--- title: King Mixture Representations url: https://www.emergentmind.com/topics/king-mixture-representations type: topic --- # King Mixture Representations to=arxiv_search.query 天天乐购彩票json {"query":"id:2512.24414 OR \"Exact finite mixture representations for species sampling processes\"","max_results":5,"sort_by":"submittedDate","sort_order":"descending"}【อ่านข้อความเต็มjson to=arxiv_search.query েচ্ছিন্ন qq的天天中彩票 [{"arxiv_id":"2512.24414","title":"Exact finite mixture representations for species sampling processes","authors":["Salvatore Favaro","Vincenzo Vitale"],"summary":"Random probability measures, together with their constructions, representations, and associated algorithms, play a central role in modern Bayesian inference. A key class is that of proper species sampling processes, which offer a relatively simple yet versatile framework that extends naturally to non-exchangeable settings. We revisit this class from a computational perspective and show that they admit exact finite mixture representations. In particular, we prove that any proper species sampling process can be written, at the prior level, as a finite mixture with a latent truncation variable and reweighted atoms, while preserving its distributional features exactly. These finite formulations can be used as drop-in replacements in Bayesian mixture models, recasting posterior computation in terms of familiar finite-mixture machinery. This yields straightforward MCMC implementations and tractable expressions, while avoiding ad hoc truncations and model-specific constructions. The resulting representation preserves the full generality of the original infinite-dimensional priors while enabling practical gains in algorithm design and implementation.","categories":["stat.ME","math.PR"],"published":"2025-12-30","pdf_url":"http://arxiv.org/pdf/2512.24414v1","abs_url":"http://arxiv.org/abs/2512.24414v1"}] to=arxiv_search.query 天天彩աիjson {"query":"id:2206.06404 OR id:2206.11094 OR id:2606.12455","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} 开元棋牌 to=arxiv_search.query 天天中彩票公司 񹚂json [{"arxiv_id":"2606.12455","title":"King Function for Shifted Gaussian: Laguerre Structure, Spectral Theory and Density","authors":["Guangyao Zhang"],"summary":"We study King function arising as radial kernels in the laboratory-frame spherical harmonic expansion of shifted Gaussian distributions. We first clarify their relation with the co-moving Laguerre hierarchy by means of a King-Laguerre expansion. We then derive the King differential equation and show that the associated self-adjoint operator in a Gaussian-weighted Hilbert space is unitarily equivalent to the free radial Schrodinger operator on the half-line. This yields the spectral representation and generalized eigenfunction. Finally, we prove that real-parameter King function, lies in the resolvent set, form a dense non-orthogonal system in a natural radial velocity space, providing an approximation-theoretic basis for King mixture representations. Weighted L1-integrability criteria and closed-form moment formulas are also derived, justifying the normalization of King function.","categories":["math-ph","math.SP","physics.plasm-ph"],"published":"2026-06-05","pdf_url":"http://arxiv.org/pdf/2606.12455v1","abs_url":"http://arxiv.org/abs/2606.12455v1"},{"arxiv_id":"2206.11094","title":"Discrete mixture representations of parametric distribution families: geometry and statistics","authors":["Lutz Mattner","Stephan Schrempp"],"summary":"We investigate existence and properties of discrete mixture representations $P_\\theta =\\sum_{i\\in E} w_\\theta(i) \\, Q_i$ for a given family $P_\\theta$, $\\theta\\in\\Theta$, of probability measures. The noncentral chi-squared distributions provide a classical example. We obtain existence results and results about geometric and statistical aspects of the problem, the latter including loss of Fisher information, Rao-Blackwellization, asymptotic efficiency and nonparametric maximum likelihood estimation of the mixing probabilities.","categories":["math.ST","math.PR","62F10, 62F11, 60E05, 62F12"],"published":"2022-06-22","pdf_url":"http://arxiv.org/pdf/2206.11094v1","abs_url":"http://arxiv.org/abs/2206.11094v1"},{"arxiv_id":"2206.06404","title":"Compositional Mixture Representations for Vision and Text","authors":["Sebastian D. A. Lippl","Jörn-Henrik