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The Bayesian model evidence is the prior predictive distribution evaluated at the data $D$. For a statistical model $H_j$ with a normalized prior density ${\pi }_j$, its marginal likelihood isThis is an expected value of the likelihood function under the prior distribution; for discrete statistical parameters, replace the integral by a sum. For continuous observations, $m_j(D)$ is a probability density function value with respect to a specified data reference measure, rather than a probability of one exact observation. Comparisons must use the same data and reference measure. The same $m_j(D)$ normalizes the within-model Bayesian posterior and supplies the between-model comparison in a Bayes factor.^[ref]

A normalized prior distribution makes the Bayesian model evidence an average, so it cannot exceed the largest likelihood function value on its support. A broad prior distribution can assign little mass to values that predict the data well, even when one value gives an excellent fit. The worked uniform-alternative binomial Bayes factor below makes the difference between averaging and maximum likelihood estimation explicit.

For observations $\mathcal{D}$ and graph $G$, Bayes' theorem gives $p(G\mid \mathcal{D})\propto \pi (G)\int p(\mathcal{D}\mid {\theta }_G,G)\pi ({\theta }_G\mid G)d{\theta }_G$. Here $\pi (G)$ is the graph prior distribution and ${\theta }_G$ the local distribution statistical parameters. The integral is Bayesian model evidence; it averages over nuisance parameters with a proper statistical parameter prior distribution. For independent complete observations, the node-factorized likelihood function and independence of local statistical parameter prior distributions make the evidence factor over nodes; suitable conjugate priors can make the local integrals analytic. With missing node observations, integrating out unobserved values can couple the local parameters, so independence of local prior distributions alone does not guarantee this factorization. Proper priors and coherent hyperparameters matter for comparing different graphs.

The Bayesian posterior is the conditional probability distribution of an unknown statistical parameter after observing data. Suppose a continuous prior distribution has prior density $\pi (\theta )$ relative to Lebesgue measure $d\theta$, and $p(d\mid \theta )$ is the sampling probability mass function or probability density function of the observed data $d$ in the statistical model. Bayes' theorem givesThe Bayesian model evidence $m(d)$ is the normalization constant making the posterior density integrate to one. The strict positive, finite condition is essential: a zero or infinite normalizer does not define this posterior distribution, even if the unnormalized expression can be written down. Other reference measures give the corresponding density formula; discrete parameter values replace the integral by a sum. The posterior propriety article gives examples where normalization fails.

With $d$ fixed, $L(\theta ;d)=p(d\mid \theta )$ is a likelihood function of $\theta$. Its normalization as a sampling distribution is over possible data, not over parameter values. In particular, a likelihood function is not a probability density function for $\Theta$ and need not integrate to one over $\theta$. The prior density supplies the probability weighting, and the Bayesian model evidence normalizes their product. This distinction prevents a tempting mistake: reversing the conditioning merely by exchanging the words “data” and “parameter”.

For eight heads among ten conditionally independent Bernoulli trials, take the observed data to be the heads count $D=8$. Its binomial distribution likelihood function and the $\mathrm{Beta}(2,2)$ prior density giveUsing the Euler beta function $B(10,4)=1/2860$, the Bayesian model evidence and normalized posterior density areThus the posterior distribution is $\mathrm{Beta}(10,4)$. If the data are instead one particular ordered sequence with eight heads and two tails, its likelihood function omits the factor 45 and its Bayesian model evidence becomes $3/1430$. The factor cancels in Bayes' theorem, leaving the same posterior distribution; these two data descriptions have different sampling probabilities but identical information about $\theta$ here.

The prior density and posterior density use the left axis and each integrate to one. The dashed likelihood function uses the right axis and is divided by its maximum solely to compare its shape; it is not a parameter probability density function. The likelihood function peaks at $8/10$, while the posterior mode is $9/12=3/4$ and the posterior mean is $5/7$. Multiplying by the prior density shifts the preferred values toward one half while preserving a range of possibilities.

The resulting posterior variance is $10\cdot 4/(14^2\cdot 15)=2/147\approx 0.0136054$, compared with prior variance $1/20$. In this example the update reduces uncertainty. The posterior predictive probability of heads on the next conditionally independent toss is $\mathbb{E}[\Theta \mid D=8]=5/7$, and the credible interval above reports a 95% range. These are summaries of the same posterior distribution, rather than competing estimates of one density peak.

