Dempster-Shafer's theory models the uncertainty present in the data and model and helps in building robust machine learning models.

Knowing the Galactic 3D dust distribution is relevant for understanding many processes in the interstellar medium and for correcting many astronomical observations for dust absorption and emission. Here, we aim for a 3D reconstruction of the Galactic dust distribution with an increase in the number of meaningful resolution elements by orders of magnitude with respect to previous reconstructions, while taking advantage of the dust's spatial correlations to inform the dust map. We use iterative grid refinement to define a log-normal process in spherical coordinates. This log-normal process assumes a fixed correlation structure, which was inferred in an earlier reconstruction of Galactic dust. Our map is informed through 111 Million data points, combining data of PANSTARRS, 2MASS, Gaia DR2 and ALLWISE. The log-normal process is discretized to 122 Billion degrees of freedom, a factor of 400 more than our previous map. We derive the most probable posterior map and an uncertainty estimate using natural gradient descent and the Fisher-Laplace approximation. The dust reconstruction covers a quarter of the volume of our Galaxy, with a maximum coordinate distance of $16\,\text{kpc}$, and meaningful information can be found up to at distances of $4\,$kpc, still improving upon our earlier map by a factor of 5 in maximal distance, of $900$ in volume, and of about eighteen in angular grid resolution. Unfortunately, the maximum posterior approach chosen to make the reconstruction computational affordable introduces artifacts and reduces the accuracy of our uncertainty estimate. Despite of the apparent limitations of the presented 3D dust map, a good part of the reconstructed structures are confirmed by independent maser observations. Thus, the map is a step towards reliable 3D Galactic cartography and already can serve for a number of tasks, if used with care.

Dempster-Shafer's theory models the uncertainty present in the data and model and helps in building robust machine learning models.

Initial work on variational autoencoders assumed independent latent variables with simple distributions. Subsequent work has explored incorporating more complex distributions and dependency structures: including normalizing flows in the encoder network allows latent variables to entangle non-linearly, creating a richer class of distributions for the approximate posterior, and stacking layers of latent variables allows more complex priors to be specified for the generative model. This work explores incorporating arbitrary dependency structures, as specified by Bayesian networks, into VAEs. This is achieved by extending both the prior and inference network with graphical residual flows - residual flows that encode conditional independence by masking the weight matrices of the flow's residual blocks. We compare our model's performance on several synthetic datasets and show its potential in data-sparse settings.

We describe a stochastic, dynamical system capable of inference and learning in a probabilistic latent variable model. The most challenging problem in such models - sampling the posterior distribution over latent variables - is proposed to be solved by harnessing natural sources of stochasticity inherent in electronic and neural systems. We demonstrate this idea for a sparse coding model by deriving a continuous-time equation for inferring its latent variables via Langevin dynamics. The model parameters are learned by simultaneously evolving according to another continuous-time equation, thus bypassing the need for digital accumulators or a global clock. Moreover we show that Langevin dynamics lead to an efficient procedure for sampling from the posterior distribution in the 'L0 sparse' regime, where latent variables are encouraged to be set to zero as opposed to having a small L1 norm. This allows the model to properly incorporate the notion of sparsity rather than having to resort to a relaxed version of sparsity to make optimization tractable. Simulations of the proposed dynamical system on both synthetic and natural image datasets demonstrate that the model is capable of probabilistically correct inference, enabling learning of the dictionary as well as parameters of the prior.

This is a complete Free course for statistics. In this course, you will learn how to estimate parameters of a population using sample statistics, hypothesis testing and confidence intervals, t-tests and ANOVA, correlation and regression, and chi-squared test. This course is taught by industry professionals and you will learn by doing various exercises.

Gaussian Processes (GPs) are a versatile and popular method in Bayesian Machine Learning. A common modification are Sparse Variational Gaussian Processes (SVGPs) which are well suited to deal with large datasets. While GPs allow to elegantly deal with Gaussian-distributed target variables in closed form, their applicability can be extended to non-Gaussian data as well. These extensions are usually impossible to treat in closed form and hence require approximate solutions. This paper proposes to approximate the inverse-link function, which is necessary when working with non-Gaussian likelihoods, by a piece-wise constant function. It will be shown that this yields a closed form solution for the corresponding SVGP lower bound. In addition, it is demonstrated how the piece-wise constant function itself can be optimized, resulting in an inverse-link function that can be learnt from the data at hand.

As a data scientist, I am curious about knowing different analytical processes from a probabilistic point of view. There are two most popular ways of looking into any event, namely Bayesian and Frequentist . When Frequentist researchers look at any event from frequency of occurrence, Bayesian researchers focus more on probability of events happening. I will try to cover as much theory as possible with illustrative examples and sample codes so that readers can learn and practice simultaneously. As we all know Baye's rule is one of the most popular probability equation, which is defined as: P(a given b) P(a intersection b) / P(b) ….. (1) Here a and b are events that have taken place.

In the era of big data, it is necessary to split extremely large data sets across multiple computing nodes and construct estimators using the distributed data. When designing distributed estimators, it is desirable to minimize the amount of communication across the network because transmission between computers is slow in comparison to computations in a single computer. Our work provides a general framework for understanding the behavior of distributed estimation under communication constraints for nonparametric problems. We provide results for a broad class of models, moving beyond the Gaussian framework that dominates the literature. As concrete examples we derive minimax lower and matching upper bounds in the distributed regression, density estimation, classification, Poisson regression and volatility estimation models under communication constraints. To assist with this, we provide sufficient conditions that can be easily verified in all of our examples.

In this case, standard stochastic interpolation tools like the common SABR model often cannot be calibrated to observed implied volatility smiles, due to data being only available for the at-the-money quote of the respective underlying swaption. Here, we propose to infer the geometry of the full unknown implied volatility cube by learning stochastic latent representations of implied volatility cubes via variational autoencoders, enabling inference about the missing volatility data conditional on the observed data by an approximate Gibbs sampling approach. Imputed estimates of missing quotes can afterwards be used to fit a standard stochastic volatility model. Since training data for the employed variational autoencoder model is usually sparsely available, we test the robustness of the approach for a model trained on synthetic data on real market quotes and we show that SABR interpolated volatilites calibrated to reconstructed volatility cubes with artificially imputed missing values differ by not much more than two basis points compared to SABR fits calibrated to the complete cube. Moreover, we show how the imputation can be used to successfully set up delta-neutral portfolios for hedging purposes.