Jacobsen","Janis Keuper"],"summary":"Learning a common representation space between vision and language allows deep networks to relate objects in the image to the corresponding semantic meaning. We present a model that learns a shared Gaussian mixture representation imposing the compositionality of the text onto the visual domain without having explicit location supervision. By combining the spatial transformer with a representation learning approach we learn to split images into separately encoded patches to associate visual and textual representations in an interpretable manner. On variations of MNIST and CIFAR10, our model is able to perform weakly supervised object detection and demonstrates its ability to extrapolate to unseen combination of objects.","categories":["cs.CV","cs.LG"],"published":"2022-06-13","pdf_url":"http://arxiv.org/pdf/2206.06404v1","abs_url":"http://arxiv.org/abs/2206.06404v1"}] “King mixture representations” is a nonstandard label applied to several distinct mixture-theoretic constructions. Its most precise usage in current Bayesian nonparametric work refers to the exact finite mixture representation of a proper species sampling process (SSP), obtained by introducing a latent truncation variable and reweighted atoms in a way that is compatible with Kingman’s paintbox representation of exchangeable partitions [2512.24414]. The term is not standard in the general literature on discrete mixture representations [2206.11094]. Distinct, non-equivalent usages also appear in multimodal representation learning, where images and text are embedded as Gaussian mixtures in a shared latent space [2206.06404], and in mathematical physics, where mixtures of King functions are used to approximate shifted-Gaussian radial structures [2606.12455]. ## 1. Terminological status and scope In the Bayesian nonparametric setting, the phrase denotes an exact finite-mixture reparametrization of an infinite-dimensional prior. The supplied synthesis explicitly introduces the name “King mixture representations” to emphasize compatibility with Kingman’s representation of exchangeable partitions. In that sense, the object of study is not a new stochastic process class, but a finite-mixture representation of any proper SSP that preserves the original law exactly at the prior level [2512.24414]. In broader mixture theory, the phrase has no established canonical meaning. The paper on discrete mixture representations of parametric families states that “The term ‘King Mixture Representations’ is not standard in the mixture literature and does not appear in the paper.” Its focus is instead the general identity $$ P_\theta=\sum_{i\in E} w_\theta(i)\,Q_i, $$ together with existence, geometry, Fisher-information loss, Rao–Blackwellization, asymptotic efficiency, and NPMLE of the mixing probabilities [2206.11094]. The phrase is therefore best understood as context-dependent. A concise classification is: | Context | Core object | Status of the term | |---|---|---| | Proper SSPs and exchangeable partitions | Exact finite mixture with latent truncation \(K\) | Explicitly introduced in the supplied synthesis | | General discrete mixture theory | \(P_\theta=\sum_i w_\theta(i)Q_i\) | Not standard | | Shifted-Gaussian radial analysis | Mixtures of King functions \(\mathcal K_l\) | Approximation-theoretic usage | This suggests that the SSP usage is the most technically specific sense of the term, while the others are analogical or domain-specific extensions. ## 2. Species sampling processes, exchangeability, and Kingman structure Let \((X,\mathcal B_X)\) be a Polish space. A general SSP is a random probability measure $$ G(A)=\sum_{j=1}^\infty w_j\delta_{\theta_j}(A)+w_0G_0(A),\qquad A\in\mathcal B_X, $$ where \(\theta_j\) are i.i.d. from a diffuse base measure \(G_0\), the nonnegative weights satisfy \(\sum_{j\ge1} w_j\le 1\) with \(w_0=1-\sum_j w_j\), and \((\theta_j)_j\) is independent of \((w_j)_j\). A proper SSP is the purely atomic case $$ \sum_{j=1}^\infty w_j=1 \quad \text{almost surely}, $$ so \(w_0=0\) and \(G\) has no residual diffuse component [2512.24414]. If \(X_i\mid G\) are i.i.d. from $$ f_G(x)=\int f(x\mid \theta)\,G(d\theta), $$ the sample induces an exchangeable random partition \(\Pi_n\) of \(\{1,\dots,n\}\) by tying observations that draw the same atom. The law of \(\Pi_n\) is described by the