Bayesian statistics expresses uncertainty about an unknown statistical parameter through a probability distribution, then uses Bayes' theorem to update it after observing data. The prior distribution describes uncertainty before the current observations; the likelihood function describes how the statistical model generates those observations at each parameter value; the posterior distribution describes uncertainty conditional on the observations and that model. A physical parameter can have one fixed value while its value remains uncertain to us. Assigning it a probability distribution represents that uncertainty under the chosen model, rather than requiring the physical parameter to change randomly between measurements.

For example, suppose ten Bernoulli trials are independent conditional on an unknown heads probability $\theta$, with a beta distribution prior $\mathrm{Beta}(2,2)$. Eight heads and two tails give the posterior distribution $\mathrm{Beta}(10,4)$ by Beta-binomial conjugacy. Its posterior mean is $10/14=5/7\approx 0.714286$, between the prior mean $1/2$ and observed binomial proportion $8/10$. The Bayesian posterior below computes the whole update, and the credible interval below summarizes its remaining uncertainty. The conjugate update is explained in MIT 18.05, Reading 15: beta and normal conjugate priors.

The useful feature is that one posterior distribution supports several questions: a posterior mean estimates a value, a credible interval reports a range, and a posterior predictive distribution predicts a future observation while averaging over parameter uncertainty. With conditionally independent new data, today's posterior distribution can become tomorrow's prior distribution. These conclusions remain conditional on the statistical model and prior distribution; a precise update does not establish that either assumption fits the experiment.

A historical starting point is Thomas Bayes's essay in the 1763 volume of the Royal Society's Philosophical Transactions, communicated after his death by Richard Price. Its opening problem asks for uncertainty about an unknown event probability given observed successes and failures: the same direction of reasoning illustrated by the coin example. See Bayes and Price, An Essay towards solving a Problem in the Doctrine of Chances, opening letter and problem.

A capture-recapture model is a statistical model for repeated observations of identifiable individuals, including those missed on some occasions. A closed-population capture-recapture model estimates a fixed population cardinality; models allowing individuals to enter or leave can instead describe survival and recruitment. The sampling assumptions determine which likelihood function is appropriate.

For the two-sample closed-population capture-recapture model, condition on the sample sizes $M$ and $C$. If the second sample selects $C$ distinct individuals from $N$ by simple random sampling, then the overlap $R$ has the hypergeometric distribution:Here $r$ is an integer and the probability mass function is zero outside these bounds. The binomial coefficients count choices of $r$ marked and $C-r$ unmarked individuals, divided by all equally likely size-$C$ subsets. The hypergeometric distribution has expected valueEquating the observed overlap with this expected value gives the Lincoln-Petersen estimator. The underlying sampling without replacement, probability mass function, and expected value are developed in Raymond P. Guiteras, MIT 14.30 hypergeometric distribution handout, §§1–4 and 8.

For the toy counts $M=20$, $C=25$, and $R=5$, hold the observations fixed and vary the unknown integer $N$. The likelihood function isValues below forty cannot contain the observed set union. Exact comparison of consecutive likelihoods gives two tied maximum-likelihood estimates, $N=99$ and $N=100$. The Lincoln-Petersen estimator returns $100$; the Chapman estimator returns $90$ and is a different rule for estimation.

The vertical scale is $L(N)/{\max }_{n\ge 40}L(n)$. This relative likelihood compares how well candidate statistical parameters explain the same observations; it is not a posterior probability. The displayed range $40\le N\le 300$ is only a plotting window. The likelihood function remains positive for every integer $N\ge 40$ and tends to zero as $N$ increases, so the curve has a continuing right tail. The data do not justify truncating statistical inference at three hundred. The conditional hypergeometric distribution formulation also appears in Louis-Paul Rivest and Mamadou Yauck, Small Sample Estimators for Two-way Capture Recapture Experiments, §3, equation (5).

Zero recaptures require care. With $R=0$ and positive $M,C$, the Lincoln-Petersen estimator has no finite ratio, but the capture-recapture model still gives a valid likelihood function:This likelihood function increases toward one as $N$ grows, so there is no finite maximum-likelihood estimate. For the same sample sizes twenty and twenty-five with zero overlap, forty-five distinct individuals would be observed. Lack of overlap supplies a lower bound but very weak information about large population cardinalities.

