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For $p,q>0$, express the product of the two Gamma integrals as the nonnegative double integralUse $r=x+y$ and $u=x/(x+y)$, with inverse $x=ru$, $y=r(1-u)$ and Jacobian determinant of absolute value $r$. The new domain of integration is $r>0$, $0<u<1$, and the integrand with this factor becomes $r^{p+q-1}{e}^{-r}{u}^{p-1}(1-u{)}^{q-1}$. Tonelli's theorem and separation of the two integrals therefore give $\Gamma (p+q)B(p,q)$. The positive real parameter hypothesis ensures that all the original integrals are finite.

For complex parameters with positive real parts, the same proof first gives the undivided identity $\Gamma (p+q)B(p,q)=\Gamma (p)\Gamma (q)$. Absolute integrability follows by taking real parts in the exponents, so Fubini's theorem applies. The nonvanishing of the Gamma function then permits division by $\Gamma (p+q)$.

The shortest binary linear recurrence producing the prefix $11001011$ has length three:Its connection polynomial is $1+D^2+D^3$.

Suppose terminal utility is nondecreasing and concave, admissible trading plans are closed under convex combinations, and extra initial cash can be retained in a positive bank account. Assume the expected-utility value is finite. Then the remaining-horizon portfolio value is nondecreasing and concave in initial wealth.

For concavity, combine admissible plans from initial wealths $x,y$ with weights $\lambda ,1-\lambda$. Linear self-financing budget equations give terminal wealth $\lambda X_T+(1-\lambda )Y_T$ from initial wealth $\lambda x+(1-\lambda )y$. Concavity of the utility function gives the corresponding expected-utility inequality. Use optimal plans when attained, or arbitrarily near-optimal plans and pass to their suprema. For monotonicity, retaining the extra initial cash gives no smaller terminal wealth.

The convexity hypothesis is substantive: discrete portfolio restrictions need not permit the mixed plan, so a linear wealth equation alone does not prove concavity of the optimization value.

The minimum distance of a BCH code is at least its design distance. A hypothetical word of smaller weight yields a square Vandermonde system from the consecutive-root equations; its support elements are distinct, so the determinant is nonzero and every word coefficient must vanish.

For error locations $i_1,\dots ,i_{\nu }$ in a cyclic code with primitive root $\alpha$, the error locator polynomial is $\sigma (Z)={\prod }_h(1-{\alpha }^{i_h}Z)$. Its roots encode the error positions.

A beta prior $\mathrm{Beta}(\alpha ,\beta )$ combined with $s$ Bernoulli successes and $f$ failures gives posterior $\mathrm{Beta}(\alpha +s,\beta +f)$.

A gamma prior with shape $\alpha$ and rate $\beta$ combined with $n$ independent observations from an exponential distribution of rate $\theta$ gives the posteriorUnder squared-error loss, the Bayes estimator is its posterior mean $(\alpha +n)/(\beta +{\sum }_i{X}_i)$.

For conditionally independent Poisson observations of mean $\theta$ and a gamma distribution prior of shape $r>0$ and rate $\lambda >0$, the likelihood function contributes ${\theta }^{{\sum }_i{x}_i}{e}^{-n\theta }$. Multiplying by the prior density gives a posterior distribution with shape $r+{\sum }_i{x}_i$ and rate $\lambda +n$. The single-observation update is shape $r+X$ and rate $\lambda +1$.

The posterior mean is the convex combinationThus the prior mean $r/\lambda$ and the sample maximum-likelihood estimate $\bar{x}$ bracket it. Integrating the next conditional Poisson distribution against this posterior distribution gives the Poisson-gamma mixture, a negative binomial distribution; its spread includes uncertainty in $\theta$ as well as the conditional count variation.

An improper prior is a nonnegative prior kernel with infinite total mass. It can still produce a proper posterior after multiplication by the likelihood and normalization.

A point-null mixture prior assigns positive mass to one parameter value and distributes the remaining mass continuously over alternatives. Posterior point mass is obtained by dividing the null component of the marginal density by the full marginal density.

