The goal of the variational autoencoder (VAE) is to learn a probability distribution $Pr(\mathbf{x})$ over a multi-dimensional variable $\mathbf{x}$. There are two main reasons for modelling distributions. First, we might want to draw samples (generate) from the distribution to create new plausible values of $\mathbf{x}$. Second, we might want to measure the likelihood that a new vector $\mathbf{x} {*}$ was created by this probability distribution. In fact, it turns out that the variational autoencoder is well-suited to the former task but not for the latter. It is common to talk about the variational autoencoder as if it is the model of $Pr(\mathbf{x})$. However, this is misleading; the variational autoencoder is a neural architecture that is designed to help learn the model for $Pr(\mathbf{x})$.

We review the math and code needed to fit a Gaussian Process (GP) regressor to data. We conclude with a demo of a popular application, fast function minimization through GP-guided search. The gif below illustrates this approach in action -- the red points are samples from the hidden red curve. Using these samples, we attempt to leverage GPs to find the curve's minimum as fast as possible. Appendices contain quick reviews on (i) the GP regressor posterior derivation, (ii) SKLearn's GP implementation, and (iii) GP classifiers. Follow @efavdb Follow us on twitter for new submission alerts!

We review the Metropolis algorithm -- a simple Markov Chain Monte Carlo (MCMC) sampling method -- and its application to estimating posteriors in Bayesian statistics. A simple python example is provided. Follow @efavdb Follow us on twitter for new submission alerts! One of the central aims of statistics is to identify good methods for fitting models to data. Notice that if we could solve for this function, we would be able to identify which parameter values are most likely -- those that are good candidates for a fit.

The full code is available as an IPython notebook on github. Assuming a multivariate normal distribution for the residuals in linear regression allows us to construct test statistics and therefore specify uncertainty in our fits. A t-test judges the explanatory power of a predictor in isolation, although the standard error that appears in the calculation of the t-statistic is a function of the other predictors in the model. On the other hand, an F-test is a global test that judges the explanatory power of all the predictors together, and we've seen that parsimony in choosing predictors can improve the quality of the overall regression. We've also seen that multicollinearity can throw off the results of individual t-tests as well as obscure the interpretation of the signs of the fitted coefficients. A symptom of multicollinearity is when none of the individual coefficients are significant but the overall F-test is significant.