Meta tags:
Headings (most frequently used words):
likelihood, interpretation, and, likelihoods, probability, distribution, function, ratio, relative, discrete, continuous, example, the, contents, definition, that, eliminate, nuisance, parameters, products, of, log, background, see, also, notes, references, further, reading, external, links, in, general, for, mixed, distributions, regularity, conditions, profile, conditional, marginal, partial, graph, equations, exponential, families, historical, remarks, interpretations, under, different, foundations, relationship, between, density, functions, region, gamma, frequentist, bayesian, likelihoodist, aic, based,
Text of the page (most frequently used words):
the (559), likelihood (267), theta (142), textstyle (134), and (112), #function (94), probability (90), log (83), for (80), that (68), #parameter (59), mid (53), statistics (47), partial (45), displaystyle (44), mathbf (43), with (42), maximum (39), edit (39), this (38), given (38), statistical (37), model (35), beta (34), mathcal (33), data (32), are (31), frac (30), from (29), can (27), text (26), density (26), ratio (26), parameters (26), inference (24), distribution (24), value (24), not (24), isbn (23), bayesian (22), test (22), likelihoods (21), estimation (20), doi (20), left (20), right (20), analysis (19), interval (18), sample (18), university (18), press (18), theorem (17), interpretation (17), which (17), alpha (16), all (15), models (15), sum (15), estimate (15), where (15), relative (15), two (15), boldsymbol (15), information (14), exponential (14), posterior (14), bayes (14), max (14), used (14), one (14), discrete (14), articles (13), population (13), estimator (13), statistic (13), random (13), fisher (13), evidence (13), when (13), see (13), region (13), but (13), hat (13), mathop (13), operatorname (13), arg (13), about (12), may (12), regression (12), confidence (12), based (12), also (12), values (12), then (12), often (12), coin (12), product (11), continuous (11), more (11), being (11), independent (11), each (11), support (11), wikipedia (10), terms (10), using (10), conditional (10), general (10), linear (10), variable (10), family (10), space (10), theory (10), cambridge (10), observed (10), set (10), any (10), have (10), events (10), nuisance (10), conditions (10), observation (10), use (9), correlation (9), series (9), distributions (9), equations (9), prior (9), plot (9), 978 (9), springer (9), fixed (9), has (9), example (9), such (9), than (9), defined (9), event (9), only (9), time (8), tests (8), factor (8), variance (8), new (8), marginal (8), intervals (8), functions (8), measure (8), same (8), difference (8), there (8), eta (8), lim (8), toggle (7), under (7), august (7), original (7), different (7), science (7), vector (7), multivariate (7), score (7), frequentist (7), transformation (7), chart (7), york (7), mathematical (7), journal (7), asymptotic (7), profile (7), section (7), single (7), possible (7), rule (7), logarithm (7), unknown (7), other (7), since (7), called (7), result (7), thus (7), derivative (7), written (7), maximizing (7), above (7), mass (7), non (6), was (6), 2019 (6), contain (6), econometrics (6), specific (6), way (6), aic (6), most (6), approximation (6), size (6), component (6), matrix (6), john (6), 1985 (6), wiley (6), inverse (6), society (6), jstor (6), sufficient (6), likelihoodist (6), help (6), true (6), main (6), term (6), does (6), aligned (6), individual (6), ell (6), these (6), known (6), over (6), interest (6), mathsf (6), int (6), page (5), citations (5), empty (5), research (5), rank (5), first (5), least (5), families (5), simple (5), normality (5), parametric (5), testing (5), point (5), normalization (5), chapman (5), hall (5), oxford (5), part (5), introduction (5), 1922 (5), royal (5), foundations (5), history (5), surface (5), 1080 (5), principle (5), within (5), will (5), points (5), some (5), into (5), while (5), assuming (5), choice (5), learn (5), remove (5), another (5), case (5), because (5), knowledge (5), respect (5), derivatives (5), gamma (5), graph (5), particular (5), definition (5), lambda (5), considered (5), regularity (5), assumed (5), infty (5), constant (5), outcome (5), heads (5), subsection (5), table (4), contents (4), search (4), view (4), additional (4), march (4), 2025 (4), technical (4), index (4), portal (4), design (4), methods (4), squares (4), box (4), components (4), standard (4), mixed (4), coefficient (4), power (4), method (4), order (4), sampling (4), experiment (4), central (4), range (4), links (4), 1997 (4), edwards (4), further (4), entropy (4), tools (4), relation (4), english (4), 1982 (4), note (4), uniqueness (4), robert (4), always (4), sons (4), laplace (4), its (4), samples (4), interpreted (4), context (4), adding (4), instead (4), make (4), they (4), been (4), whose (4), wilks (4), distributed (4), figure (4), how (4), message (4), please (4), improve (4), corresponding (4), interpretations (4), what (4), should (4), proposed (4), basis (4), form (4), between (4), similar (4), background (4), observations (4), rather (4), langle (4), rangle (4), common (4), products (4), almost (4), corresponds (4), without (4), equal (4), viewed (4), article (4), approaches (4), eliminate (4), usually (4), diagnostic (4), quad (4), personal (4), hide (4), move (4), sidebar (4), topic (3), you (3), foundation (3), 2026 (3), expanded (3), style (3), too (3), mathematics (3), social (3), control (3), process (3), medical (3), hazard (3), proportional (3), survival (3), frequency (3), cross (3), equation (3), cluster (3), partition (3), generalized (3), adaptive (3), errors (3), pearson (3), credible (3), median (3), squared (3), prediction (3), unbiased (3), mean (3), moments (3), optimal (3), shape (3), empirical (3), natural (3), study (3), error (3), missing (3), cox (3), scatter (3), bar (3), 2018 (3), paradigm (3), 2014 (3), 1996 (3), hdl (3), 521 (3), 1972 (3), 2013 (3), 2nd (3), eds (3), 1214 (3), ronald (3), 2008 (3), 222 (3), 1921 (3), small (3), curvature (3), biometrika (3), roots (3), 1975 (3), american (3), joint (3), 471 (3), 269 (3), 1995 (3), notes (3), 1971 (3), 1111 (3), 2517 (3), 6161 (3), mountain (3), pass (3), existence (3), references (3), estimates (3), combined (3), large (3), both (3), cases (3), estimated (3), many (3), logarithms (3), several (3), those (3), calculated (3), depends (3), generally (3), quantity (3), four (3), below (3), established (3), hypothesis (3), were (3), fact (3), problem (3), maximization (3), procedure (3), begin (3), ldots (3), end (3), work (3), simply (3), second (3), stationary (3), dimensional (3), exists (3), mathbb (3), equiv (3), gradient (3), negative (3), equivalent (3), key (3), follows (3), probabilities (3), sometimes (3), conditioning (3), properties (3), allow (3), real (3), twice (3), therefore (3), suppose (3), ratios (3), states (3), odds (3), need (3), positive (3), integral (3), leq (3), mathrm (3), fair (3), observing (3), fairness (3), parameterized (3), gives (3), languages (2), contact (2), privacy (2), policy (2), available (2), commons (2), hidden (2), categories (2), lacking (2), april (2), sections (2), issues (2), short (2), description (2), wikidata (2), retrieved (2), system (2), engineering (2), studies (2), clinical (2), hazards (2), limit (2), spectral (2), domain (2), autoregressive (2), structural (2), seasonal (2), adjustment (2), stationarity (2), normal (2), principal (2), manova (2), contingency (2), categorical (2), degrees (2), freedom (2), anova (2), nonparametric (2), effects (2), validation (2), residuals (2), determination (2), moment (2), van (2), alternative (2), hodges (2), lehmann (2), sign (2), selection (2), anderson (2), chi (2), goodness (2), fit (2), wald (2), lagrange (2), multiplier (2), multiple (2), powerful (2), hypotheses (2), bootstrap (2), rao (2), minimum (2), estimators (2), distance (2), estimating (2), efficiency (2), decision (2), location (2), scale (2), designs (2), stochastic (2), trial (2), scientific (2), randomized (2), controlled (2), methodology (2), collection (2), reduction (2), cleaning (2), scaling (2), transform (2), tables (2), dispersion (2), count (2), deviation (2), geometric (2), arithmetic (2), free (2), dictionary (2), external (2), ahlquist (2), strategies (2), ward (2), michael (2), royall (2), richard (2), 412 (2), charles (2), approach (2), gary (2), 1992 (2), 8018 (2), 4443 (2), johns (2), hopkins (2), 4614 (2), azzalini (2), reading (2), chap (2), practical (2), theoretic (2), akaike (2), criterion (2), fienberg (2), 9781118341544 (2), 3rd (2), 2003 (2), 1983 (2), good (2), 2011 (2), 161 (2), 2440 (2), bibcode (2), proceedings (2), philosophical (2), princeton (2), computing (2), formula (2), drawn (2), 1093 (2), biomet (2), 