|
Krichevsky–Trofimov Estimator
In information theory, given an unknown stationary source with alphabet ''A'' and a sample ''w'' from , the Krichevsky–Trofimov (KT) estimator produces an estimate ''p''''i''(''w'') of the probability of each symbol ''i'' ∈ ''A''. This estimator is optimal in the sense that it minimizes the worst-case regret asymptotically. For a binary alphabet and a string ''w'' with ''m'' zeroes and ''n'' ones, the KT estimator ''p''''i''(''w'') is defined as: : \begin p_0(w) &= \frac, \\ pt p_1(w) &= \frac. \end See also * Rule of succession * Dirichlet-multinomial distribution 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 distribu ... References Information theory Data compression {{probability-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. A key measure in information theory is entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a die (with six equally likely outcomes). Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. Important sub-fields of information theory include s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stationary Process
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance also do not change over time. If you draw a line through the middle of a stationary process then it should be flat; it may have 'seasonal' cycles, but overall it does not trend up nor down. Since stationarity is an assumption underlying many statistical procedures used in time series analysis, non-stationary data are often transformed to become stationary. The most common cause of violation of stationarity is a trend in the mean, which can be due either to the presence of a unit root or of a deterministic trend. In the former case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting. In the latter case of a deterministic trend, the process is cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regret (decision Theory)
In decision theory, on making decisions under uncertainty—should information about the best course of action arrive ''after'' taking a fixed decision—the human emotional response of regret is often experienced, and can be measured as the value of difference between a made decision and the optimal decision. The theory of regret aversion or anticipated regret proposes that when facing a decision, individuals might ''anticipate'' regret and thus incorporate in their choice their desire to eliminate or reduce this possibility. Regret is a negative emotion with a powerful social and reputational component, and is central to how humans learn from experience and to the human psychology of risk aversion. Conscious anticipation of regret creates a feedback loop that transcends regret from the emotional realm—often modeled as mere human behavior—into the realm of the rational choice behavior that is modeled in decision theory. Description Regret theory is a model in theoretical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule Of Succession
In probability theory, the rule of succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem. The formula is still used, particularly to estimate underlying probabilities when there are few observations or for events that have not been observed to occur at all in (finite) sample data. Statement of the rule of succession If we repeat an experiment that we know can result in a success or failure, ''n'' times independently, and get ''s'' successes, and ''n − s'' failures, then what is the probability that the next repetition will succeed? More abstractly: If ''X''1, ..., ''X''''n''+1 are conditionally independent random variables that each can assume the value 0 or 1, then, if we know nothing more about them, :P(X_=1 \mid X_1+\cdots+X_n=s)=. Interpretation Since we have the prior knowledge that we are looking at an experiment for which both success and failure are possible, our estimate is as if we had obs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet-multinomial Distribution
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 (after George Pólya). It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector \boldsymbol, 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. It reduces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. A key measure in information theory is entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a die (with six equally likely outcomes). Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. Important sub-fields of information theory include s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
