In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time between production errors, or length along a roll of fabric in the weaving manufacturing process. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It is closely related to the concepts of an ancillary statistic which contains no information about the model parameters, and of a complete statistic which only contains information about the parameters and no ancillary information.
In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use:
- With a shape parameter k and a scale parameter θ
- With a shape parameter and a rate parameter
Unsupervised learning is a framework in machine learning where, in contrast to supervised learning, algorithms learn patterns exclusively from unlabeled data. Other frameworks in the spectrum of supervisions include weak- or semi-supervision, where a small portion of the data is tagged, and self-supervision. Some researchers consider self-supervised learning a form of unsupervised learning.
In statistics, an expectation–maximization (EM) algorithm is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the E step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step. It can be used, for example, to estimate a mixture of gaussians, or to solve the multiple linear regression problem.
The information bottleneck method is a technique in information theory introduced by Naftali Tishby, Fernando C. Pereira, and William Bialek. It is designed for finding the best tradeoff between accuracy and complexity (compression) when summarizing a random variable X, given a joint probability distribution p(X,Y) between X and an observed relevant variable Y - and self-described as providing "a surprisingly rich framework for discussing a variety of problems in signal processing and learning".
In mathematical statistics, the Kullback–Leibler (KL) divergence, denoted , is a type of statistical distance: a measure of how much a model probability distribution Q is different from a true probability distribution P. Mathematically, it is defined as
In probability and statistics, the Dirichlet distribution, often denoted , is a family of continuous multivariate probability distributions parameterized by a vector of positive reals. It is a multivariate generalization of the beta distribution, hence its alternative name of multivariate beta distribution (MBD). Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution.
A Boltzmann machine, named after Ludwig Boltzmann is a spin-glass model with an external field, i.e., a Sherrington–Kirkpatrick model, that is a stochastic Ising model. It is a statistical physics technique applied in the context of cognitive science. It is also classified as a Markov random field.
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 as well as unknown parameters and latent variables, with various sorts of relationships among the three types of random variables, as might be described by a graphical model. As typical in Bayesian inference, the parameters and latent variables are grouped together as "unobserved variables". Variational Bayesian methods are primarily used for two purposes:
- To provide an analytical approximation to the posterior probability of the unobserved variables, in order to do statistical inference over these variables.
- To derive a lower bound for the marginal likelihood of the observed data. This is typically used for performing model selection, the general idea being that a higher marginal likelihood for a given model indicates a better fit of the data by that model and hence a greater probability that the model in question was the one that generated the data.
In probability theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite support of non-negative integers. It is also called the Dirichlet compound multinomial distribution (DCM) or multivariate Pólya distribution. It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector , and an observation drawn from a multinomial distribution with probability vector p and number of trials n. The Dirichlet parameter vector captures the prior belief about the situation and can be seen as a pseudocount: observations of each outcome that occur before the actual data is collected. The compounding corresponds to a Pólya urn scheme. It is frequently encountered in Bayesian statistics, machine learning, empirical Bayes methods and classical statistics as an overdispersed multinomial distribution.
The auxiliary particle filter is a particle filtering algorithm introduced by Pitt and Shephard in 1999 to improve some deficiencies of the sequential importance resampling (SIR) algorithm when dealing with tailed observation densities.
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1.
In probability theory and statistics, the noncentral beta distribution is a continuous probability distribution that is a noncentral generalization of the (central) beta distribution.
A restricted Boltzmann machine (RBM) is a generative stochastic artificial neural network that can learn a probability distribution over its set of inputs.
Probalign is a sequence alignment tool that calculates a maximum expected accuracy alignment using partition function posterior probabilities. Base pair probabilities are estimated using an estimate similar to Boltzmann distribution. The partition function is calculated using a dynamic programming approach.
t-distributed stochastic neighbor embedding (t-SNE) is a statistical method for visualizing high-dimensional data by giving each datapoint a location in a two or three-dimensional map. It is based on Stochastic Neighbor Embedding originally developed by Geoffrey Hinton and Sam Roweis, where Laurens van der Maaten and Hinton proposed the t-distributed variant. It is a nonlinear dimensionality reduction technique for embedding high-dimensional data for visualization in a low-dimensional space of two or three dimensions. Specifically, it models each high-dimensional object by a two- or three-dimensional point in such a way that similar objects are modeled by nearby points and dissimilar objects are modeled by distant points with high probability.
In machine learning, a deep belief network (DBN) is a generative graphical model, or alternatively a class of deep neural network, composed of multiple layers of latent variables, with connections between the layers but not between units within each layer.
Mixture of experts (MoE) is a machine learning technique where multiple expert networks (learners) are used to divide a problem space into homogeneous regions. MoE represents a form of ensemble learning.
An energy-based model (EBM) is an application of canonical ensemble formulation from statistical physics for learning from data. The approach prominently appears in generative artificial intelligence.
