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):
tation 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 space for the maximum likelihood estimator to exist 4 while the continuity assumption is usually met the compactness assumption about the parameter space is often not as the bounds of the true parameter values might be unknown in that case concavity of the likelihood function plays a key role more specifically if the likelihood function is twice continuously differentiable on the k dimensional parameter space θ textstyle theta assumed to be an open connected subset of r k textstyle mathbb r k there exists a unique maximum θ θ textstyle hat theta in theta if the matrix of second partials h θ 2 l θ i θ j i j 1 1 n i n j displaystyle mathbf h theta equiv left frac partial 2 l partial theta _ i partial theta _ j right _ i j 1 1 n_ mathrm i n_ mathrm j is negative definite for every θ θ textstyle theta in theta at which the gradient l l θ i i 1 n i textstyle nabla l equiv left frac partial l partial theta _ i right _ i 1 n_ mathrm i vanishes and if the likelihood function approaches a constant on the boundary of the parameter space θ textstyle partial theta i e lim θ θ l θ 0 displaystyle lim _ theta to partial theta l theta 0 which may include the points at infinity if θ textstyle theta is unbounded mäkeläinen and co authors prove this result using morse theory while informally appealing to a mountain pass property 5 mascarenhas restates their proof using the mountain pass theorem 6 in the proofs of consistency and asymptotic normality of the maximum likelihood estimator additional assumptions are made about the probability densities that form the basis of a particular likelihood function these conditions were first established by chanda 7 in particular for almost all x textstyle x and for all θ θ textstyle theta in theta log f θ r 2 log f θ r θ s 3 log f θ r θ s θ t displaystyle frac partial log f partial theta _ r quad frac partial 2 log f partial theta _ r partial theta _ s quad frac partial 3 log f partial theta _ r partial theta _ s partial theta _ t exist for all r s t 1 2 k textstyle r s t 1 2 ldots k in order to ensure the existence of a taylor expansion second for almost all x textstyle x and for every θ θ textstyle theta in theta it must be that f θ r f r x 2 f θ r θ s f r s x 3 f θ r θ s θ t h r s t x displaystyle left frac partial f partial theta _ r right f_ r x quad left frac partial 2 f partial theta _ r partial theta _ s right f_ rs x quad left frac partial 3 f partial theta _ r partial theta _ s partial theta _ t right h_ rst x where h textstyle h is such that h r s t z d z m textstyle int _ infty infty h_ rst z dz leq m infty this boundedness of the derivatives is needed to allow for differentiation under the integral sign and lastly it is assumed that the information matrix i θ log f θ r log f θ s f d z displaystyle mathbf i theta int _ infty infty frac partial log f partial theta _ r frac partial log f partial theta _ s f dz is positive definite and i θ textstyle left mathbf i theta right is finite this ensures that the score has a finite variance 8 the above conditions are sufficient but not necessary that is a model that does not meet these regularity conditions may or may not have a maximum likelihood estimator of the properties mentioned above further in case of non independently or non identically distributed observations additional properties may need to be assumed in bayesian statistics almost identical regularity conditions are imposed on the likelihood function in order to proof asymptotic normality of the posterior probability 9 10 and therefore to justify a laplace approximation of the posterior in large samples 11 likelihood ratio and relative likelihood edit see also pseudo r squared likelihood ratio edit this section is about the likelihood ratio in general for the use of likelihood ratios in interpreting diagnostic tests see likelihood ratios in diagnostic testing for the statistical test to compare goodness of fit see likelihood ratio test a likelihood ratio is the ratio of any two specified likelihoods frequently written as λ θ 1 θ 2 x l θ 1 x l θ 2 x displaystyle lambda theta _ 1 theta _ 2 mid x frac mathcal l theta _ 1 mid x mathcal l theta _ 2 mid x the likelihood ratio is central to likelihoodist statistics the law of likelihood states that the degree to which data considered as evidence supports one parameter value versus another is measured by the likelihood ratio in frequentist inference the likelihood ratio is the basis for a test statistic the so called likelihood ratio test by the neyman pearson lemma this is the most powerful test for comparing two simple hypotheses at a given significance level numerous other tests can be viewed as likelihood ratio tests or approximations thereof 12 the asymptotic distribution of the log likelihood ratio considered as a test statistic is given by wilks theorem the likelihood ratio is also of central importance in bayesian inference where it is known as the bayes factor and is used in bayes rule stated in terms of odds bayes rule states that the posterior odds of two alternatives a 1 displaystyle a_ 1 and a 2 displaystyle a_ 2 given an event b displaystyle b is the prior odds times the likelihood ratio as an equation o a 1 a 2 b o a 1 a 2 λ a 1 a 2 b displaystyle o a_ 1 a_ 2 mid b o a_ 1 a_ 2...
|