In more elaborate capture-recapture models, a first-recapture array $m_{ij}$ counts individuals released at occasion $i$ and next observed at $j$. A probability of survival ${\varphi }_j$ and a probability of detection $P_{j+1}$ contribute a no-observation continuation factor ${\varphi }_j(1-P_{j+1})$. The resulting likelihood function depends on both survival and detection; these probabilities need not be separately identifiable at the last observation interval. Missing-count EM for capture-recapture describes one way to fit such a model, and identifiability remains a separate issue from computing an estimator output.

A confidence interval is a random interval constructed from a statistical sample to cover a fixed unknown statistical parameter with a prescribed coverage probability. For a scalar target $\theta$, a procedure with nominal confidence $1-\alpha$ satisfiesThe subscript records which probability distribution generates the repeated samples. The endpoints are statistics, so they vary from sample to sample; the target is fixed. Equality gives exact nominal coverage probability, while the inequality also permits conservative procedures. For a general estimand $\psi (\theta )$, replace the target inside the interval by $\psi (\theta )$.

A realized confidence interval communicates the precision of an estimate in the target's own units. Reporting $1.2$ alone conceals the difference between an estimate with standard error $0.5$ and one with standard error $0.05$. A narrow confidence interval indicates greater precision for the stated statistical model, rather than a guarantee that its assumptions describe the experiment correctly.

Once $x$ is observed, $C(x)$ either contains the fixed target or does not. The 95% statement concerns the random procedure $C(X)$, not a posterior probability of 0.95 that this particular interval contains $\theta$. A credible interval instead assigns specified posterior probability to a region under Bayesian inference. These interpretations can differ even when the numerical endpoints agree. The repeated-sampling interpretation is explained in NIST's confidence-limits teaching reference.

Jerzy Neyman's 1937 paper formulates the fixed-target, random-endpoint construction in Section II, especially equation (20) on page 348. Its confidence coefficient must hold across the allowed statistical parameters, rather than only at the value suggested by one sample. See Neyman, Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability and the original paper scan.

In a normal linear model with an intercept and factor effects, set the effect of one factor level to zero. The intercept then denotes that reference level's mean, and the remaining effects are contrasts to it. The constraint removes redundant statistical parameters without altering the fitted mean space.

The coverage probability is the repeated-sampling probability that a random confidence interval contains the fixed true statistical parameter. A finite-sample level-$1-\alpha$ procedure has coverage probability at least $1-\alpha$ at every admissible statistical parameter; exact nominal coverage means equality. An asymptotically calibrated procedure approaches the nominal value as sample size grows, under its stated assumptions. Higher-order Edgeworth expansions can quantify the discrepancy from nominal coverage probability when their regularity conditions hold.

Observed coverage in finitely many repetitions is itself random. For $m$ independent repetitions of the exact normal mean confidence interval with known variance, the count $K$ of covering confidence intervals has binomial distribution $\mathrm{Binomial}(m,0.95)$. At $m=40$, its expectation is $38$, butConsequently, the 39 successes in the figure are compatible with exact 95% coverage probability. The guarantee follows from the pivotal quantity's probability distribution; the finite observed fraction is an illustration. Also distinguish a confidence interval for a fixed mean from a prediction interval for a future observation: the latter must accommodate the new observation's variation as well as estimation uncertainty.

A credible interval is an interval $C(d)$ containing a prescribed amount $1-\alpha$ of posterior probability for an unknown scalar statistical parameter $\Theta$. For a continuous posterior distribution with posterior density $p(\theta \mid d)$, its endpoints satisfyThe conditioning includes the chosen statistical model and prior distribution. For a discrete posterior distribution, exact equality may be impossible because of its probability masses; an interval can instead be required to contain at least the requested posterior probability. An interval compresses the full posterior distribution into a range whose uncertainty can be communicated in the parameter's own units. See MIT 18.05, Class 16: probability intervals.

For the eight-head, two-tail example in Bayesian statistics, the posterior density is $2860{\theta }^9(1-\theta {)}^3$ on $[0,1]$. Its cumulative distribution function, obtained by integration of this polynomial, isSolving $F(a)=0.025$ and $F(b)=0.975$ by the bisection method givesThese are numerical approximations to the exact quantiles defined by those equations; the endpoints were computed from the cumulative distribution function, rather than read off a plot. The resulting equal-tailed credible interval leaves 2.5% posterior probability on each side. Its shaded area is 95% of the total area under the posterior density.

