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For independent exponential observations with rate $\theta$, the MLE is $1/\overline{X}$ and has asymptotic variance ${\theta }^2/n$.

For a binomial count $K\sim \mathrm{Bin}(n,p)$, $n\ge 1$, with $p\in [0,1]$, the maximum-likelihood estimator is $\hat{p}=K/n$. For $0<K<n$, differentiating $K\log p+(n-K)\log (1-p)$ gives a unique interior maximum. If $K=0$ or $K=n$, the maximum is respectively at zero or one. Excluding those endpoints from the parameter space means these samples have only a supremum, not an attained maximum. The estimate is unbiased and has variance $p(1-p)/n$, by the binomial distribution moments.

The log-likelihood is the natural logarithm of the likelihood function:For positive finite likelihood, the natural logarithm is a strictly increasing function, so the maximizing parameter values are unchanged. Set $\ell =-\infty$ where $L=0$; the same ordering then includes zero-likelihood boundaries whenever a positive finite maximum exists. For independent and identically distributed random variables, a product becomes a sum,This makes derivatives simpler and avoids forming products of tiny numbers in floating-point arithmetic. These equal-maximizer and additive formulas underlie the worked calculations in MIT 18.05, Reading 10b, section 3.1.

For the ordered sample of eight successes and two failures in likelihood function, on $0<p<1$,The score equation ${\ell }^{\prime }(p)=0$ gives $8(1-p)=2p$, hence $\hat{p}=8/10=0.8$. The negative second derivative proves strict concavity, so this is the unique interior maximum. Both counts are positive, so $L(0)=L(1)=0$, while $L(p)>0$ inside: the same point is the unique global maximum on $[0,1]$. Counting only successes adds $\log 45$ to the log-likelihood, leaving both derivatives and the MLE unchanged. For all-success or all-failure samples, the boundary analysis in maximum-likelihood estimator applies instead of an interior score equation.

The two panels below compare $k=2,5,8$ successes out of ten Bernoulli trials. Their peaks are at $\hat{p}=k/10$. The left panel uses relative height; the right panel uses the log-likelihood difference from each peak. These are the same comparison on two scales:A difference of $-2$ means relative height $e^{-2}$, and a difference of $-10$ means $e^{-10}$; the log-likelihood scale makes small relative heights easier to see. Neither vertical axis is a parameter probability density function. The zero likelihood endpoints have log-likelihood $-\infty$ and are outside the finite vertical range in the right panel.

Multiplying a likelihood function by a positive parameter-independent constant adds a constant to its log-likelihood. Subtracting its peak removes that constant. This is why the ordered-data and count-data curves coincide in this figure after rescaling. When differentiable, the gradient of the log-likelihood is the score function; the negative Hessian matrix is the observed information. A solution of a score equation still needs checking against other stationary points and boundaries: maximizing the log-likelihood does not automatically make it concave.

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.

An ideal $\mathfrak{m}$ of a commutative unital ring $R$ is maximal exactly when $R/\mathfrak{m}$ is a field.

The maximal spectrum $\mathrm{MaxSpec}R$ is the set of maximal ideals of a commutative ring $R$.

The residue field at a maximal ideal $\mathfrak{m}$ is the quotient field $\kappa (\mathfrak{m})=R/\mathfrak{m}$. At a prime ideal $\mathfrak{p}$, the residue field is the fraction field of $R/\mathfrak{p}$.

After neglecting the decaying mode, the linear growth between conformal times ${\tau }_1$ and ${\tau }_2$ is

In cosmic time during matter domination, $a(t)\propto t^{2/3}$ and the pressureless density contrast obeysIts growing and decaying solutions are $t^{2/3}$ and $t^{-1}$.

For a constant matrix $A$ and a forcing vector-valued function $g$, considerWhen the component Laplace transforms converge and the Laplace transform of a derivative applies, setting $Y=\mathcal{L}y$ and $G=\mathcal{L}g$ givesChoose $s$ in the convergence half-plane and outside the spectrum of $A$. The resolvent $(sI-A{)}^{-1}$ is the Laplace transform of the matrix exponential $e^{At}$ for sufficiently large real part. Inverting gives the variation-of-constants formulaThe matrix acts on the forcing from the left, so there is no implicit commutation of a time-dependent forcing with $A$.

If $A$ is an $m$-by-$n$ matrix and $B$ is an $n$-by-$m$ matrix over the same field, both products are square matrices andThe products can even have different sizes: for $A=(12)$ and $B=(34{)}^T$, $AB=(11)$ and $BA=\left(\begin{array}{cc}3& 6\\ 4& 8\end{array}\right)$, but both have trace $11$.^[ref]

Grouping all but one factor proves that $\mathrm{tr}(A_1{A}_2\cdots A_k)$ is unchanged by a cyclic permutation whenever the cyclic products are defined and square. In particular, every commutator has trace zero, since $\mathrm{tr}(AB-BA)=0$ by linearity.

Cyclic rotation does not allow arbitrary reordering. Forone has $ABC=A$, so $\mathrm{tr}(ABC)=1$, but $AC=0$, so $\mathrm{tr}(ACB)=0$. Rotating $ABC$ to $BCA$ or $CAB$ preserves its trace; swapping just $B$ and $C$ need not.

