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Bayesian Quadrature
Bayesian quadrature is a method for approximating intractable integration problems. It falls within the class of probabilistic numerics, probabilistic numerical methods. Bayesian quadrature views numerical integration as a Bayesian inference task, where function evaluations are used to estimate the integral of that function. For this reason, it is sometimes also referred to as "Bayesian probabilistic numerical integration" or "Bayesian numerical integration". The name "Bayesian cubature" is also sometimes used when the integrand is multi-dimensional. A potential advantage of this approach is that it provides probabilistic uncertainty quantification for the value of the integral. Bayesian quadrature Numerical integration Let f:\mathcal \rightarrow \mathbb be a function defined on a domain \mathcal (where typically \mathcal\subseteq \mathbb^d). In numerical integration, function evaluations f(x_1), \ldots, f(x_n) at distinct locations x_1, \ldots, x_n in \mathcal are used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Quadrature Animation
Thomas Bayes ( ; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister. Bayesian ( or ) may be either any of a range of concepts and approaches that relate to statistical methods based on Bayes' theorem Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes) gives a mathematical rule for inverting Conditional probability, conditional probabilities, allowing one to find the probability of a cause given its effect. For exampl ..., or a follower of these methods. Bayes * * * * * * * * * * * – sometimes called ''Bayes' rule'' or ''Bayesian updating'' * * * * * * Bayesian *'' Bayesian'', a superyacht sunk off Palermo in 2024 * * * * * * * * * * * * * * * * * * (BIC) * Widely applicable Bayesian information criterion (WBIC) * * * * * * (BMA) * (BMC) * * * * * * * (BAYOMA) * * * * * * * * * * * * * * * * * * * * * * (PBE) * * * * * See also * * * *, a cryptanalytic process * * * * (BUGS) * * (BATMAN) * * * *, a genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Active Learning (machine Learning)
Active learning is a special case of machine learning in which a learning algorithm can interactively query a human user (or some other information source), to label new data points with the desired outputs. The human user must possess knowledge/expertise in the problem domain, including the ability to consult/research authoritative sources when necessary. In statistics literature, it is sometimes also called optimal experimental design. The information source is also called ''teacher'' or ''oracle''. There are situations in which unlabeled data is abundant but manual labeling is expensive. In such a scenario, learning algorithms can actively query the user/teacher for labels. This type of iterative supervised learning is called active learning. Since the learner chooses the examples, the number of examples to learn a concept can often be much lower than the number required in normal supervised learning. With this approach, there is a risk that the algorithm is overwhelmed by un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Monte Carlo Method
In numerical analysis, the quasi-Monte Carlo method is a method for numerical integration and solving some other problems using low-discrepancy sequences (also called quasi-random sequences or sub-random sequences) to achieve variance reduction. This is in contrast to the regular Monte Carlo method or Monte Carlo integration, which are based on sequences of pseudorandom numbers. Monte Carlo and quasi-Monte Carlo methods are stated in a similar way. The problem is to approximate the integral of a function ''f'' as the average of the function evaluated at a set of points ''x''1, ..., ''x''''N'': : \int_ f(u)\,u \approx \frac\,\sum_^N f(x_i). Since we are integrating over the ''s''-dimensional unit cube, each ''x''''i'' is a vector of ''s'' elements. The difference between quasi-Monte Carlo and Monte Carlo is the way the ''x''''i'' are chosen. Quasi-Monte Carlo uses a low-discrepancy sequence such as the Halton sequence, the Sobol sequence, or the Faure sequence, whereas Mont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Integration
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals. There are different methods to perform a Monte Carlo integration, such as uniform sampling, stratified sampling, importance sampling, sequential Monte Carlo (also known as a particle filter), and mean-field particle methods. Overview In numerical integration, methods such as the trapezoidal rule use a deterministic approach. Monte Carlo integration, on the other hand, employs a non-deterministic approach: each realization provides a different outcome. In Monte Carlo, the final outcome is an approximation of the correct value with respective error bars, and the correct value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Process