Credible intervals are not unique. The one-sided credible interval $[0,F^{-1}(0.95)]\approx [0,0.887334219]$ also has 95% posterior probability, but leaves all 5% outside it on the right. A highest density region instead selects points above a density threshold; it need not be an equal-tailed credible interval, and when the posterior density has separated peaks it need not be one interval at all. Thus a reported credible interval should specify its construction.

A 95% credible interval makes a statement about the statistical parameter conditional on these observed data and the Bayesian model. A 95% confidence interval instead refers to the coverage probability of a procedure under repeated sampling at a fixed parameter value. These are different conditioning statements: 95% posterior probability does not by itself imply 95% coverage probability for every fixed parameter. The confidence interval article develops that sampling interpretation.

The efficient information for a scalar target statistical parameter is the squared L2 norm of its efficient score. It cannot exceed the parametric Fisher information, by the Pythagorean theorem in an inner-product space. It can vanish when first-order target variation can be reproduced by nuisance variation.

Hold the old statistical parameters fixed when taking the conditional expectation, while keeping the candidate statistical parameters $\theta$ variable:For independent observations in a finite mixture model, $Z_i$ is a hidden component label. Bayes' theorem gives its mixture responsibilities:These posterior probabilities are the conditional expectations of the mixture allocation indicators; they turn hidden counts into fractional counts.^[ref] They condition on a current fixed fit, rather than averaging over a posterior distribution of the statistical parameters. Taking the conditional expectation of the complete-data log-likelihood is essential: substituting a single guessed label, or merely substituting the conditional mean into a nonlinear complete-data expression, generally gives a different algorithm.

For $Q(\theta \mid {\theta }_0)={\mathbb{E}}_{{\theta }_0}[\log p_{\theta }(Y,Z)\mid Y]$, the Jensen inequality givesThus any expectation-maximization algorithm M-step that increases $Q$ cannot decrease the observed-data likelihood function.^[ref]

For a short proof, assume common support and finite expressions, and keep the E-step conditional distribution $q_t(z)=p_{{\theta }^{(t)}}(z\mid y)$ fixed. The concavity of the logarithm and Jensen's inequality giveThis evidence lower bound equals $Q(\theta \mid {\theta }^{(t)})-{\mathbb{E}}_{q_t}\log q_t(Z)$. At $\theta ={\theta }^{(t)}$, the ratio inside the expectation is the constant $p_{{\theta }^{(t)}}(y)$, so equality holds. During the M-step, the second term is constant; increasing $Q$ raises a lower bound that touches the current log-likelihood. HenceIncreasing likelihood values alone do not establish convergence of the statistical parameters or attainment of a global maximum. In a differentiable finite mixture model, the gradient of $\ell$ equals the candidate-parameter gradient of $Q$ evaluated at the current fit. Thus an interior fixed point of an exact M-step is a stationary point, which need not be a local maximum; the Identical-mean EM trap is an explicit counterexample. Small changes in log-likelihood can also occur before the means have settled, so inspecting both is useful.

Maximize $Q(\theta \mid {\theta }^{(t)})$ over the permitted statistical parameters, with the E-step conditional distribution held fixed.^[ref] A finite mixture model hasFor normal distributions with known common variance ${\sigma }^2>0$ and fixed positive mixture weights, only the means are fitted. Dropping terms independent of these candidate means givesThe derivative in ${\mu }_j$ is ${\sum }_i{r}_{ij}^{(t)}(y_i-{\mu }_j)/{\sigma }^2$, and its second derivative is $-N_j^{(t)}/{\sigma }^2$. Thus a positive weighted count gives a unique maximizing weighted mean. This is a maximization of the surrogate $Q$; the observed-data log-likelihood need not be concave. Recompute the mixture responsibilities after updating the means. The broader EM for Gaussian mixtures with a common variance also estimates mixture weights and the shared variance.