203 (2), version (2), rolle (2), 01621459 (2), association (2), valued (2), 1977 (2), unique (2), consistent (2), before (2), pdf (2), kass (2), kalbfleisch (2), sprott (2), applied (2), 2347496 (2), 2307 (2), direct (2), 189 (2), christian (2), monfort (2), alain (2), 40551 (2), econometric (2), concentrated (2), concentrating (2), 267 (2), advanced (2), zellner (2), implications (2), methodological (2), 1979 (2), chanda (2), consistency (2), 2333005 (2), mascarenhas (2), lemma (2), mäkeläinen (2), needs (2), expansion (2), sampled (2), describing (2), successive (2), their (2), sets (2), acceptably (2), high (2), give (2), sense (2), generated (2), asymptotically (2), serves (2), merit (2), includes (2), inline (2), specified (2), following (2), among (2), bayesianism (2), frequentism (2), against (2), rational (2), knowing (2), express (2), our (2), incomplete (2), concept (2), confused (2), mentioned (2), obey (2), bears (2), distinct (2), paper (2), introduced (2), historical (2), remarks (2), here (2), widehat (2), zero (2), number (2), maximize (2), take (2), words (2), inner (2), taking (2), coordinates (2), change (2), yields (2), exp (2), big (2), particularly (2), useful (2), include (2), factors (2), involving (2), again (2), easier (2), precision (2), shown (2), open (2), application (2), differentiable (2), find (2), just (2), compute (2), calculus (2), nabla (2), computation (2), whether (2), independence (2), overall (2), equals (2), addition (2), added (2), interpreting (2), surprisal (2), content (2), contrast (2), convenient (2), plays (2), role (2), concavity (2), integrated (2), identically (2), variables (2), cdot (2), full (2), considering (2), occurs (2), leads (2), concentration (2), maximizes (2), could (2), partitioned (2), subset (2), them (2), depending (2), compared (2), certain (2), coverage (2), directly (2), comprise (2), 100 (2), comparing (2), discussed (2), importance (2), approximations (2), proof (2), proofs (2), made (2), exist (2), every (2), finite (2), definite (2), rst (2), various (2), assumption (2), masses (2), amounts (2), dominating (2), radon (2), nikodym (2), 4pt (2), specifying (2), relationship (2), let (2), via (2), flip (2), landing (2), mapsto (2), argument (2), essay (2), nested (2), related (2), appearance (2), upload (2), file (2), changes (2), read (2), create (2), account (2), donate (2), menu (2), add, mobile, cookie, statement, developers, code, conduct, legal, safety, contacts, disclaimers, apply, site, agree, registered, trademark, profit, organization, wikimedia, inc, creative, attribution, sharealike, license, rendered, parsoid, last, edited, utc, https, org, php, title, likelihood_function, oldid, 1368389852, wikiproject, category, kriging, geostatistics, geographic, environmental, cartography, spatial, psychometrics, official, national, accounts, jurimetrics, demography, crime, census, actuarial, identification, reliability, quality, probabilistic, chemometrics, epidemiology, trials, bioinformatics, biostatistics, applications, nelson, aalen, hitting, accelerated, failure, aft, kaplan, meier, whittle, wavelet, fourier, autoregression, var, heteroskedasticity, arch, arima, jenkins, arma, xcf, pacf, autocorrelation, acf, breusch, godfrey, durbin, watson, ljung, johansen, dickey, fuller, granger, causality, break, cointegration, smoothing, trend, decomposition, elliptical, classification, discriminant, canonical, cochran, mantel, haenszel, mcnemar, graphical, cohen, kappa, ancova, poisson, regressions, binomial, logistic, bernoulli, homoscedasticity, heteroscedasticity, robust, isotonic, semiparametric, nonlinear, predictors, ordinary, template, splines, mars, simultaneous, confounding, der, waerden, ordered, jonckheere, terpstra, friedman, kruskal, wallis, mann, whitney, signed, wilcoxon, bic, shapiro, wilk, jarque, bera, lilliefors, darling, kolmogorov, smirnov, student, comparisons, randomization, permutation, uniformly, tails, jackknife, resampling, tolerance, pivot, plug, scheffé, blackwellization, sensitivity, robustness, asymptotics, divergence, loss, functional, sufficiency, completeness, monotone, specification, quasi, sectional, cohort, observational, down, assignment, interaction, factorial, blocking, experiments, questionnaire, opinion, poll, stratified, survey, replication, effect, detrending, differencing, preprocessing, dimensionality, truncation, winsorizing, outlier, unit, min, standardization, feature, anscombe, stabilizing, yeo, johnson, transformations, processing, line, ecdf, heatmap, violin, stem, leaf, display, run, radar, pie, histogram, forest, fan, correlogram, biplot, graphics, spearman, kendall, dependence, grouped, summary, skewness, kurtosis, percentile, interquartile, variation, average, absolute, mode, lehmer, heinz, heronian, harmonic, cubic, contraharmonic, center, descriptive, outline, statlect, planetmath, look, wiktionary, 