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Pseudocount
In statistics, additive smoothing, also called Laplace smoothing or Lidstone smoothing, is a technique used to smooth count data, eliminating issues caused by certain values having 0 occurrences. Given a set of observation counts \mathbf = \langle x_1, x_2, \ldots, x_d \rangle from a d-dimensional multinomial distribution with N trials, a "smoothed" version of the counts gives the estimator : \hat\theta_i = \frac \qquad (i = 1, \ldots, d), where the smoothed count \hat x_i = N \hat\theta_i, and the "pseudocount" ''α'' > 0 is a smoothing parameter, with ''α'' = 0 corresponding to no smoothing (this parameter is explained in below). Additive smoothing is a type of shrinkage estimator, as the resulting estimate will be between the empirical probability ( relative frequency) x_i/N and the uniform probability 1/d. Common choices for ''α'' are 0 (no smoothing), (the Jeffreys prior), or 1 (Laplace's rule of succession), but the parameter may also be set empir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "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. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive Statistical parameter, parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as exponents of the variable and its complement to 1, respectively, and control the shape parameter, shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior of percentages and proportions. In Bayesian inference, the beta distribution is the conjugate prior distribution, conjugate prior probability distribution for the Bernoulli distribution, Bernoulli, binomial distribution, binomial, negative binomial distribution, negative binomial, and geometric distribution, geometric distributions. The formulation of the beta dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Indifference
The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities. The principle of indifference states that in the absence of any relevant evidence, agents should distribute their credence (or "degrees of belief") equally among all the possible outcomes under consideration. In Bayesian probability, this is the simplest non-informative prior. Examples The textbook examples for the application of the principle of indifference are coins, dice, and cards. In a macroscopic system, at least, it must be assumed that the physical laws that govern the system are not known well enough to predict the outcome. As observed some centuries ago by John Arbuthnot (in the preface of ''Of the Laws of Chance'', 1692), :It is impossible for a Die, with such determin'd force and direction, not to fall on such determin'd side, only I don't know the force and direction which makes it fall on such determin'd side, and therefore I call it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cromwell's Rule
Cromwell's rule, named by statistician Dennis Lindley, states that the use of prior probabilities of 1 ("the event will definitely occur") or 0 ("the event will definitely not occur") should be avoided, except when applied to statements that are logically true or false, such as equaling 4. The reference is to Oliver Cromwell, who wrote to the General Assembly of the Church of Scotland on 3 August 1650, shortly before the Battle of Dunbar, including a phrase that has become well known and frequently quoted: As Lindley puts it, assigning a probability should "leave a little probability for the moon being made of green cheese; it can be as small as 1 in a million, but have it there since otherwise an army of astronauts returning with samples of the said cheese will leave you unmoved". Similarly, in assessing the likelihood that tossing a coin will result in either a head or a tail facing upwards, there is a possibility, albeit remote, that the coin will land on its edge a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PPM Compression Algorithm
Prediction by partial matching (PPM) is an adaptive statistical data compression technique based on context modeling and prediction. PPM models use a set of previous symbols in the uncompressed symbol stream to predict the next symbol in the stream. PPM algorithms can also be used to cluster data into predicted groupings in cluster analysis. Theory Predictions are usually reduced to symbol rankings. Each symbol (a letter, bit or any other amount of data) is ranked before it is compressed, and the ranking system determines the corresponding codeword (and therefore the compression rate). In many compression algorithms, the ranking is equivalent to probability mass function estimation. Given the previous letters (or given a context), each symbol is assigned with a probability. For instance, in arithmetic coding the symbols are ranked by their probabilities to appear after previous symbols, and the whole sequence is compressed into a single fraction that is computed according to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hidden Markov Model
A hidden Markov model (HMM) is a Markov model in which the observations are dependent on a latent (or ''hidden'') Markov process (referred to as X). An HMM requires that there be an observable process Y whose outcomes depend on the outcomes of X in a known way. Since X cannot be observed directly, the goal is to learn about state of X by observing Y. By definition of being a Markov model, an HMM has an additional requirement that the outcome of Y at time t = t_0 must be "influenced" exclusively by the outcome of X at t = t_0 and that the outcomes of X and Y at t < t_0 must be conditionally independent of at given at time . Estimation of the parameters in an HMM can be performed using maximum likelihood estimation. For linear chain HMMs, the Baum–Welch algorithm can be used to estimate parameters. Hidden Markov models are known for their applications to thermodynamics, statistical mechanics, physics, chem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artificial Neural Network