In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. It is often used in Bayesian inference to describe the prior knowledge about the distribution of random variables—how likely it is that the random variables are distributed according to one or another particular distribution. As an example, a bag of 100 real-world dice is a ''random probability mass function (random pmf)''—to sample this random pmf you put your hand in the bag and draw out a die, that is, you draw a pmf. A bag of dice manufactured using a crude process 100 years ago will likely have probabilities that deviate wildly from the uniform pmf, whereas a bag of state-of-the-art dice used by Las Vegas casinos may have barely perceptible imperfe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel Methods For Vector Output
Kernel methods are a well-established tool to analyze the relationship between input data and the corresponding output of a function. Kernels encapsulate the properties of functions in a Kernel trick, computationally efficient way and allow algorithms to easily swap functions of varying complexity. In typical machine learning algorithms, these functions produce a scalar output. Recent development of kernel methods for functions with vector-valued output is due, at least in part, to interest in simultaneously solving related problems. Kernels which capture the relationship between the problems allow them to ''borrow strength'' from each other. Algorithms of this type include multi-task learning (also called multi-output learning or vector-valued learning), Inductive transfer, transfer learning, and co-kriging. Multi-label classification can be interpreted as mapping inputs to (binary) coding vectors with length equal to the number of classes. In Gaussian processes, kernels are called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel Embedding Of Distributions
In machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS).A. Smola, A. Gretton, L. Song, B. Schölkopf. (2007)A Hilbert Space Embedding for Distributions. ''Algorithmic Learning Theory: 18th International Conference''. Springer: 13–31. A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis.L. Song, K. Fukumizu, F. Dinuzzo, A. Gretton (2013)Kernel Embeddings of Conditional Distributions: A unified kernel framework for n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral :\int_a^b f(x) \, dx to a given degree of accuracy. If is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure ('' quadrature'' or ''squaring''), as in the quadrature of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matérn Covariance Function
In statistics, the Matérn covariance, also called the Matérn kernel, is a covariance function used in spatial statistics, geostatistics, machine learning, image analysis, and other applications of multivariate statistical analysis on metric spaces. It is named after the Swedish forestry statistician Bertil Matérn. It specifies the covariance between two measurements as a function of the distance d between the points at which they are taken. Since the covariance only depends on distances between points, it is stationary. If the distance is Euclidean distance, the Matérn covariance is also isotropic. Definition The Matérn covariance between measurements taken at two points separated by ''d'' distance units is given by Rasmussen, Carl Edward and Williams, Christopher K. I. (2006Gaussian Processes for Machine Learning/ref> : C_\nu(d) = \sigma^2\frac^\nu K_\nu\Bigg(\sqrt\frac\Bigg), where \Gamma is the gamma function, K_\nu is the modified Bessel function of the second kind, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adaptive Quadrature
Adaptive quadrature is a numerical integration method in which the integral of a function f(x) is approximated using static quadrature rules on adaptively refined subintervals of the region of integration. Generally, adaptive algorithms are just as efficient and effective as traditional algorithms for "well behaved" integrands, but are also effective for "badly behaved" integrands for which traditional algorithms may fail. General scheme Adaptive quadrature follows the general scheme 1. procedure integrate ( f, a, b, τ ) 2. Q \approx \int_a^b f(x)\,\mathrmx 3. \varepsilon \approx \left, Q - \int_a^b f(x)\,\mathrmx\ 4. if ''ε'' > ''τ'' then 5. m = (a + b) / 2 6. Q = integrate(f, a, m, τ/2) + integrate(f, m, b, τ/2) 7. endif 8. return Q An approximation Q to the integral of f(x) over the interval ,b/math> is computed (line 2), as well as an error estimate \varepsilon (line 3). If the estimated error is larger than the required tol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic Numerics
Probabilistic numerics is aactivefield of study at the intersection of applied mathematics, statistics, and machine learning centering on the concept of uncertainty in computation. In probabilistic numerics, tasks in numerical analysis such as finding numerical solutions for integration, linear algebra, optimization and simulation and differential equations are seen as problems of statistical, probabilistic, or Bayesian inference. Introduction A numerical method is an algorithm that ''approximates'' the solution to a mathematical problem (examples below include the solution to a linear system of equations, the value of an integral, the solution of a differential equation, the minimum of a multivariate function). In a ''probabilistic'' numerical algorithm, this process of approximation is thought of as a problem of ''estimation'', ''inference'' or ''learning'' and realised in the framework of probabilistic inference (often, but not always, Bayesian inference). Formally, this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