316, 63682, deeper, dive, london, 04411, rohde, berlin, 319, 10460, introductory, lindsey, 139, 852359, mark, vecer, jan, february, 2021, 10419, 258120, 3390, risks9020031, risks, markets, martingale, 1989, 36697, unifying, political, likehood, king, boos, dennis, stefanski, construction, 124, 4817, 1007, 4818, 1_2, essential, adelchi, 60650, burnham, 2002, verlag, multimodel, sakamoto, ishiguro, kitagawa, 1986, reidel, atkinson, celebration, sox, higgins, owens, chapters, 1002, making, gelman, carlin, stern, dunson, vehtari, rubin, crc, lindley, 1980, viewpoint, jaynes, logic, jeffreys, griffin, 1950, weighing, bandyopadhyay, forster, north, holland, publishing, philosophy, stephen, 1030037905, 1930, 535, 15206, 1017, s0305004100016297, 1930pcps, 528f, 528, klemens, ben, 329, modeling, techniques, 604, 368, 91208, 1280, jfm, 15172, 1098, rsta, 0009, 1922rspta, 309f, 309, 594, transactions, theoretical, probable, deduced, metron, 1999, 2676741, 1009212248, 214, hald, 2007, shorter, raja, 1960, admitting, 207, rai, kamta, ryzin, 1510, 03610928208828325, 1505, communications, tarone, gruenhage, 352, 904, 10480321, 903, foutz, 357, 148, 10479926, 147, solution, papadopoulos, alecos, september, stack, exchange, why, put, mle, vos, paul, 82668, geometrical, 276, 0400509, 1973, 328, 25049882, 311, sankhyā, indian, venzon, moolgavkar, 1988, 387, 90777, glim, international, conference, generalised, aitkin, murray, bolker, benjamin, 691, 12522, 187, ecological, pickles, andrew, norwich, hutchins, 86094, 190, gourieroux, 175, 170, davidson, russell, 1993, loglikelihood, 506011, mackinnon, james, harvard, 674, 00560, 125, 127, amemiya, takeshi, wen, hsiang, wei, taichung, taiwan, chapter, 2017, tunghai, course, pawitan, yudi, 2001, 850765, modelling, hudson, 262, tb00877, 256, rossi, held, sabanés, bové, davison, 2000, 9780412606502, buse, 157, 00031305, 10482817, 153, statistician, expository, tierney, luke, kadane, joseph, 1990, validity, expansions, geisser, elsevier, 488, 444, 88376, 473, chen, chan, limiting, 546, tb01384, 540, heyde, johnstone, processes, tb01071, 184, greenberg, edward, webster, 09077, bridge, literature, 1954, maxima, regarding, constrained, minimizers, 1159, 15896597, s2cid, 02331934, 2010, 527973, 1121, optimization, timo, schmidt, klaus, styan, george, 1981, 767, 2240844, aos, 1176345516, 758, annals, gouriéroux, shao, jun, third, 423, 422, billingsley, patrick, arnold, 98165, pseudolikelihood, previous, increases, shrinks, eventually, either, very, nearly, entire, essentially, separate, together, somewhere, midst, adjacent, draw, ordinates, surrounds, differ, converts, differences, lies, inside, art, choosing, keeping, narrow, calculation, would, assigned, chosen, accurate, heuristically, makes, render, actually, having, happened, quantifies, heuristic, showing, post, hoc, itself, summarizes, already, selected, best, list, precise, introducing, lacks, although, speak, proposition, remains, entity, amount, brought, even, due, structure, yet, low, vice, versa, contexts, seen, multiplied, normalized, statisticians, consensus, paradigms, described, subsections, likelihoodism, axiomatic, adopted, treatment, phylogenetics, invention, reaction, earlier, reasoning, his, meaning, stress, spite, emphasis, laid, upon, still, tendency, treat, though, sort, measures, belief, appropriate, expectation, sir, laws, predicting, games, chance, whereas, however, psychological, judgment, resemblance, concepts, wholly, late, formal, refer, papers, published, today, quoting, middle, denotes, complete, solved, cdots, finding, looks, daunting, much, simpler, minus, computed, correspond, switching, writing, differentiate, exponentiation, evaluated, determines, indicates, serve, implicitly, going, consequence, sequence, xrightarrow, surely, neighborhood, well, euclidean, allows, basic, requires, differential, smooth, plotted, coined, tested, generalizes, univariate, curve