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Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability".
A Bayesian network is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.
In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a sample of the desired distribution by recording states from the chain. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Various algorithms exist for constructing chains, including the Metropolis–Hastings algorithm.
Minimum description length (MDL) is a model selection principle where the shortest description of the data is the best model. MDL methods learn through a data compression perspective and are sometimes described as mathematical applications of Occam's razor. The MDL principle can be extended to other forms of inductive inference and learning, for example to estimation and sequential prediction, without explicitly identifying a single model of the data.
A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning.
In the study of complex networks, a network is said to have community structure if the nodes of the network can be easily grouped into sets of nodes such that each set of nodes is densely connected internally. In the particular case of non-overlapping community finding, this implies that the network divides naturally into groups of nodes with dense connections internally and sparser connections between groups. But overlapping communities are also allowed. The more general definition is based on the principle that pairs of nodes are more likely to be connected if they are both members of the same community(ies), and less likely to be connected if they do not share communities. A related but different problem is community search, where the goal is to find a community that a certain vertex belongs to.
In statistics and machine learning, ensemble methods use multiple learning algorithms to obtain better predictive performance than could be obtained from any of the constituent learning algorithms alone. Unlike a statistical ensemble in statistical mechanics, which is usually infinite, a machine learning ensemble consists of only a concrete finite set of alternative models, but typically allows for much more flexible structure to exist among those alternatives.
A constrained conditional model (CCM) is a machine learning and inference framework that augments the learning of conditional models with declarative constraints. The constraint can be used as a way to incorporate expressive prior knowledge into the model and bias the assignments made by the learned model to satisfy these constraints. The framework can be used to support decisions in an expressive output space while maintaining modularity and tractability of training and inference.
Probabilistic programming (PP) is a programming paradigm in which probabilistic models are specified and inference for these models is performed automatically. It represents an attempt to unify probabilistic modeling and traditional general purpose programming in order to make the former easier and more widely applicable. It can be used to create systems that help make decisions in the face of uncertainty.
A Bayesian Confidence Propagation Neural Network (BCPNN) is an artificial neural network inspired by Bayes' theorem, which regards neural computation and processing as probabilistic inference. Neural unit activations represent probability ("confidence") in the presence of input features or categories, synaptic weights are based on estimated correlations and the spread of activation corresponds to calculating posterior probabilities. It was originally proposed by Anders Lansner and Örjan Ekeberg at KTH Royal Institute of Technology. This probabilistic neural network model can also be run in generative mode to produce spontaneous activations and temporal sequences.
Inductive programming (IP) is a special area of automatic programming, covering research from artificial intelligence and programming, which addresses learning of typically declarative and often recursive programs from incomplete specifications, such as input/output examples or constraints.
Stan is a probabilistic programming language for statistical inference written in C++. The Stan language is used to specify a (Bayesian) statistical model with an imperative program calculating the log probability density function.
Quantum machine learning is the integration of quantum algorithms within machine learning programs. The most common use of the term refers to machine learning algorithms for the analysis of classical data executed on a quantum computer, i.e. quantum-enhanced machine learning. While machine learning algorithms are used to compute immense quantities of data, quantum machine learning utilizes qubits and quantum operations or specialized quantum systems to improve computational speed and data storage done by algorithms in a program. This includes hybrid methods that involve both classical and quantum processing, where computationally difficult subroutines are outsourced to a quantum device. These routines can be more complex in nature and executed faster on a quantum computer. Furthermore, quantum algorithms can be used to analyze quantum states instead of classical data. Beyond quantum computing, the term "quantum machine learning" is also associated with classical machine learning methods applied to data generated from quantum experiments, such as learning the phase transitions of a quantum system or creating new quantum experiments. Quantum machine learning also extends to a branch of research that explores methodological and structural similarities between certain physical systems and learning systems, in particular neural networks. For example, some mathematical and numerical techniques from quantum physics are applicable to classical deep learning and vice versa. Furthermore, researchers investigate more abstract notions of learning theory with respect to quantum information, sometimes referred to as "quantum learning theory".
PyMC is a Python package for Bayesian statistical modeling and probabilistic machine learning which focuses on advanced Markov chain Monte Carlo and variational fitting algorithms. It is a rewrite from scratch of the previous version of the PyMC software. Unlike PyMC2, which had used Fortran extensions for performing computations, PyMC relies on Aesara, a Python library that allows to define, optimize, and efficiently evaluate mathematical expressions involving multi-dimensional arrays. From version 3.8 PyMC relies on ArviZ to handle plotting, diagnostics, and statistical checks. PyMC and Stan are the two most popular probabilistic programming tools. PyMC is an open source project, developed by the community and fiscally sponsored by NumFOCUS.
In machine learning, hyperparameter optimization or tuning is the problem of choosing a set of optimal hyperparameters for a learning algorithm. A hyperparameter is a parameter whose value is used to control the learning process. By contrast, the values of other parameters are learned.
In machine learning, a variational autoencoder (VAE), is an artificial neural network architecture introduced by Diederik P. Kingma and Max Welling, belonging to the families of probabilistic graphical models and variational Bayesian methods.
ArviZ is a Python package for exploratory analysis of Bayesian models.
An energy-based model (EBM) is a form of generative model (GM) imported directly from statistical physics to learning. GMs learn an underlying data distribution by analyzing a sample dataset. Once trained, a GM can produce other datasets that also match the data distribution. EBMs provide a unified framework for many probabilistic and non-probabilistic approaches to such learning, particularly for training graphical and other structured models.
Probabilistic numerics is a scientific field at the intersection of statistics, machine learning and applied mathematics, where tasks in numerical analysis including finding numerical solutions for integration, linear algebra, optimisation and differential equations are seen as problems of statistical, probabilistic, or Bayesian inference.
Bayesian quadrature is a numerical method for solving numerical integration problems which falls within the class of 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.
References
- ↑ Saad, Feras A.; Cusumano-Towner, Marco F.; Schaechtle, Ulrich; Rinard, Martin C.; Mansinghka, Vikash K. (January 2019). "Bayesian Synthesis of Probabilistic Programs for Automatic Data Modeling". Proc. ACM Program. Lang. 3 (POPL): 37:1–37:32. arXiv: 1907.06249 . Bibcode:2019arXiv190706249S. doi:10.1145/3290350. ISSN 2475-1421. S2CID 53697125.
- ↑ "Talking Machines: Probabilistic programming, with Ben Vigoda | Robohub". robohub.org. Retrieved 2017-03-04.
- ↑ Lake, Brenden M.; Salakhutdinov, Ruslan; Tenenbaum, Joshua B. (2015-12-11). "Human-level concept learning through probabilistic program induction". Science. 350 (6266): 1332–1338. Bibcode:2015Sci...350.1332L. doi: 10.1126/science.aab3050 . ISSN 0036-8075. PMID 26659050.
