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Multinomial Naive Bayes
In statistics, naive Bayes classifiers are a family of simple " probabilistic classifiers" based on applying Bayes' theorem with strong (naive) independence assumptions between the features (see Bayes classifier). They are among the simplest Bayesian network models, but coupled with kernel density estimation, they can achieve high accuracy levels. Naive Bayes classifiers are highly scalable, requiring a number of parameters linear in the number of variables (features/predictors) in a learning problem. Maximum-likelihood training can be done by evaluating a closed-form expression, which takes linear time, rather than by expensive iterative approximation as used for many other types of classifiers. In the statistics literature, naive Bayes models are known under a variety of names, including simple Bayes and independence Bayes. All these names reference the use of Bayes' theorem in the classifier's decision rule, but naive Bayes is not (necessarily) a Bayesian method. Introdu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Efficacy
Efficacy is the ability to perform a task to a satisfactory or expected degree. The word comes from the same roots as ''effectiveness'', and it has often been used synonymously, although in pharmacology a pragmatic clinical trial#Efficacy versus effectiveness, distinction is now often made between efficacy and effectiveness. The word ''efficacy'' is used in pharmacology and medicine to refer both to the maximum response achievable from a pharmaceutical drug in research settings, and to the capacity for sufficient therapeutic effect or beneficial change in clinical settings. Pharmacology In pharmacology, efficacy () is the maximum response achievable from an applied or dosed agent, for instance, a small molecule drug. Intrinsic activity is a relative term for a drug's efficacy relative to a drug with the highest observed efficacy. It is a purely descriptive term that has little or no mechanistic interpretation. In order for a drug to have an effect, it needs to bind to its t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum A Posteriori
In Bayesian statistics, a maximum a posteriori probability (MAP) estimate is an estimate of an unknown quantity, that equals the mode of the posterior distribution. The MAP can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. It is closely related to the method of maximum likelihood (ML) estimation, but employs an augmented optimization objective which incorporates a prior distribution (that quantifies the additional information available through prior knowledge of a related event) over the quantity one wants to estimate. MAP estimation can therefore be seen as a regularization of maximum likelihood estimation. Description Assume that we want to estimate an unobserved population parameter \theta on the basis of observations x. Let f be the sampling distribution of x, so that f(x\mid\theta) is the probability of x when the underlying population parameter is \theta. Then the function: :\theta \mapsto f(x \mid \theta) \! is known as th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Rule
In decision theory, a decision rule is a function which maps an observation to an appropriate action. Decision rules play an important role in the theory of statistics and economics, and are closely related to the concept of a strategy (game theory), strategy in game theory. In order to evaluate the usefulness of a decision rule, it is necessary to have a loss function detailing the outcome of each action under different states. Formal definition Given an observable random variable ''X'' over the probability space \scriptstyle (\mathcal,\Sigma, P_\theta), determined by a parameter ''θ'' ∈ ''Θ'', and a set ''A'' of possible actions, a (deterministic) decision rule is a function ''δ'' : \scriptstyle\mathcal→ ''A''. Examples of decision rules * An estimator is a decision rule used for estimating a parameter. In this case the set of actions is the parameter space, and a loss function details the cost of the discrepancy between the true value of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Classification
In statistics, classification is the problem of identifying which of a set of categories (sub-populations) an observation (or observations) belongs to. Examples are assigning a given email to the "spam" or "non-spam" class, and assigning a diagnosis to a given patient based on observed characteristics of the patient (sex, blood pressure, presence or absence of certain symptoms, etc.). Often, the individual observations are analyzed into a set of quantifiable properties, known variously as explanatory variables or ''features''. These properties may variously be categorical (e.g. "A", "B", "AB" or "O", for blood type), ordinal (e.g. "large", "medium" or "small"), integer-valued (e.g. the number of occurrences of a particular word in an email) or real-valued (e.g. a measurement of blood pressure). Other classifiers work by comparing observations to previous observations by means of a similarity or distance function. An algorithm that implements classification, especially in a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, the data-generating process. A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" ( Herman Adèr quoting Kenneth Bollen). All statistical hypothesis tests and all statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference. Introduction Informally, a statistical model can be thought of as a statistical assumption (or set of statistical assumptions) with a certain property: that the assumption allows us to calculate the probability of any event. As an example, consider a pair of ordinary six-sid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proportionality (mathematics)