, intersection, analogous, convenience, intuitive, expressed, total, weight, supported, extent, unsurprising, adds, denoted, lowercase, uppercase, purposes, notably, objective, logarithmically, concave, strictly, increasing, improper, uniform, important, situation, replacement, happening, land, adaption, occur, restriction, ranks, numerical, after, fitting, residual, totals, exact, hypergeometric, results, depend, graphically, slicing, along, ridge, creating, graphed, better, isometric, instance, normally, consequently, derived, frisch, waugh, lovell, projection, reduce, dimensions, expressing, replacing, correspondence, determined, explicitly, reduces, computational, burden, focuses, few, others, developed, reduced, slightly, formulation, suited, approximately, 954, regions, geq, greater, threshold, percentages, actual, standardized, plausibilities, found, denominator, standardizing, assess, performing, medicine, stated, alternatives, times, numerous, thereof, significance, level, neyman, degree, supports, versus, measured, law, frequently, compare, pseudo, identical, imposed, justify, necessary, meet, independently, assumptions, densities, ensure, must, boundedness, needed, lastly, ensures, differentiation, taylor, specifically, continuously, vanishes, infinity, unbounded, authors, prove, informally, appealing, property, restates, morse, boundary, partials, connected, verified, global, utmost, suffices, continuity, met, compactness, bounds, might, compact, extreme, contributions, commensurate, arises, proportionality, extended, consideration, consists, distinguish, dealt, manner, discussion, uses, counting, construct, mixture, otherwise, comparable, provides, fundamental, justified, observe, claims, statements, consisting, removed, verifying, despite, absolutely, calculate, calculations, displayed, integrate, hence, now, flips, saying, conclusion, reached, equivalently, imagine, flipping, tosses, consider, expresses, lands, tossed, perfectly, hht, deterministic, specify, truth, potentially, disastrous, consequences, prosecutor, fallacy, notation, avoided, indicate, regarded, conditioned, realization, possibly, differently, editor, feelings, presents, rewriting, encyclopedic, like, reflection, argumentative, converse, approximated, indication, hessian, predictive, averaging, schwarz, evaluation, lower, bound, posteriori, approximate, variational, markov, chain, monte, carlo, hierarchical, conjugate, building, indifference, cromwell, coherence, bernstein, von, mises, removing, details, understandable, experts, readers, understand, encyclopedia, item, projects, printable, download, print, export, switch, legacy, parser, get, shortened, url, cite, permanent, link, actions, talk, українська, ไทย, தமிழ், sunda, slovenščina, русский, português, polski, norsk, bokmål, nederlands, lietuvių, 한국어, 日本語, italiano, עברית, galego, français, فارسی, español, deutsch, català, বাংলা, български, العربية, top, special, pages, recent, community, contribute, current, navigation, jump,
Text of the page (random words):
account log in personal tools donate create account log in contents move to sidebar hide top 1 definition toggle definition subsection 1 1 discrete probability distribution 1 1 1 example 1 2 continuous probability distribution 1 2 1 relationship between the likelihood and probability density functions 1 3 in general 1 4 likelihoods for mixed continuous discrete distributions 1 5 regularity conditions 2 likelihood ratio and relative likelihood toggle likelihood ratio and relative likelihood subsection 2 1 likelihood ratio 2 2 relative likelihood function 2 2 1 likelihood region 3 likelihoods that eliminate nuisance parameters toggle likelihoods that eliminate nuisance parameters subsection 3 1 profile likelihood 3 2 conditional likelihood 3 3 marginal likelihood 3 4 partial likelihood 4 products of likelihoods 5 log likelihood toggle log likelihood subsection 5 1 graph 5 2 likelihood equations 5 3 exponential families 5 3 1 example the gamma distribution 6 background and interpretation toggle background and interpretation subsection 6 1 historical remarks 6 2 interpretations under different foundations 6 2 1 frequentist interpretation 6 2 2 bayesian interpretation 6 2 3 likelihoodist interpretation 6 2 4 aic based interpretation 7 see also 8 notes 9 references 10 further reading 11 external links toggle the table of contents likelihood function 26 