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In this way, it is possible to backpropagate the gradient without involving stochastic variable during the update. The scheme of a variational autoencoder after the reparameterization trick. In Variational Autoencoders (VAEs), the VAE objective function, known as the Evidence Lower Bound (ELBO), is given by:
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 ...
In Bayesian inference, the Bernstein–von Mises theorem provides the basis for using Bayesian credible sets for confidence statements in parametric models.It states that under some conditions, a posterior distribution converges in total variation distance to a multivariate normal distribution centered at the maximum likelihood estimator ^ with covariance matrix given by (), where is the true ...
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 ...
Statistical inference makes propositions about a population, using data drawn from the population with some form of sampling.Given a hypothesis about a population, for which we wish to draw inferences, statistical inference consists of (first) selecting a statistical model of the process that generates the data and (second) deducing propositions from the model.
Many variational autoencoders applications and extensions have been used to adapt the architecture to other domains and improve its performance. β {\displaystyle \beta } -VAE is an implementation with a weighted Kullback–Leibler divergence term to automatically discover and interpret factorised latent representations.
While classical variational problems, such as the brachistochrone problem, can be solved using the Hamilton–Jacobi–Bellman equation, [8] the method can be applied to a broader spectrum of problems. Further it can be generalized to stochastic systems, in which case the HJB equation is a second-order elliptic partial differential equation. [9]
In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if 1 is a root of the process's characteristic equation .