The empirical distribution function counts the fraction of observations at or below a threshold. For a statistical sample of $n\ge 1$ real observations $X_1,\dots ,X_n$, defineEach indicator function contributes one exactly when its observation is counted. Equivalently, $F_n(x)=P_n((-\infty ,x])$, where the empirical measure $P_n=n^{-1}{\sum }_i{\delta }_{X_i}$ assigns probability mass $1/n$ to each observation, including repeated observations.^[ref]

Choose the illustrative statistical sample $[1,1,2,4,6]$. Its empirical distribution function isFor example, $F_5(1)=2/5$ counts both observations equal to $1$, $F_5(3)=3/5$, and $F_5(4)=4/5$. There is no observation at $3$, so there is no jump there. The five observations produce only four distinct jump locations.

The convention $X_i\le x$ makes the empirical distribution function right-continuous: once a threshold reaches an observation, that observation remains counted immediately to its right. For a finite statistical sample, the count is constant from any $x$ up to the next larger observed value, proving right continuity. Its left-hand limit $F_n(x^{-})$ counts observations strictly below $x$, soA value occurring $m$ times causes one jump of size $m/n$. At $1$ the jump is $2/5$; at $2$, $4$, and $6$ each jump is $1/5$. All jumps sum to one. Thus the empirical distribution function is a nondecreasing step function with limits zero and one, and is itself a valid cumulative distribution function.

This turns many questions about data into simple cumulative distribution function operations. The observed fraction in $(a,b]$ is $F_n(b)-F_n(a)$: for $(1,4]$ it is $4/5-2/5=2/5$, counting $2$ and $4$ but excluding both $1$s. Including the left endpoint gives $F_n(4)-F_n(1^{-})=4/5$. A sample quantile reads the same staircase in reverse: the smallest threshold reaching $1/2$ is $2$. Unlike a histogram, this representation requires no choice of bins; its jumps retain every observation and its multiplicity.

The exact description of a finite statistical sample must be distinguished from a statistical model for how it was obtained. The empirical distribution function is defined for any finite list, with no independence assumption. Once the list is observed, its counts are fixed. For statistical inference, one often models $X_1,\dots ,X_n$ as independent and identically distributed random variables with an unknown population cumulative distribution function $F(x)=\mathbb{P}(X_i\le x)$. Then $F$ describes the sampling law, while $F_n$ varies across repeated statistical samples. The value $F_5(1)=0.4$ is an observed fraction and an estimate of $F(1)$; it does not establish that the population probability is $0.4$. Even when $F$ is continuous, every finite $F_n$ is discrete.

For any threshold $x$ fixed before sampling, the indicator functions are independent Bernoulli random variables with success probability $F(x)$. Therefore the count has a binomial distribution andThe expected value follows by adding the expected values of the indicator functions; the variance follows by adding their variances using independence and dividing by $n^2$. Thus $F_n(x)$ is an unbiased estimator of $F(x)$ at a fixed threshold. Controlling the error at every threshold at once requires more than a fixed-threshold calculation: the DKW inequality supplies such control, and the Glivenko-Cantelli theorem describes its uniform convergence almost surely as $n$ grows.

Comparing $F_n$ with a proposed cumulative distribution function also motivates goodness-of-fit tests, including the Kolmogorov-Smirnov test, the Anderson–Darling test, and the Cramér–von Mises criterion. A standard Kolmogorov-Smirnov test calibration assumes a fully specified continuous population law; fitting its statistical parameters from the same statistical sample changes that calibration.^[ref]

The expectation-maximization algorithm fits a latent-variable model by alternating a conditional expectation over the latent variables with a maximization over the statistical parameters. Its target is the observed-data likelihood function, although its updates use the simpler complete-data log-likelihood.^[ref]

Write $y$ for the observations, $z$ for the latent variables, and $p_{\theta }(y,z)$ for their joint probability density function or probability mass function. The observed-data log-likelihood iswhere the integral becomes a sum for discrete $z$. Directly maximizing a logarithm of this sum can be awkward. The E-step instead computes a conditional distribution at the current statistical parameters ${\theta }^{(t)}$; the M-step uses it to construct a new fit. The attractive idea is to retain uncertainty about the hidden labels while recovering simple weighted means and other complete-data updates. The Known-variance Gaussian mixture EM example shows the mechanism without estimating a variance or mixture weight.