languages العربية български বাংলা català deutsch español فارسی français galego עברית italiano 日本語 한국어 lietuvių nederlands norsk bokmål polski português русский slovenščina sunda தமிழ் ไทย українська 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia function related to statistics and probability theory this article may be too technical for most readers to understand please help improve it to make it understandable to non experts without removing the technical details august 2025 learn how and when to remove this message part of a series on bayesian statistics posterior likelihood prior evidence background bayesian inference bayesian probability bayes theorem bernstein von mises theorem coherence cox s theorem cromwell s rule likelihood principle principle of indifference principle of maximum entropy model building conjugate prior linear regression empirical bayes hierarchical model posterior approximation markov chain monte carlo laplace s approximation integrated nested laplace approximations variational inference approximate bayesian computation estimators bayes estimator credible interval maximum a posteriori estimation evidence approximation evidence lower bound nested sampling model evaluation bayes factor schwarz criterion model averaging posterior predictive mathematics portal v t e a likelihood function often simply called the likelihood gives the relative merit of various statistical models for describing a data set often the models being compared are parameterized by a parameter with the parameter often written as θ or they are parameterized by multiple parameters given as the components of a vector for a probability function or probability density function pr x θ that gives the probability or probability density of data x for a given model specifying parameter θ the likelihood is any function of θ equal to c pr x θ for some positive value c in maximum likelihood estimation the model parameter s or argument that maximizes the likelihood function serves as a point estimate for the unknown parameter while the fisher information often approximated by the likelihood s hessian matrix at the maximum gives an indication of the estimate s precision in contrast in bayesian statistics the estimate of interest is the converse of the likelihood the so called posterior probability of the parameter given the observed data which is calculated via bayes rule 1 definition edit this section is written like a personal reflection personal essay or argumentative essay that states a wikipedia editor s personal feelings or presents an original argument about a topic please help improve it by rewriting it in an encyclopedic style august 2025 learn how and when to remove this message the likelihood function parameterized by a possibly multivariate parameter θ textstyle theta is usually defined differently for discrete and continuous probability distributions a more general definition is discussed below given a probability density or mass function x f x θ displaystyle x mapsto f x mid theta where x textstyle x is a realization of the random variable x textstyle x the likelihood function is θ f x θ displaystyle theta mapsto f x mid theta often written l θ x displaystyle mathcal l theta mid x in other words when f x θ textstyle f x mid theta is viewed as a function of x textstyle x with θ textstyle theta fixed it is a probability density function and when viewed as a function of θ textstyle theta with x textstyle x fixed it is a likelihood function in the frequentist paradigm the notation f x θ textstyle f x mid theta is often avoided and instead f x θ textstyle f x theta or f x θ textstyle f x theta are used to indicate that θ textstyle theta is regarded as a fixed unknown quantity rather than as a random variable being conditioned on the likelihood function does not specify the probability that θ textstyle theta is the truth given the observed sample x x textstyle x x such an interpretation is a common error with potentially disastrous consequences see prosecutor s fallacy discrete probability distribution edit let x textstyle x be a discrete random variable with probability mass function p textstyle p depending on a parameter θ textstyle theta then the function l θ x p θ x p θ x x pr x x θ θ displaystyle mathcal l theta mid x p_ theta x p_ theta x x text pr x x mid theta theta considered as a function of θ textstyle theta a possible value of the deterministic