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constant. Two sequences are inversely proportional if corresponding elements have a constant product, also called the coefficient of proportionality. This definition is commonly extended to related varying quantities, which are often called ''variables''. This meaning of ''variable'' is not the common meaning of the term in mathematics (see variable (mathematics)); these two different concepts share the same name for historical reasons. Two functions f(x) and g(x) are ''proportional'' if their ratio \frac is a constant function. If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., (for details see Ratio). Proportionality is closely rela ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutually Independent
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other. When dealing with collections of more than two events, two notions of independence need to be distinguished. The events are called pairwise independent if any two events in the collection are independent of each other, while mutual independence (or collective independence) of events means, informally speaking, that each event is independent of any combination of other events in the collection. A similar notion exists for collections of random variables. Mutual independence implies pairwise independence, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Independence
In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of conditional probability, as a special case where the probability of the hypothesis given the uninformative observation is equal to the probability without. If A is the hypothesis, and B and C are observations, conditional independence can be stated as an equality: :P(A\mid B,C) = P(A \mid C) where P(A \mid B, C) is the probability of A given both B and C. Since the probability of A given C is the same as the probability of A given both B and C, this equality expresses that B contributes nothing to the certainty of A. In this case, A and B are said to be conditionally independent given C, written symbolically as: (A \perp\!\!\!\perp B \mid C). The concept of conditional independence is essential to graph-based theories of statistical inference, as it establ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain Rule (probability)
In probability theory, the chain rule (also called the general product rule) permits the calculation of any member of the joint distribution of a set of random variables using only conditional probabilities. The rule is useful in the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities. Chain rule for events Two events The chain rule for two random events A and B says P(A \cap B) = P(B \mid A) \cdot P(A). Example This rule is illustrated in the following example. Urn 1 has 1 black ball and 2 white balls and Urn 2 has 1 black ball and 3 white balls. Suppose we pick an urn at random and then select a ball from that urn. Let event A be choosing the first urn: P(A) = P(\overline) = 1/2. Let event B be the chance we choose a white ball. The chance of choosing a white ball, given that we have chosen the first urn, is P(B, A) = 2/3. Event A \cap B would be their intersection: choosing the first urn and a white ball from it. The pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joint Probability
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions, i.e. the distributions of each of the individual random variables. It also encodes the conditional probability distributions, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s). In the formal mathematical setup of measure theory, the joint distribution is given by the pushforward measure, by the map obtained by pairing together the given random variables, of the sample space's probability measure. In the case of real-valued random variables, the joint distribution, as a particular multivariate distribution, may be expressed by a multivariate cumu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Probability Table
In statistics, the conditional probability table (CPT) is defined for a set of discrete and mutually dependent random variables to display conditional probabilities of a single variable with respect to the others (i.e., the probability of each possible value of one variable if we know the values taken on by the other variables). For example, assume there are three random variables x_1,x_2, x_3 where each has K states. Then, the conditional probability table of x_1 provides the conditional probability values P(x_1=a_k\mid x_2,x_3) – where the vertical bar , means “given the values of” – for each of the ''K'' possible values a_k of the variable x_1 and for each possible combination of values of x_2,\, x_3. This table has K^3 cells. In general, for M variables x_1,x_2,\ldots,x_M with K_i states for each variable x_i, the CPT for any one of them has the number of cells equal to the product K_1K_2\cdots K_M. A conditional probability table can be put into matrix form. As an exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
