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Using the distribution semantics, a probability distribution is defined over the two-valued well-founded models of the atoms in the program. The probability of a model is defined as P ( M ) = ∏ l ∈ M P ( l ) {\displaystyle P(M)=\prod _{l\in M}P(l)} where the product runs over all the literals in the model M {\displaystyle M} .
The softmax function, also known as softargmax [1]: 184 or normalized exponential function, [2]: 198 converts a vector of K real numbers into a probability distribution of K possible outcomes. It is a generalization of the logistic function to multiple dimensions, and is used in multinomial logistic regression .
The Birnbaum–Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times. The chi distribution. The noncentral chi distribution; The chi-squared distribution, which is the sum of the squares of n independent Gaussian random variables.
A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss, a roll of a die) and the probabilities are encoded by a discrete list of the probabilities of the outcomes; in this case the discrete probability distribution is known as probability mass function.
PPLs often extend from a basic language. For instance, Turing.jl [12] is based on Julia, Infer.NET is based on .NET Framework, [13] while PRISM extends from Prolog. [14] However, some PPLs, such as WinBUGS, offer a self-contained language that maps closely to the mathematical representation of the statistical models, with no obvious origin in another programming language.
The Dirichlet distribution is the conjugate prior distribution of the categorical distribution (a generic discrete probability distribution with a given number of possible outcomes) and multinomial distribution (the distribution over observed counts of each possible category in a set of categorically distributed observations).
For example, if the population is represented by bit strings of length 4, the EDA can represent the population of promising solution using a single vector of four probabilities (p1, p2, p3, p4) where each component of p defines the probability of that position being a 1. Using this probability vector it is possible to create an arbitrary number ...
Consider a statistical application where a user needs to know key percentage points of a given distribution. For example, they require the median and 25% and 75% quartiles as in the example above or 5%, 95%, 2.5%, 97.5% levels for other applications such as assessing the statistical significance of an observation whose distribution is known ...