In machine learning, a neural network (also artificial neural network or neural net, abbreviated ANN or NN) is a computational model inspired by the structure and functions of biological neural networks. A neural network consists of connected units or nodes called '' artificial neurons'', which loosely model the neurons in the brain. Artificial neuron models that mimic biological neurons more closely have also been recently investigated and shown to significantly improve performance. These are connected by ''edges'', which model the synapses in the brain. Each artificial neuron receives signals from connected neurons, then processes them and sends a signal to other connected neurons. The "signal" is a real number, and the output of each neuron is computed by some non-linear function of the sum of its inputs, called the '' activation function''. The strength of the signal at each connection is determined by a ''weight'', which adjusts during the learning process. Typically, ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Machine Learning
Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of Computational statistics, statistical algorithms that can learn from data and generalise to unseen data, and thus perform Task (computing), tasks without explicit Machine code, instructions. Within a subdiscipline in machine learning, advances in the field of deep learning have allowed Neural network (machine learning), neural networks, a class of statistical algorithms, to surpass many previous machine learning approaches in performance. ML finds application in many fields, including natural language processing, computer vision, speech recognition, email filtering, agriculture, and medicine. The application of ML to business problems is known as predictive analytics. Statistics and mathematical optimisation (mathematical programming) methods comprise the foundations of machine learning. Data mining is a related field of study, focusing on exploratory data analysi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Event (probability Theory)
In probability theory, an event is a subset of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event consisting of only a single outcome is called an or an ; that is, it is a singleton set. An event that has more than one possible outcome is called a compound event. An event S is said to if S contains the outcome x of the experiment (or trial) (that is, if x \in S). The probability (with respect to some probability measure) that an event S occurs is the probability that S contains the outcome x of an experiment (that is, it is the probability that x \in S). An event defines a complementary event, namely the complementary set (the event occurring), and together these define a Bernoulli trial: did the event occur or not? Typically, when the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sample (statistics)
In this statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample (termed sample for short) of individuals from within a population (statistics), statistical population to estimate characteristics of the whole population. The subset is meant to reflect the whole population, and statisticians attempt to collect samples that are representative of the population. Sampling has lower costs and faster data collection compared to recording data from the entire population (in many cases, collecting the whole population is impossible, like getting sizes of all stars in the universe), and thus, it can provide insights in cases where it is infeasible to measure an entire population. Each observation measures one or more properties (such as weight, location, colour or mass) of independent objects or individuals. In survey sampling, weights can be applied to the data to adjust for the sample design, particularly in stratified samplin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halting Problem
In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. The halting problem is ''Undecidable problem, undecidable'', meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs. The problem comes up often in discussions of computability since it demonstrates that some functions are mathematically Definable set, definable but not Computable function, computable. A key part of the formal statement of the problem is a mathematical definition of a computer and program, usually via a Turing machine. The proof then shows, for any program that might determine whether programs halt, that a "pathological" program exists for which makes an incorrect determination. Specifically, is the program that, when called with some input, passes its own s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model (abstract)
The term conceptual model refers to any model that is formed after a wikt:concept#Noun, conceptualization or generalization process. Conceptual models are often abstractions of things in the real world, whether physical or social. Semantics, Semantic studies are relevant to various stages of process of concept formation, concept formation. Semantics is fundamentally a study of concepts, the meaning that thinking beings give to various elements of their experience. Overview Concept models and conceptual models The value of a conceptual model is usually directly proportional to how well it corresponds to a past, present, future, actual or potential state of affairs. A concept model (a model of a concept) is quite different because in order to be a good model it need not have this real world correspondence. In artificial intelligence, conceptual models and conceptual graphs are used for building expert systems and knowledge-based systems; here the analysts are concerned to repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
