A common applied statistics task involves building regression models to characterize non-linear relationships between variables. When we write a function that takes continuous values as inputs, we are essentially implying an infinite vector that only returns values (indexed by the inputs) when the function is called upon to do so. To make this notion of a "distribution over functions" more concrete, let's quickly demonstrate how we obtain realizations from a Gaussian process, which result in an evaluation of a function over a set of points. We are going generate realizations sequentially, point by point, using the lovely conditioning property of mutlivariate Gaussian distributions.

Written by Chris Fonnesbeck, Assistant Professor of Biostatistics, Vanderbilt University Medical Center. You can view, fork, and play with this project on the Domino data science platform. A common applied statistics task involves building regression models to characterize non-linear relationships between variables. It is possible to fit such models by assuming a particular non-linear functional form, such as a sinusoidal, exponential, or polynomial function, to describe one variable's response to the variation in another. Unless this relationship is obvious from the outset, however, it involves possibly extensive model selection procedures to ensure the most appropriate model is retained. Alternatively, a non-parametric approach can be adopted by defining a set of knots across the variable space and use a spline or kernel regression to describe arbitrary non-linear relationships.

A common applied statistics task involves building regression models to characterize non-linear relationships between variables. It is possible to fit such models by assuming a particular non-linear functional form, such as a sinusoidal, exponential, or polynomial function, to describe one variable's response to the variation in another. Unless this relationship is obvious from the outset, however, it involves possibly extensive model selection procedures to ensure the most appropriate model is retained. Alternatively, a non-parametric approach can be adopted by defining a set of knots across the variable space and use a spline or kernel regression to describe arbitrary non-linear relationships. However, knot layout procedures are somewhat ad hoc and can also involve variable selection. A third alternative is to adopt a Bayesian non-parametric strategy, and directly model the unknown underlying function.

Mullachery, Vikram, Khera, Aniruddh, Husain, Amir

This paper describes and discusses Bayesian Neural Network (BNN). The paper showcases a few different applications of them for classification and regression problems. BNNs are comprised of a Probabilistic Model and a Neural Network. The intent of such a design is to combine the strengths of Neural Networks and Stochastic modeling. Neural Networks exhibit continuous function approximator capabilities. Stochastic models allow direct specification of a model with known interaction between parameters to generate data. During the prediction phase, stochastic models generate a complete posterior distribution and produce probabilistic guarantees on the predictions. Thus BNNs are a unique combination of neural network and stochastic models with the stochastic model forming the core of this integration. BNNs can then produce probabilistic guarantees on it's predictions and also generate the distribution of parameters that it has learnt from the observations. That means, in the parameter space, one can deduce the nature and shape of the neural network's learnt parameters. These two characteristics makes them highly attractive to theoreticians as well as practitioners. Recently there has been a lot of activity in this area, with the advent of numerous probabilistic programming libraries such as: PyMC3, Edward, Stan etc. Further this area is rapidly gaining ground as a standard machine learning approach for numerous problems