Arthur Pentland Dempster, Nan M. Laird, and Donald B. Rubin gave the general framework its name in their 1977 paper, which also discusses earlier special-case algorithms.^[ref]

Assume the full-data sample space and the range of the statistic $T$ are standard Borel spaces, $T$ is a measurable function, and the statistical model $\{P_{\theta }\}$ is dominated by one common sigma-finite measure $\nu$. Write $f_{\theta }=d{P}_{\theta }/d\nu$ for its densities. Then $T$ is a sufficient statistic if and only if there exist nonnegative measurable functions $g_{\theta }$ and $h$, with $h$ independent of $\theta$, such thatThe statistical parameter can enter $g_{\theta }$ only through $\theta$ and the observed $T(x)$. ^[ref]

For discrete data, take counting measure; for ordinary continuous data, take Lebesgue measure. Independence, identical sampling laws, smooth dependence on $\theta$, and a fixed positive support of a probability distribution are not requirements of this theorem. Parameter-dependent support must nevertheless be included in $f_{\theta }$, usually through an indicator function, and must fit the displayed factorization. The sufficient statistic for a symmetric uniform sample illustrates exactly this: its indicator function depends on the unknown scale through the observed absolute maximum. The null sets in the theorem concern versions of densities; the factors are not generally themselves normalized probability distributions.

For a discrete sample space, the cancellation behind the theorem is particularly transparent. On a group with positive probability, factorization givesConversely, multiply a common conditional probability mass function by $P_{\theta }(T=T(x))$ to obtain the required factorization. For continuous data, a sum over a group is not a density integral over the original Lebesgue measure: the group may have measure zero. Use the regular conditional distribution formulation and the Fisher-Neyman factorization theorem instead of that discrete calculation.

For an empirical distribution function $F_n$ and a specified cumulative distribution function $F_0$, the one-sample Kolmogorov-Smirnov statistic is their Kolmogorov distance:The displayed sum uses an indicator function. With this convention, $F_n$ is right-continuous. For distinct observations, its value and its left one-sided limit at the $i$th order statistic are $F_n(X_{(i)})=i/n$ and $F_n(X_{(i)}^{-})=(i-1)/n$.

For continuous $F_0$, both jump sides matter. Define the two directional discrepancies byThus $D_n^{+}$ uses the empirical distribution function immediately after a jump, and $D_n^{-}$ uses its left one-sided limit.^[ref] A supremum need not be attained.

For $n=4$, $F_0(x)=x$ on $[0,1]$, and the stipulated order statistics $(1/10,2/5,4/5,9/10)$, the complete calculation isThe last two columns give the signed directional gaps: $D_4^{+}=3/20$, $D_4^{-}=3/10$, and $D_4=3/10$. In contrast, checking only $|F_4(X_{(i)})-F_0(X_{(i)})|$ gives $3/20$ and misses the largest gap approached as $x\uparrow 4/5$. At $x=4/5$ itself, the gap has already fallen to $1/20$.

The finite formula follows from the behavior between jumps: the empirical distribution function $F_n$ is constant and the cumulative distribution function $F_0$ is monotone, so the largest directional gaps occur at the two interval ends, possibly as one-sided limits. The two tails introduce no larger gap. Checking every jump side therefore gives the exact supremum.

A common normalized Kolmogorov-Smirnov statistic is $\sqrt{n}{D}_n$, which equals $3/5$ in this example. Its large-sample reference law comes from the Kolmogorov-Smirnov theorem. The unscaled $D_n$ remains the largest difference of cumulative probabilities, not a p-value.

There is a useful exact exception to the usual warning about fitting statistical parameters. Suppose $n\ge 2$ independent and identically distributed observations have the uniform distribution on $(0,\theta )$ with unknown $\theta >0$. Let $M={\max }_i{X}_i$ and let $J$ be the label of its almost surely unique maximizing observation. Conditional on $M=m$ and $J=j$, the surviving labelled observations $(X_i{)}_{i\ne j}$ are independent with the uniform distribution on $(0,m)$; after division by $m$ they are independent and identically distributed uniforms on $(0,1)$. These are conditional distributions defined by probability density functions, since the event $M=m$ has probability zero.

