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Here $X$ has the binomial distribution $\mathrm{Bin}(n,\theta )$, so, up to a factor independent of $\theta$, the likelihood function isFor $0<X<n$, differentiating the log-likelihood givesand the boundary cases give the same formula. Thus the binomial proportion maximum-likelihood estimator is

The noise has the product Laplace distribution densitywhich is normalized because ${\int }_{\mathbb{R}}{e}^{-2|t|}dt=1$. By translation invariance of Lebesgue measure, the conditional law of $A(u)+N$ has densityHence an associated likelihood function isThe map $(u,y)\mapsto y-A(u)$ is measurable because $A$ is measurable and vector subtraction is continuous. Composition with the continuous norm and exponential functions proves that $L$ is jointly measurable.

LetThus $d_t$ patients recover during month $t$, while $r_t-d_t$ survive that month without recovery. The likelihood function is

The independent $\mathrm{Beta}(T,1)$ prior densities are proportional to $q_t^{T-1}$. Beta-binomial conjugacy therefore leaves the coordinates posteriorly independent, with

LetIgnoring the immaterial choice of density values at the sample points, the likelihood function isBetween consecutive order statistics, $N(\theta )$ is constant while $(1+1/\theta {)}^{N(\theta )}$ decreases with $\theta$. A maximum can therefore be moved to the left endpoint of one of these intervals. With an equivalent version using $1_{\{x\le \theta \}}$, the maximum is attained at a sample order statistic. Thus the maximum-likelihood estimator coincides with one of $X_1,\dots ,X_n$.

Write the Cepheid measurement as $\hat{\mu }=\mu +{ϵ}_{\mu }$, where ${ϵ}_{\mu }\sim N(0,{\sigma }_{\mu }^2)$, and define the calibrated absolute magnitude dataThus $q=(q_1,\dots ,q_K{)}^T$ has a multivariate normal distribution with mean $M_{0}1$ and covariance matrix

The Hubble law and the definition of distance modulus giveConsequently, withthe Hubble-flow observations are independent random variables satisfying $x_i\sim N(M_0-\theta ,{\sigma }_{\mathrm{tot}}^2)$. Apart from a factor independent of the parameters, the joint likelihood function isThe expression extends by continuity to a singular limiting covariance such as ${\sigma }_I=0$.

The likelihood function and prior distribution give the posterior densityCompleting the square in the quadratic form givesThus Gaussian conjugacy for a normal linear model yields

Let $Y_i$ be the first team's goal count in match $i$, let $x_i^T$ be row $i$ of the design matrix, and let $\beta$ be the regression coefficient vector. The fitted Poisson regression assumes that the $Y_i$ are independent random variables withThe likelihood function maximized by glm is therefore

All three sets of statistics are independent. With $s_1={\hat{S}}_1$, $s_{2g}={\hat{S}}_2^g$, and $s_{3i}={\hat{S}}_3^i$, their likelihood function, up to a parameter-independent factor, is

Approximate changing covariates by constant values on observation intervals. The generator matrix is then constant on each interval, with transition matrix $\exp (Q_j\Delta t_j)$. Multiplying the relevant entries gives a conditional likelihood function for discretely observed states.

For an Inhomogeneous Poisson process whose known positive intensity changes from ${\lambda }_1$ to ${\lambda }_2$ at an unknown time $\theta \in (0,T)$, the ordered arrival-time likelihood function isUnder a uniform prior, each interval between successive arrivals has an exponential posterior density with rate in the exponent ${\lambda }_2-{\lambda }_1$. Its integrated weights allow exact interval selection followed by a truncated exponential draw; when the two rates coincide, the Bayesian posterior is uniform and the data contain no change-time information.

A pseudo-marginal Metropolis–Hastings algorithm replaces an intractable likelihood function by a nonnegative unbiased likelihood estimator and includes its auxiliary randomness in the state. If $\mathbb{E}[\hat{L}(\theta ,U)]=L(\theta )$ for $U\sim m_{\theta }$, the extended target is proportional to $p(\theta )\hat{L}(\theta ,u)m_{\theta }(u)$. An independence proposal for new auxiliary randomness cancels the $m_{\theta }$ factors in the Metropolis–Hastings acceptance probability. On rejection the old estimate must be retained. Integrating the extended target gives the desired marginal posterior distribution exactly.

The Gaussian model $Y=X\beta +Zb+\epsilon$ with $b\sim N(0,D)$ and independent $\epsilon \sim N(0,{\sigma }^{2}I)$ has marginal law $N(X\beta ,ZD{Z}^T+{\sigma }^{2}I)$. Its group coefficients are integrated out of the marginal likelihood function. The special random-slope model uses predictor values in the grouping columns of $Z$.

Restricted maximum likelihood estimates covariance matrix parameters from linear combinations of the data whose distribution does not involve the fixed mean coefficients. It differs from a restricted maximum-likelihood estimator obtained by maximizing an ordinary likelihood function under a null hypothesis.

For $|\varphi |<1$, independent $N(0,{\sigma }^2)$ innovations and stationary initial variance ${\sigma }^2/(1-{\varphi }^2)$, the full likelihood function isConditioning on $X_1$ gives a different likelihood.

An unbiased likelihood estimator has expectation equal to the likelihood function at each parameter. A pseudo-marginal algorithm requires this estimator to be nonnegative. Importance sampling provides $\hat{L}=M^{-1}{\sum }_j\ell (F_j)p_{\theta }(F_j)/g_{\theta }(F_j)$ for independent random variables $F_j\sim g_{\theta }$, provided the proposal covers the target support. Relative variance affects the mixing time even though unbiasedness guarantees the correct marginal target.

A zeros trick implements a positive likelihood function $L(\theta )$ by adding an observed zero with Poisson distribution mean $K-\log L(\theta )$. If this mean is nonnegative throughout the parameter support, its likelihood is $e^{-K}L(\theta )$ and yields the intended Bayesian posterior. The constant $K$ must be independent of the parameter; arbitrary clipping changes the target.