, and magnetic resonance imaging.

Bayesian model comparison (BMC) offers a principled approach for assessing the relative merits of competing computational models and propagating uncertainty into model selection decisions. However, BMC is often intractable for the popular class of hierarchical models due to their high-dimensional nested parameter structure. To address this intractability, we propose a deep learning method for performing BMC on any set of hierarchical models which can be instantiated as probabilistic programs. Since our method enables amortized inference, it allows efficient re-estimation of posterior model probabilities and fast performance validation prior to any real-data application. In a series of extensive validation studies, we benchmark the performance of our method against the state-of-the-art bridge sampling method and demonstrate excellent amortized inference across all BMC settings. We then use our method to compare four hierarchical evidence accumulation models that have previously been deemed intractable for BMC due to partly implicit likelihoods. In this application, we corroborate evidence for the recently proposed L\'evy flight model of decision-making and show how transfer learning can be leveraged to enhance training efficiency. Reproducible code for all analyses is provided.

Likelihood-free inference involves inferring parameter values given observed data and a simulator model. The simulator is computer code which takes parameters, performs stochastic calculations, and outputs simulated data. In this work, we view the simulator as a function whose inputs are (1) the parameters and (2) a vector of pseudo-random draws. We attempt to infer all these inputs conditional on the observations. This is challenging as the resulting posterior can be high dimensional and involve strong dependence. We approximate the posterior using normalizing flows, a flexible parametric family of densities. Training data is generated by likelihood-free importance sampling with a large bandwidth value epsilon, which makes the target similar to the prior. The training data is "distilled" by using it to train an updated normalizing flow. The process is iterated, using the updated flow as the importance sampling proposal, and slowly reducing epsilon so the target becomes closer to the posterior. Unlike most other likelihood-free methods, we avoid the need to reduce data to low dimensional summary statistics, and hence can achieve more accurate results. We illustrate our method in two challenging examples, on queuing and epidemiology.

Likelihood-free inference methods typically make use of a distance between simulated and real data. A common example is the maximum mean discrepancy (MMD), which has previously been used for approximate Bayesian computation, minimum distance estimation, generalised Bayesian inference, and within the nonparametric learning framework. The MMD is commonly estimated at a root-$m$ rate, where $m$ is the number of simulated samples. This can lead to significant computational challenges since a large $m$ is required to obtain an accurate estimate, which is crucial for parameter estimation. In this paper, we propose a novel estimator for the MMD with significantly improved sample complexity. The estimator is particularly well suited for computationally expensive smooth simulators with low- to mid-dimensional inputs. This claim is supported through both theoretical results and an extensive simulation study on benchmark simulators.

High-dimensional feature selection is a central problem in a variety of application domains such as machine learning, image analysis, and genomics. In this paper, we propose graph-based tests as a useful basis for feature selection. We describe an algorithm for selecting informative features in high-dimensional data, where each observation comes from one of $K$ different distributions. Our algorithm can be applied in a completely nonparametric setup without any distributional assumptions on the data, and it aims at outputting those features in the data, that contribute the most to the overall distributional variation. At the heart of our method is the recursive application of distribution-free graph-based tests on subsets of the feature set, located at different depths of a hierarchical clustering tree constructed from the data. Our algorithm recovers all truly contributing features with high probability, while ensuring optimal control on false-discovery. Finally, we show the superior performance of our method over other existing ones through synthetic data, and also demonstrate the utility of the method on two real-life datasets from the domains of climate change and single cell transcriptomics.