exchangeable partition probability function (EPPF) \(p(n_1,\dots,n_k)\), where \(k\) is the number of blocks and \(n_i\) are the block sizes. For classical Gibbs-type SSPs, including Dirichlet and Pitman–Yor priors, the EPPF admits closed form and determines predictive rules [2512.24414]. Kingman’s paintbox provides the structural backdrop. Kingman’s theorem states that any exchangeable random partition arises by sampling i.i.d. “colors” with frequencies \((p_j)\) from a random paintbox. Proper SSPs realize such partitions with paintbox frequencies \((w_j)\). In particular, Poisson–Dirichlet and Pitman–Yor priors are paintbox partitions with specific distributions on \((w_j)\). In the SSP usage of the term, “King” refers to this Kingman compatibility rather than to a new probabilistic object [2512.24414]. ## 3. Exact finite mixture representation The central theorem for proper SSPs introduces a latent truncation variable \(K\) and a family of reweighted atoms. Let $$ G(A\mid w,\theta)=\sum_{j=1}^\infty w_j\delta_{\theta_j}(A), $$ with \(\sum_{j=1}^\infty w_j=1\) almost surely, and let \(\{\xi_j\}_{j\ge1}\) be a strictly decreasing sequence in \((0,1)\) with \(\xi_j\downarrow 0\). For \(k\ge1\), define $$ s_k=\sum_{i=1}^k \xi_i^{-1}w_i, \qquad P(K=k\mid w)=(\xi_k-\xi_{k+1})\,s_k. $$ Conditionally on \((w,\theta,K=k)\), define $$ \tilde w_j=\frac{\xi_j^{-1}w_j}{s_k},\qquad j=1,\dots,k, $$ and the random finite measure $$ G^*(A\mid \tilde w,\theta,K=k)=\sum_{j=1}^k \tilde w_j\delta_{\theta_j}(A). $$ The theorem proves $$ G^* \overset d= G, $$ that is, equality in distribution at the prior level [2512.24414]. The representation is exact rather than asymptotic. It preserves the predictive distributions, the induced partition law, and the EPPF of the original infinite random measure. The proof is based on a telescoping identity and Tonelli’s theorem; in the summary this identity is written as $$ \sum_{k=1}^\infty\sum_{j=1}^k (\xi_k-\xi_{k+1})\,\xi_j^{-1}w_j\delta_{\theta_j} = \sum_{j=1}^\infty w_j\delta_{\theta_j}. $$ The support of \(K\) is \(\mathbb N\), with pmf \(P(K=k\mid w)\). Its law depends jointly on the schedule \(\{\xi_j\}\) and the SSP weights \(w\), and arises from a slice-like decomposition over intervals \((\xi_{k+1},\xi_k]\) [2512.24414]. For proper SSPs the finite representation remains purely atomic: no residual \(G_0\) term appears in \(G^*\). If an SSP has \(w_0>0\), the theorem applies to its proper atomic part, while the diffuse residual remains unaffected. The representation is therefore a prior-level identity that converts an infinite paintbox into a mixture over finite paintboxes of random size \(K\), without any ad hoc truncation [2512.24414]. ## 4. Canonical instances: Dirichlet, Pitman–Yor, and geometric stick-breaking For the Dirichlet process, described in the synthesis as Poisson–Dirichlet \(PD(\theta)\) with \(\theta=\alpha>0\), the EPPF is $$ p(n_1,\dots,n_k)=\frac{\alpha^k}{(\alpha)_n}\prod_{i=1}^k (n_i-1)!, $$ where \((a)_n=a(a+1)\cdots(a+n-1)\) is the rising factorial. For \(PY(\theta,\sigma)\), with \(\theta>-\sigma\) and \(\sigma\in(0,1)\), $$ p(n_1,\dots,n_k)=\frac{(\theta+\sigma)_{k-1,\sigma}}{(\theta+1)_{n-1}}\prod_{i=1}^k (1-\sigma)_{n_i-1}, $$ where \((a)_{m,\sigma}=\prod_{r=0}^{m-1}(a+r\sigma)\) is the generalized rising factorial [2512.24414]. Both priors admit stick-breaking weights $$ w_1=V_1,\qquad w_j=V_j\prod_{ij\}\). For the DP this becomes $$ v_j\mid \cdots \sim \mathrm{Beta}(1+n_j+m_j,\; \alpha+h_j). $$ If \(\xi_j\) is deterministic and independent of the stick-breaking variables, then $$ v_j\mid \cdots \sim \mathrm{Beta}(a_j+n_j,\; b_j+r_j), $$ where \(r_j=\sum_i \mathbf 1\{z_i>j\}\). For GSB, $$ v\mid \cdots \sim \mathrm{Beta}\!