but unknown parameter θ textstyle theta is the likelihood function given the outcome x textstyle x of the random variable x textstyle x sometimes the probability of the value x textstyle x of x textstyle x for the parameter value θ textstyle theta is written as p x x θ or p x x θ the likelihood is the probability that a particular outcome x textstyle x is observed when the true value of the parameter is θ textstyle theta equivalent to the probability mass on x textstyle x it is not a probability density over the parameter θ textstyle theta the likelihood l θ x textstyle mathcal l theta mid x should not be confused with p θ x textstyle p theta mid x which is the posterior probability of θ textstyle theta given the data x textstyle x example edit figure 1 the likelihood function p h 2 textstyle p_ text h 2 for the probability of a coin landing heads up without prior knowledge of the coin s fairness given that we have observed hh figure 2 the likelihood function p h 2 1 p h textstyle p_ text h 2 1 p_ text h for the probability of a coin landing heads up without prior knowledge of the coin s fairness given that we have observed hht consider a simple statistical model of a coin flip a single parameter p h textstyle p_ text h that expresses the fairness of the coin the parameter is the probability that a coin lands heads up h when tossed p h textstyle p_ text h can take on any value within the range 0 0 to 1 0 for a perfectly fair coin p h 0 5 textstyle p_ text h 0 5 imagine flipping a fair coin twice and observing two heads in two tosses hh assuming that each successive coin flip is i i d then the probability of observing hh is p hh p h 0 5 0 5 2 0 25 displaystyle p text hh mid p_ text h 0 5 0 5 2 0 25 equivalently the likelihood of observing hh assuming p h 0 5 textstyle p_ text h 0 5 is l p h 0 5 hh 0 25 displaystyle mathcal l p_ text h 0 5 mid text hh 0 25 this is not the same as saying that p p h 0 5 hh 0 25 textstyle p p_ text h 0 5 mid text hh 0 25 a conclusion which could only be reached via bayes theorem given knowledge about the marginal probabilities p p h 0 5 textstyle p p_ text h 0 5 and p hh textstyle p text hh now suppose that the coin is not a fair coin but instead that p h 0 3 textstyle p_ text h 0 3 then the probability of two heads on two flips is p hh p h 0 3 0 3 2 0 09 displaystyle p text hh mid p_ text h 0 3 0 3 2 0 09 hence l p h 0 3 hh 0 09 displaystyle mathcal l p_ text h 0 3 mid text hh 0 09 more generally for each value of p h textstyle p_ text h we can calculate the corresponding likelihood the result of such calculations is displayed in figure 1 the integral of l textstyle mathcal l over 0 1 is 1 3 likelihoods need not integrate or sum to one over the parameter space continuous probability distribution edit let x textstyle x be a random variable following an absolutely continuous probability distribution with density function f textstyle f a function of x textstyle x which depends on a parameter θ textstyle theta then the function l θ x f θ x displaystyle mathcal l theta mid x f_ theta x considered as a function of θ textstyle theta is the likelihood function of θ textstyle theta given the outcome x x textstyle x x again l textstyle mathcal l is not a probability density or mass function over θ textstyle theta despite being a function of θ textstyle theta given the observation x x textstyle x x relationship between the likelihood and probability density functions edit this section may contain original research please improve it by verifying the claims made and adding inline citations statements consisting only of original research should be removed august 2026 learn how and when to remove this message the use of the probability density in specifying the likelihood function above is justified as follows given an observation x j textstyle x_ j the likelihood for the interval x j x j h textstyle x_ j x_ j h where h 0 textstyle h 0 is a constant is given by l θ x x j x j h textstyle mathcal l theta mid x in x_ j x_ j h observe that a r g m a x θ l θ x x j x j h a r g m a x θ 1 h l θ x x j x j h displaystyle mathop operatorname arg max _ theta mathcal l theta mid x in x_ j x_ j h mathop operatorname arg max _ theta frac 1 h mathcal l theta mid x in x_ j x_ j h since h textstyle h is positive and constant because a r g m a x θ 1 h l θ x x j x j h a r g m a x θ 1 h pr x j x x j h θ a r g m a x θ 1 h x j x j h f x θ d x