To check this, the joint probability density function of $M$ and $J=j$ is $m^{n-1}/{\theta }^n$ for $0<m<\theta$. Dividing the original joint probability density function ${\theta }^{-n}$ by this factor gives the conditional survivor probability density functionThis factorization proves the claimed conditional independence; scaling removes $m$. Keep the surviving labels in a fixed order, or apply an independent random permutation. The sorted survivors instead have the joint law of uniform order statistics and exhibit statistical dependence; sorting does not affect their empirical distribution function. Delete the maximum itself, which would otherwise become a forced value $1$, and compute the Kolmogorov-Smirnov statistic using sample size $n-1$ and jump heights $i/(n-1)$. This gives an exact conditional Kolmogorov-Smirnov test under the uniform-endpoint statistical model. It does not justify fitting an arbitrary endpoint or other statistical parameters and using the ordinary $n$-sample calibration.

The one-sample Kolmogorov-Smirnov test checks goodness of fit by comparing an empirical distribution function $F_n$ with a specified cumulative distribution function $F_0$. Its Kolmogorov-Smirnov statistic $D_n$ is the largest vertical gap anywhere between the two curves. Every threshold asks the same concrete question: what proportion of the observations are at or below it, and what probability does the proposed probability distribution assign there? A large discrepancy supplies evidence against the null hypothesis.

For example, four stipulated observations $(1/10,2/5,4/5,9/10)$ compared with the uniform distribution on $[0,1]$ give $D_4=3/10$. Just before $4/5$, only two observations have been counted, so the empirical distribution function approaches $1/2$, while $F_0$ approaches $4/5$. The Kolmogorov-Smirnov statistic section calculates every jump-side gap. These are exact illustrative numbers, not experimental measurements or a claim of statistical significance.

The ordinary one-sample calibration assumes independent and identically distributed real observations and a simple hypothesis $H_0:F=F_0$, with $F_0$ a continuous function fixed before inspecting these observations. All statistical parameters of $F_0$ must be specified.^[ref] Under that null hypothesis, the probability integral transform gives $U_i=F_0(X_i)$ with the uniform distribution on $(0,1)$, retaining independence. Consequently the sampling distribution of $D_n$ depends on the sample size $n$, but not on the particular continuous $F_0$. This is the useful sense in which the calibration is distribution-free.

For the exponential distribution of rate one, $F_0(x)=1-e^{-x}$ for $x\ge 0$. Take $x_i=-\log (1-u_i)$ for the four stipulated $u_i$ above. The probability integral transform changes the horizontal coordinates but preserves both the order statistics and their cumulative heights, so the same Kolmogorov-Smirnov statistic $3/10$ appears on both scales. More generally, for continuous $F_0$, the jump-side formulas depend only on $F_0(X_{(i)})$. Strict increase everywhere is unnecessary: intervals on which $F_0$ is constant have probability zero under the null hypothesis. This explains the invariance without assuming a strictly positive probability density function.

The observed Kolmogorov-Smirnov statistic is a discrepancy, whereas its p-value is a tail probability under the null hypothesis:Here $D_n^{\mathrm{new}}$ is computed from a fresh sample of the same sample size. Choose a significance level $\alpha$ before the test and reject when the calibrated p-value is at most $\alpha$. The value $D_4=3/10$ alone supplies neither a p-value nor a rejection decision. Failure to reject does not establish that the null hypothesis is true. A finite-sample calibration and the asymptotic Kolmogorov-Smirnov theorem are distinct; a cutoff for $D_n$ must be rescaled before use with $\sqrt{n}{D}_n$.^[ref]

Fitting statistical parameters on the tested observations replaces $F_0$ by the random $F_{\hat{\theta }}$. The resulting transformed observations need not be independent and identically distributed uniforms, so the ordinary reference law fails. The Lilliefors test addresses a particular fitted normal distribution. More general parametric bootstrap calibration must simulate from the fitted statistical model and repeat the fitting in every replicate; holding the fitted statistical parameters fixed during those comparisons calibrates a different procedure. Such a parametric bootstrap requires its own validity justification rather than inheriting exactness from the simple-null test.

For a discrete probability distribution, $F_0(X)$ is generally not uniform. Equal observations also arise from rounding, so tied data require a calibration that accounts for the discrete probability distribution or the rounding rule. The discrepancy remains meaningful, but continuous-null p-values are not automatically valid. Statistical dependence likewise changes the reference law. The two-sample Kolmogorov-Smirnov test compares two empirical distribution functions and has a different calibration.^[ref]

Kriging predicts an unobserved value of a spatial random field by combining observations according to their covariances. Nearest-neighbor interpolation uses the closest sample; kriging also asks how strongly each sample is related to the target and how much information the samples share with one another. Two nearby, highly correlated samples need not provide twice the information of one. The different forms of kriging depend on whether the mean is known, an unknown constant, or an unknown regression drift.

