Codex @codex  0

For one observation from a continuous uniform distribution on $[0,\theta ]$ and prior density $\theta e^{-\theta }$, the likelihood cancels the factor $\theta$ and leaves $e^{-\theta }{1}_{\{\theta \ge x\}}$. Normalization gives posterior density $e^{-(\theta -x)}{1}_{\{\theta \ge x\}}$, for $x\ge 0$. Its mean is $x+1$ and variance one. The Bayes estimator under squared error loss is therefore $x+1$, unchanged by multiplying the loss by a positive constant.

A prior predictive check simulates parameters from the prior distribution and observations from their sampling model, then examines whether those observations are plausible before conditioning on data. This reveals unintended information from apparently broad priors, especially after nonlinear transformations or when scales and correlations are coupled. It differs from a posterior predictive check, which uses the posterior distribution.

For a common success chance with beta prior and conditionally independent binomial observations of sizes $m_i$, put $M={\sum }_i{m}_i$ and $s={\sum }_i{x}_i$. Beta-binomial conjugacy gives posterior mean $(\alpha +s)/(\alpha +\beta +M)$, equal to $Z(s/M)+(1-Z)\alpha /(\alpha +\beta )$. Multiplying by the next exposure predicts its expected count. The factor tends to one as exposure increases and to zero as prior concentration increases at fixed prior mean.

For a common gamma distribution (shape $\alpha$, rate $\beta$) frequency prior distribution and conditionally independent Poisson distribution yearly counts of means $w_j\Theta$, observed average amounts $y_j$ determine counts $k_j=w_j{y}_j/c_j$. The posterior distribution shape is $\alpha +{\sum }_j{k}_j$ and rate is $\beta +{\sum }_j{w}_j$. The future per-policy mean is the display, equal to credibility factor $W/(W+\beta )$ times ${\sum }_j(w_j/W)(c_{n+1}/c_j)y_j$ plus the complementary weight times $c_{n+1}\alpha /\beta$. The data must lie on their count lattice; impossible noninteger recovered counts have zero likelihood.

// Destination: probability-theory.bigb

conditionally independent normal distribution observations of mean $\theta$ and known variance ${\sigma }_1^2$, with normal prior mean $\mu$ and variance ${\sigma }_2^2$, give normal-normal conjugacy with known observation variance. Their posterior mean has the displayed exact credibility estimate form. It minimizes posterior squared-error loss and predicts the conditional expectation of the next observation.

A Bayes act minimizes $\int L(\theta ,a)\Pi (d\theta )$ for a specified parameter distribution $\Pi$. Applying the same minimization to the posterior after observing data gives a Bayes decision rule. The optimal data-based Bayes risk is the expectation of the optimal posterior no-data loss.

For loss $L(\theta ,a)=w(\theta )|\theta -a|$, assume $w\ge 0$, $0<\mathbb{E}[w(\Theta )\mid x]<\infty$ and $\mathbb{E}[w(\Theta )|\Theta |\mid x]<\infty$. The posterior expected loss is a positive constant times absolute-error loss under the posterior distribution reweighted by $w$. Thus its minimizers are the medians of that weighted posterior distribution. More precisely, the weighted cumulative distribution function $F_w$ satisfies $F_w(a^{-})\le 1/2\le F_w(a)$; the one-sided slopes of the convex loss give this condition.

For relative absolute loss on positive parameters, $w(\theta )=1/\theta$. With a gamma distribution posterior of shape two and rate $b>0$, reweighting produces an exponential distribution of rate $b$, so the estimate is $(\log 2)/b$. The weighted first-moment condition ensures finite loss; without it every finite action can have infinite loss, making a median characterization uninformative.

For integrable real $g$, including points where $g<0$ decreases the integral and including points where $g>0$ increases it. Thus the negative set minimizes the integral, up to null sets and arbitrary choices where $g=0$. This elementary signed-integral fact derives the Bayes classifier and cost-sensitive Bayes classifier.

With zero cost for correct classifications, assign class one when the displayed inequality holds and class two otherwise. It compares conditional expected losses; ties may be resolved arbitrarily. For positive costs and densities, the log likelihood ratio threshold is $\log (c(1\mid 2){\pi }_2/(c(2\mid 1){\pi }_1))$.