displaystyle begin aligned mathop operatorname arg max _ theta frac 1 h mathcal l theta mid x in x_ j x_ j h mathop operatorname arg max _ theta frac 1 h pr x_ j leq x leq x_ j h mid theta mathop operatorname arg max _ theta frac 1 h int _ x_ j x_ j h f x mid theta dx end aligned where f x θ textstyle f x mid theta is the probability density function it follows that a r g m a x θ l θ x x j x j h a r g m a x θ 1 h x j x j h f x θ d x displaystyle mathop operatorname arg max _ theta mathcal l theta mid x in x_ j x_ j h mathop operatorname arg max _ theta frac 1 h int _ x_ j x_ j h f x mid theta dx the first fundamental theorem of calculus provides that lim h 0 1 h x j x j h f x θ d x f x j θ displaystyle lim _ h to 0 frac 1 h int _ x_ j x_ j h f x mid theta dx f x_ j mid theta then a r g m a x θ l θ x j a r g m a x θ lim h 0 l θ x x j x j h a r g m a x θ lim h 0 1 h x j x j h f x θ d x a r g m a x θ f x j θ displaystyle begin aligned mathop operatorname arg max _ theta mathcal l theta mid x_ j mathop operatorname arg max _ theta left lim _ h to 0 mathcal l theta mid x in x_ j x_ j h right 4pt mathop operatorname arg max _ theta left lim _ h to 0 frac 1 h int _ x_ j x_ j h f x mid theta dx right 4pt mathop operatorname arg max _ theta f x_ j mid theta end aligned therefore a r g m a x θ l θ x j a r g m a x θ f x j θ displaystyle mathop operatorname arg max _ theta mathcal l theta mid x_ j mathop operatorname arg max _ theta f x_ j mid theta and so maximizing the probability density at x j textstyle x_ j amounts to maximizing the likelihood of the specific observation x j textstyle x_ j in general edit in measure theoretic probability theory the density function is defined as the radon nikodym derivative of the probability distribution relative to a common dominating measure 2 the likelihood function is this density interpreted as a function of the parameter rather than the random variable 3 thus we can construct a likelihood function for any distribution whether discrete continuous a mixture or otherwise likelihoods are comparable e g for parameter estimation only if they are radon nikodym derivatives with respect to the same dominating measure the above discussion of the likelihood for discrete random variables uses the counting measure under which the probability density at any outcome equals the probability of that outcome likelihoods for mixed continuous discrete distributions edit the above can be extended in a simple way to allow consideration of distributions which contain both discrete and continuous components suppose that the distribution consists of a number of discrete probability masses p k θ textstyle p_ k theta and a density f x θ textstyle f x mid theta where the sum of all the p textstyle p s added to the integral of f textstyle f is always one assuming that it is possible to distinguish an observation corresponding to one of the discrete probability masses from one which corresponds to the density component the likelihood function for an observation from the continuous component can be dealt with in the manner shown above for an observation from the discrete component the likelihood function for an observation from the discrete component is simply l θ x p k θ displaystyle mathcal l theta mid x p_ k theta where k textstyle k is the index of the discrete probability mass corresponding to observation x textstyle x because maximizing the probability mass or probability at x textstyle x amounts to maximizing the likelihood of the specific observation the fact that the likelihood function can be defined in a way that includes contributions that are not commensurate the density and the probability mass arises from the way in which the likelihood function is defined up to a constant of proportionality where this constant can change with the observation x textstyle x but not with the parameter θ textstyle theta regularity conditions edit in the context of parameter estimation the likelihood function is usually assumed to obey certain conditions known as regularity conditions these conditions are assumed in various proofs involving likelihood functions and need to be verified in each particular application for maximum likelihood estimation the existence of a global maximum of the likelihood function is of the utmost importance by the extreme value theorem it suffices that the likelihood function is continuous on a compact parameter spac...
|
