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The log-likelihood has derivativeDifferentiating the normalizing identity for the family givesHence an interior maximum satisfies, and under nondegeneracy uniquely satisfies,

The log density isIts kink occurs at the unknown location $\mu$, so its parameter dependence cannot be separated into a fixed finite-dimensional statistic of $y$ and the smooth canonical form required of an exponential dispersion family. Thus the two-parameter location-scale Laplace distribution is not such a family.

For independent observations, the log-likelihood isFor every fixed $\sigma >0$, maximizing this expression in $\beta$ is equivalent to minimizing $S(\beta )$. Hencethe least absolute deviations estimator. At this value,so, in the nondegenerate case $S(\hat{\beta })>0$,

Ignoring terms independent of $\theta$, the log-likelihood isIts score function isSince ${\ell }^{\prime \prime }(\theta )=-({\sum }_i{Y}_i)/{\theta }^2\le 0$, the interior critical point is the maximum, givingThe sum of independent Poisson distributions is Poisson with mean $\theta {\sum }_i{x}_i$, soThus the maximum-likelihood estimator is also unbiased.

After integrating out the multivariate normal distribution $\beta$, the marginal distribution isUp to terms independent of ${\sigma }^2$, the log-likelihood isDifferentiating and setting the result to zero gives the maximum marginal likelihood estimatorThis is an Empirical Bayes method because the estimated hyperparameter is then inserted into the prior and posterior distributions.

For individual $i$, let $Y_i$ be the reported count, $g_i\in \{0,1\}$ the gender indicator, and $m_i\in \{0,1\}$ the minority indicator. The fitted Poisson regression isIts log-likelihood isThe maximum-likelihood estimator isHolding minority status fixed, changing the gender indicator from zero to one multiplies the fitted conditional expected value by $e^{-0.1916}=0.826$. Thus the fitted mean count for men is about $17.4\%$ lower than that for women with the same minority status.

Write $Y_i$ for the colony count and $d_i$ for the dose on plate $i$. The first fit is the Poisson regressionThus its log-likelihood isThe Poisson deviance relative to the saturated model iswhere a summand with $y_i=0$ uses $0\log 0=0$.

At dose zero the fitted expected value is $e^{{\hat{\beta }}_0}=e^{3.321995}\simeq 27.72$ eradicated colonies. Increasing dose by one unit multiplies the fitted mean by $e^{{\hat{\beta }}_1}=e^{0.0001901}\simeq 1.000190$; for example, an increase of $100$ units multiplies it by about $1.0192$. The positive fitted effect is small and its displayed two-sided $p$-value, $0.105$, gives little evidence against a zero dose coefficient.

Under independent right censoring, observation $(x_i,d_i)$ contributeswhere $d_i$ indicates an event. Multiply these contributions, take the log-likelihood, and maximize it over the parametric survival-model parameter $\eta$, numerically if the score equations have no closed form.

With constant hazard $\theta$, observation $i$ contributesWriting $D={\sum }_i{v}_i$ and $T={\sum }_i{x}_i$, the log-likelihood is $D\log \theta -\theta T$, whose derivative is $D/\theta -T$. Hencethe number of events divided by total person-time.

Ignoring terms independent of $r_0$, the log-likelihood isIts score vanishes atThe expected Fisher information isBecause a shape-three gamma variable has mean $3r_0$ and variance $3r_0^2$,The estimator is unbiased and attains the Cramér-Rao lower bound ${\mathcal{I}}_N(r_0{)}^{-1}$.

A Poisson observation is an integer-valued count modeled by a Poisson distribution of specified intensity, possibly conditionally on a latent field. Independent Poisson observations motivate Poisson data fidelity through their log-likelihood.

The unscaled residual deviance of an exponential dispersion family compares its fitted mean with the saturated statistical model at a common dispersion parameter. Its normalization removes the factor $1/\varphi$ in the log-likelihood. In Poisson regression and binomial regression, $\varphi =1$, so it is twice the saturated-minus-fitted log-likelihood. Under suitable regularity, $D/\varphi$ has approximately a chi-squared distribution with the residual degrees of freedom; this is not an automatic accurate approximation for sparse responses. Nested deviance reductions with estimated dispersion lead to approximate F-tests after scaling by the dispersion estimate.

A saturated model has enough independently varying mean parameters to match every observed response or response cell. In individual Bernoulli distribution data its fitted probabilities are $p_i=y_i$, and its maximum log-likelihood is zero under $0\log 0=0$. It defines the reference in a binomial deviance.

A score equation sets the score function equal to zero. Its solutions are stationary points of the log-likelihood, and a negative-definite likelihood Hessian identifies an interior local maximum.

For $u\ge 0$ and a positivity-preserving operator, this is the negative log-likelihood for independent Poisson observations with intensity $1+Tu$, up to constants depending only on the data and the unit background. For bounded $g\ge 0$, its scalar derivatives are $1-g/(1+s)$ and $g/(1+s{)}^2$ on $s=Tu\ge 0$. It is convex, and strictly convex in the predicted intensity when $g>0$ almost everywhere. On a unit-area domain, the Jensen inequality gives $D_g(u)\ge \Vert Tu{\Vert }_1-\Vert g{\Vert }_{\infty }\log (1+\Vert Tu{\Vert }_1)$. The shift makes $\log (1+Tu)$ integrable whenever $Tu\in L^1$; no integrability of $\log u$ is needed.

For hidden activations $h_j=\sigma (b_j+{\sum }_r{W}_{jr}{x}_r)$ and class probabilities $p_c={\mathrm{softmax}}_c(a+Vh)$, a one-hot label $t$ has log-likelihood $\ell ={\sum }_c{t}_c\log p_c$. Put ${\delta }_c=t_c-p_c$ and ${\Delta }_j=h_j(1-h_j){\sum }_c{V}_{cj}{\delta }_c$. ThenThese follow from the chain rule and give single-observation stochastic gradient descent updates for the negative log-likelihood.