A basis vector is one of the vectors in a specified basis of a vector space. Its coordinate column in that basis has one entry equal to one and all other entries zero. Columns of a matrix of a linear map are the images of domain basis vectors, expressed in the chosen codomain basis.

An ordered basis is a basis equipped with an ordering of its basis vectors. The order fixes which position of a coordinate vector corresponds to each basis vector.

A normalized basic sequence generates a spreading model when the norms of all finite scalar combinations converge as the first index tends to infinity, uniformly over increasing choices of the indices. The limiting norm on finitely supported scalar sequences is unchanged by increasing relabellings of coordinates. Completing this norm gives a Banach space with its canonical basic sequence. A diagonal application of the finite-colour infinite Ramsey theorem produces such a model from a subsequence of every normalized basic sequence.

An unconditional basic sequence is a Schauder basis for its closed span whose expansions converge after arbitrary permutations or coordinate suppressions. Equivalently, the finite coordinate projections are uniformly bounded. Failure of this property gives pairs of disjointly supported unit vectors at arbitrarily small distance, obtained by separating a finite combination with a large coordinate projection.

For $K\ge 1$, let $T_K(X)$ consist of finite normalized sequences satisfying $K^{-1}\sum |a_i|\le \Vert \sum a_i{x}_i\Vert \le \sum |a_i|$ for every scalar combination. Its tree order measures the depth of finite copies of the unit vector basis of ${\ell }^1$. A separable Banach space containing no copy of ${\ell }^1$ has a countable index because every such tree is closed and well-founded. Countable well-founded trees can be used to construct separable reflexive spaces with arbitrarily high index, precluding a separable reflexive universal space.

Every bounded sequence in a Banach space has either a weakly Cauchy sequence as a subsequence or a subsequence equivalent to the unit vector basis of ${\ell }^1$. The real-scalar proof represents vectors as continuous functions on a compact dual ball and uses infinite Ramsey theory to select independent lower and upper level sets when pointwise convergence cannot be obtained.

For a finite-dimensional associative algebra over an algebraically closed field, the Ext quiver has a vertex for each simple module isomorphism class and $a_{ij}$ arrows $i\to j$. For a basic algebra with $P_i=A{e}_i$, applying ${\mathrm{Hom}}_A(-,S_j)$ to the projective-cover sequence gives $a_{ij}=\dim e_j(J/J^2)e_i$. Our path algebra convention multiplies consecutive arrows $\alpha :i\to j$, $\beta :j\to l$ as $\beta \alpha$, making left modules have arrow maps $V_i\to V_j$. The orientation convention must be stated when comparing other definitions.

For $B=\overline{\mathbb{C}[1,x]}$ inside $A$, the displayed set is the full spectrum of $x$ in $A$. On a bounded complementary component, the maximum modulus principle gives $|p(\lambda )|\le {\max }_{{\sigma }_A(x)}|p|\le \Vert p(x)\Vert$. Evaluation $p(x)\mapsto p(\lambda )$ therefore extends to a character of an algebra on $B$, showing $\lambda \in {\sigma }_B(x)$. Conversely the unbounded resolvent component survives by resolvent components in a closed unital subalgebra. In particular, if zero lies in that component, $x^{-1}$ belongs to the norm closure of polynomials in $x$.

For a compact set $K$ in the domain, a norming bounded linear functional at $f(z)$ reduces the assertion to the scalar maximum modulus principle. If $z$ lies inside $K$, use its connected component of the interior; that component's boundary lies in $\partial K$. If $z\in \partial K$, the inequality is immediate. No smoothness or connectedness assumption on $K$ is required.

For a Banach algebra-valued holomorphic function, set $p_n(w)=\Vert f(w{)}^{2^n}{\Vert }^{1/2^n}$. These continuous functions decrease to the spectral radius by the spectral radius formula. If $M={\sup }_{\partial K}r(f(w))$, the continuous functions $\max \{p_n,M\}$ on $\partial K$ decrease to the continuous constant $M$. The Dini theorem gives uniform convergence. Apply the norm maximum principle for a Banach-space-valued holomorphic function to each power, and then let $n\to \infty$. This proof does not assume continuity of spectral radius.

If $Y$ is a proper closed subspace of a normed space $X$ and $0<\alpha <1$, there is a unit vector $x\in X$ whose distance from $Y$ exceeds $\alpha$.