A least favorable prior maximizes the Bayes risk over the allowed priors. If a Bayes rule has constant risk equal to the minimax value, its prior is least favorable.

Given observed data $x$, the posterior expected loss of action $a$ iswhere $\Pi (\cdot \mid x)$ is the posterior distribution and $L$ the loss function. Nonnegative losses define an extended nonnegative integral; meaningful comparisons require checking when finite loss is attainable. A Bayes decision rule chooses a measurable minimizer, or an approximate minimizer if an optimum is not attained.

With finite posterior second moment and squared-error loss, completing the square gives $\rho (a\mid x)=\mathrm{Var}(\Theta \mid x)+(a-\mathbb{E}[\Theta \mid x]{)}^2$. Thus the posterior mean minimizes the loss. With finite first moment and absolute-error loss, a posterior median minimizes it instead. Posterior expected loss conditions on the observed data and averages over parameter uncertainty; the frequentist risk of a decision rule fixes the parameter and averages over possible data.

For a positive parameter $\Theta$ and loss $L(d,\Theta )={\Theta }^{-1}(\Theta -d{)}^2$, assume $\mathbb{E}[\Theta \mid X]<\infty$ and $0<\mathbb{E}[{\Theta }^{-1}\mid X]<\infty$. ThenThis quadratic has the unique minimizer $d=(\mathbb{E}[{\Theta }^{-1}\mid X]{)}^{-1}$, a harmonic rather than arithmetic posterior mean. Both moment conditions matter: finiteness of the inverse moment alone does not prevent the posterior expected loss from being infinite for every finite action.

Under $L(\theta ,a)=q(\theta )(\theta -a{)}^2$ with $q\ge 0$, assume $0<\mathbb{E}[q(\Theta )\mid X]<\infty$ and $\mathbb{E}[q(\Theta ){\Theta }^2\mid X]<\infty$. Reweighting the posterior distribution by $q$ and completing the square in $a$ gives the unique Bayes estimatorIndeed, the posterior expected loss is a constant plus $\mathbb{E}[q(\Theta )\mid X](a-{\hat{\theta }}_B{)}^2$. The moment conditions justify this expansion. For $q=1$ it is the posterior mean; parameter-dependent weighting generally changes the optimum.

Under squared error loss, the Bayes estimator is the posterior mean.

A binary Bayes classifier under zero-one loss is unique up to null sets when the posterior class probabilities tie only on a null set. For two distinct nonsingular Gaussian distributions, the tie set is the zero set of a nonzero quadratic polynomial and has Lebesgue measure zero.

For two multivariate normal densities, the Bayes log posterior odds are a quadratic polynomial. Equal covariance matrices cancel the quadratic term and give linear discriminant analysis; unequal covariance matrices give quadratic discriminant analysis.

The basis constant of a Schauder basis is ${\sup }_n\Vert P_n\Vert$, where $P_n$ are its basis projections. Uniform boundedness makes this supremum finite.

If a bounded subset $K$ of a Banach space is bounded away from zero and zero lies in its weak closure, then $K$ contains a basic sequence. More generally, weak closure may be replaced by closure in any Hausdorff locally convex vector topology weaker than the weak topology.

The Uniform boundedness principle says that a pointwise bounded family of bounded linear operators from a Banach space $X$ to a normed vector space $Y$ has uniformly bounded operator norms. Being bounded pointwise means ${\sup }_T\Vert Tx\Vert <\infty$ for every fixed $x$; the conclusion is the stronger common bound ${\sup }_T\Vert T\Vert <\infty$.

For the proof, the closed sets $E_n=\{x:{\sup }_T\Vert Tx\Vert \le n\}$ cover $X$. The Baire category theorem makes one contain a ball. Subtracting two points in that ball bounds every linear operator on a ball about zero; rescaling gives the common operator norm bound. Completeness is the hypothesis that permits the Baire step. The principle turns individual scalar or vector bounds into a uniform estimate without having to compute each operator norm separately.