An MPS parent Hamiltonian is a sum of local projectors onto the orthogonal complement of the tensor's local support. The MPS is a frustration-free ground state.

An MPS tensor is injective after blocking when products of its physical matrices span the full virtual matrix algebra.

Caliper matching permits a candidate matching pair only when a prespecified distance is at most a tolerance $c$. For propensity score $e(X)$, possible rules are $|e(X_i)-e(X_j)|\le c$ or $|\mathrm{logit}e(X_i)-\mathrm{logit}e(X_j)|\le c$. These are different distances; the numeric tolerance must be interpreted on the selected scale.^[ref]

For a treated unit with propensity score $0.40$, a raw-score caliper $c=0.03$ admits a control with score $0.42$ and excludes one with score $0.46$. The matching algorithm still has to choose among admissible controls and decide whether controls may be reused. A tighter caliper excludes distant pairs but can leave more treated units unmatched. Excluding those units changes the population represented by the matched estimate; a caliper is not a guarantee of an unchanged average treatment effect on the treated. Check covariate balance after matching, and account for the paired or reused-control structure in statistical inference.

Covariate balance means that the compared treatment groups have similar distributions of specified pretreatment covariates. Exact distributional balance in a weighted population isThis is a property of the represented covariates and chosen weights. Matching and inverse probability weighting can aim to approximate it.

Equal means are weaker than equal probability distributions. For example, a covariate taking values $-1,1$ equally often in one group and $-2,2$ equally often in the other has mean zero in both, but variances one and four. A zero standardized mean difference would miss that discrepancy. Inspect relevant spreads, tails, categories and combinations as well as means.

A nonsignificant difference test does not demonstrate covariate balance: small samples may have low statistical power, while large samples can detect negligible discrepancies. Covariate balance on measured pretreatment variables also cannot establish conditional exchangeability for unmeasured potential outcomes.

If a finite simple graph is $k$-regular with $k\ge 1$ and has $n$ vertices has a maximum matching of size $m$, its unmatched vertices are independent. Counting their incident edges gives $k(n-2m)\le 2(k-1)m$, and hence $m\ge kn/(4k-2)$.

For a finite graph $G$, the matching number isIt is the cardinality of a maximum cardinality matching. The vertex capacity constraint gives $0\le \nu (G)\le \lfloor |V(G)|/2\rfloor$. The worked path graph examples have $\nu (P_4)=2$ and $\nu (P_6)=3$. A perfect matching reaches $|V(G)|/2$; the empty graph has matching number zero.

A maximal matching is a matching in a graph to which no extra edge can be added while retaining all current selected edges. Equivalently, no edge joins two unmatched vertices in a matching; those vertices form an independent set. Every maximum cardinality matching is a maximal matching, but the converse fails.

Take the path graph $P_4$ with vertices $1,2,3,4$ and edges $(1,2),(2,3),(3,4)$. A greedy algorithm that first selects the middle edge produces $M=\{(2,3)\}$. Both remaining edges touch a matched vertex, so this size-one matching in a graph is maximal. Yet $N=\{(1,2),(3,4)\}$ has two edges. No matching in a graph on four vertices has more than two edges, so $N$ is a maximum cardinality matching. The two remaining free vertices of $M$, namely $1$ and $4$, are not adjacent; that does not prevent a longer augmenting path in a matching from joining them.

Maximality is useful as a quick stopping rule and a modest guarantee. For any maximal matching $M$ and maximum cardinality matching $N$ in a finite simple graph, every edge of $N$ touches an endpoint of an edge of $M$. Assign it to one such edge of $M$. Since $N$ has disjoint endpoints, each edge of $M$ receives at most two assignments, proving $|N|\le 2|M|$. The P₄ example reaches equality. Thus maximality guarantees at least half the largest cardinality, but says nothing comparable about arbitrary edge benefits. Its selected endpoints also form a vertex cover, as explained under vertex cover.

A perfect matching is a matching in a graph that covers every vertex exactly once. Its edges have no shared endpoints, and every vertex is an endpoint of one selected edge. It is useful when a collection must be divided into disjoint pairs with no element left over.

In a bipartite graph with equally many vertices on its two sides, a perfect matching can assign every person one allowed job and every job one person. The example under permanent of a matrix has three people and three jobs, with exactly three valid assignments. The blue edges below show those three perfect matchings; a crossing in the drawing is not an extra vertex.

Every selected edge covers two vertices, so a graph with an odd integer number of vertices has no perfect matching. An even integer number is necessary but does not guarantee existence: an isolated vertex cannot be covered. A perfect matching also need not be unique. For example, the complete bipartite graph $K_{2,2}$ has two perfect matchings, corresponding to the two ways of assigning two people to two jobs.

Counting all perfect matchings is a different question from finding one. For a zero-one biadjacency matrix of a balanced bipartite graph, the count is its permanent of a matrix. The nonzero determinant criterion for a perfect matching supplies an existence test, but cancellation means that a zero determinant does not rule one out.

The finite-horizon Snell envelope of rewards $Z_n$ is defined backward by $V_N=Z_N$ and $V_n=\max \{Z_n,\mathbb{E}[V_{n+1}\mid {\mathcal{F}}_n]\}$. It is the least supermartingale dominating $Z$.