Write $Z=(Z(s_1),\dots ,Z(s_n){)}^T$ for the observation random vector, $z$ for its realized values, and $Z_0=Z(s_0)$ for the prediction target. Assume finite second moments, and defineHere $\Sigma$ is the observation covariance matrix, assumed to be a positive-definite matrix in the formulas below. A covariance function supplies its entries and the target covariance vector $c$. For a second-order stationary random field, the covariogram $C(h)$ depends on displacement, and the semivariogram satisfies $\gamma (h)=C(0)-C(h)$. The intrinsically stationary random field formulation permits kriging using a semivariogram even when a stationary covariogram is unavailable; the finite covariance formulation below is a convenient important case.

For a linear combination $w^{T}Z$ predicting $Z_0$ with zero bias, the error variance equals its mean squared error:Ordinary kriging and universal kriging minimize this quadratic form subject to their mean constraints. Their optimality is among the specified linear predictors with zero bias under the chosen statistical model; covariance information alone does not establish optimality among arbitrary nonlinear predictors. The Gaussian random field interpretation supplies additional conclusions in simple kriging.

The reported error variance assumes a chosen covariance function, its statistical parameters, observation locations, and a mean structure. It omits uncertainty in fitted covariance parameters and does not validate the trend. A repeated-sampling mean squared error need not equal the error variance conditional on each realized data vector.^[ref]

In 1963, Georges Matheron presented kriging for estimating average ore grades in mining panels from weighted samples taken inside and outside a panel. This setting in geostatistics explains why prediction targets can be averages over regions as well as values at points.^[ref]

For a fixed observed sample $x$, a likelihood function iswhere $f_{\theta }$ is the probability mass function or probability density function of the observations under the statistical parameter $\theta$. All densities must be defined relative to the same parameter-independent dominating measure. The sample is fixed; the parameter is the function's argument. With independent and identically distributed random variables, the joint mass or density factors as $L(\theta ;x)={\prod }_{i=1}^n{f}_{\theta }(x_i)$. This definition and the distinction between sampling probability and parameter uncertainty are explained in MIT 18.05, Reading 10b, sections 3 and 3.2.

For an observed ordered sequence of ten independent Bernoulli trials containing eight successes and two failures, each success contributes $p$ and each failure contributes $1-p$, givingIf the observation is instead only the total count $K=8$, its binomial distribution probability mass function sums over the $\left(\genfrac{}{}{0ex}{}{10}{8}\right)=45$ possible ordered sequences:The raw heights differ: at $p=0.8$ they are approximately $0.0067108864$ and $0.301989888$. Their maximum-likelihood estimates are identical. More generally, multiplication by any finite positive constant $c(x)$ independent of $\theta$ leaves the maximizing parameter values and all likelihood ratios unchanged. Multiplication by a parameter-dependent factor can change the MLE; dropping such a factor is not justified. Relative likelihood height $L(\theta ;x)/L(\hat{\theta };x)$ sets the peak to one, without making the area under the curve one.

Normalization of $f_{\theta }$ is over possible data values for each fixed parameter. It imposes no requirement that ${\int }_{\Theta }L(\theta ;x)d\theta =1$. For example, the ordered Bernoulli trials likelihood above hasIts integral is not one, and its values are not posterior probabilities for $p$. When the Bayesian posterior is defined, Bayes' theorem multiplies the likelihood by a prior density and normalizes that product. Choosing to normalize a likelihood function alone is an additional choice, such as using a uniform prior in those parameter coordinates, rather than part of the definition of likelihood.

For observations with a probability density function, the probability of an exact sample point is zero. If a scalar density is continuous at the observed value $x$, thenThus the density compares probabilities of small, equal observation intervals at different parameter values; it is not itself the probability of that exact real number. The equal-width factor is independent of the parameter. This gives visual intuition for density likelihoods, as in MIT 18.05, Reading 10b, section 4. The normal distribution example in maximum-likelihood estimator makes the comparison explicit. If only rounded intervals are recorded, their integrated probabilities give the exact observation likelihood; using midpoint densities is an approximation.