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The reparameterization trick (aka "reparameterization gradient estimator") is a technique used in statistical machine learning, particularly in variational inference, variational autoencoders, and stochastic optimization.
Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various sorts of relationships among the three types of random variables, as ...
Malliavin introduced Malliavin calculus to provide a stochastic proof that Hörmander's condition implies the existence of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. His calculus enabled Malliavin to prove regularity bounds for the ...
Inference engines [ edit ] PyMC implements non-gradient-based and gradient-based Markov chain Monte Carlo (MCMC) algorithms for Bayesian inference and stochastic, gradient-based variational Bayesian methods for approximate Bayesian inference.
Integrated nested Laplace approximations (INLA) is a method for approximate Bayesian inference based on Laplace's method. [1] It is designed for a class of models called latent Gaussian models (LGMs), for which it can be a fast and accurate alternative for Markov chain Monte Carlo methods to compute posterior marginal distributions.
The main flavours of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integral is the most useful for general classes of processes, but the related Stratonovich integral is frequently useful in problem formulation (particularly in engineering disciplines). The ...
Figure 1. Probabilistic parameters of a hidden Markov model (example) X — states y — possible observations a — state transition probabilities b — output probabilities. In its discrete form, a hidden Markov process can be visualized as a generalization of the urn problem with replacement (where each item from the urn is returned to the original urn before the next step). [7]
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.