\left(a+2n,\; b+\sum_i (k_i-1)\right). $$ Allocations satisfy $$ z_i\mid k_i,\cdots \propto \mathbf 1\{z_i\le k_i\}\,\xi_{z_i}^{-1}w_{z_i}f(x_i\mid \theta_{z_i}), $$ and in the stick-breaking-dependent case \(\xi_j^{-1}w_j=v_j\), so $$ z_i\mid \cdots \propto v_j f(x_i\mid \theta_j). $$ For GSB, \(\tilde w_j=1/k_i\), hence \(z_i\mid \cdots \propto f(x_i\mid \theta_j)\) over \(j\le k_i\) [2512.24414]. The \(k_i\)-updates are especially transparent. In the schedule-dependent case, $$ k_i\mid z_i,\cdots \propto \frac{(\xi_k-\xi_{k+1})\,\mathbf 1\{k\ge z_i\}}{\xi_{z_i}}. $$ If \(\xi_k-\xi_{k+1}=w_k\), then $$ p(k_i=k\mid z_i=j)=\frac{w_k}{T_j},\qquad k\ge j,\qquad T_j=\sum_{\ell\ge j} w_\ell. $$ If \(\xi_j=\exp(-\eta j)\), then $$ k_i=\left\lfloor z_i-\frac{1}{\eta}\log(U)\right\rfloor,\qquad U\sim \mathrm{Unif}(0,1). $$ If \(\xi_j=(1-\rho)\rho^{j-1}\), then \(k_i=z_i+S_i\) with \(S_i\sim \mathrm{Geom}(1-\rho)\). For GSB, \(k_i=z_i+S_i\) with \(S_i\sim \mathrm{Geom}(v)\) [2512.24414]. Empirically, the finite samplers were compared against slice samplers on a simulated four-component Normal mixture and the galaxy data. In the simulated example \((n=250)\), the true weights were \(0.5,0.2,0.2,0.1\), with means \(-4,0,5,8\) and sds \(0.8,1,0.5,1.5\). The methods were DPFinite and DPSlice, and GSBFinite and GSBSlice. For moderate/large \(\eta\) and “natural” \(\xi_j\), DPFinite and DPSlice recovered the density well and \(c_n\) aligned closely with \(4\); GSB-based methods slightly over-clustered. Runtime for \(100{,}000\) iterations, evaluated over \(500\) grid points, was: for \(n=250\), DPFinite natural \(10.51\)s, \(\eta=1.0\) \(11.06\)s, DPSlice \(\eta=1.0\) \(20.24\)s; GSBFinite natural \(18.43\)s, \(\eta=1.0\) \(11.19\)s, GSBSlice \(\eta=1.0\) \(21.49\)s. For \(n=1000\), DPFinite natural \(37.70\)s, \(\eta=1.0\) \(26.64\)s, DPSlice \(64.70\)s; GSBFinite natural \(75.00\)s, \(\eta=1.0\) \(26.87\)s, GSBSlice \(71.57\)s. On the galaxy data \((n=82)\), all methods captured multimodality; DPFinite and DPSlice stabilized around \(3\) clusters; small \(\eta\) inflated \(c_n\) slightly [2512.24414]. ## 6. Extensions, assumptions, and numerical considerations Because the finite representation is a prior-level identity, it extends pointwise to non-exchangeable or covariate-dependent settings. If \(G_t\) is proper for each \(t\), with \(\sum_j w_{j,t}=1\) almost surely, and if there exists a strictly decreasing \(\xi_{j,t}\downarrow0\), then one can introduce \(K_t\) with $$ P(K_t=k\mid w_t)=(\xi_{k,t}-\xi_{k+1,t})\,s_{k,t}, \qquad s_{k,t}=\sum_{i\le k}\xi_{i,t}^{-1}w_{i,t}, $$ and reweight $$ \tilde w_{j,t}=\frac{\xi_{j,t}^{-1}w_{j,t}}{s_{k,t}}, $$ yielding \(G_t^*\overset d=G_t\). Algorithmically, this gives time- or covariate-indexed finite mixtures with exact preservation of the original non-exchangeable prior [2512.24414]. The stated assumptions are: a proper SSP, a diffuse base measure for atoms, and independence \(\theta_j\perp w_j\). The schedule \(\xi_j\in(0,1)\) may be deterministic or random, but it must be strictly decreasing to \(0\). The scope includes all proper SSPs, including Gibbs-type priors, normalized CRMs admitting stick-breaking forms, and stick-breaking processes with dependent lengths such as DSBw [2512.24414]. The main numerical guidance concerns the choice of schedule and the management of tail probabilities. Deterministic schedules such as \(\xi_j=\exp(-\eta j)\) or \(\xi_j=(1-\rho)\rho^{j-1}\) yield simple, stable \(k_i\)-updates by closed-form inversion. Very small \(\eta\) can inflate \(c_n\), since more components become available. In the stick-breaking-dependent case, synchronization of \(k_i\)-updates and tail mass \(T_j\) is important, and stable accumulators for \(\sum w_j\) and tail sums are recommended to avoid floating-point drift [2512.24414]. A further scope qualification concerns interpretation of cluster counts. The occupied-cluster summary \(c_n\) is data-driven and should not be interpreted as a direct estimate of a “true” finite \(m\). The exact finite representation yields exact inference for SSP mixtures; when the modeling objective is consistent recovery of a finite \(m\), the supplied synthesis states that mixtures of finite mixtures with \(\sigma<0\) and explicit priors on \(m\) are appropriate [2512.24414]. ## 7. Related and non-equivalent uses Outside Bayesian nonparametrics, the same label has been attached to different kinds of mixture constructions. In multimodal learning, one such usage refers to a shared Gaussian mixture latent space for images and text. There the image-induced mixture is $$ p(z\mid x)=\sum_{k=1}^{K}\pi_k(x)\,\mathcal N(z;\mu_k(x),\Sigma_k(x)), $$ while the text-induced mixture is $$ q(z\mid y)=\frac{1}{|y|}\sum_{i=1}^{|y|}\mathcal N(z;\mu_{\text{text}}(w_i),\Sigma_{\text{text}}(w_i)). $$ Textual compositionality is represented by mixing token-level Gaussians, and the training objective is $$ \mathcal L(\theta,\phi)=\mathcal L_{\text{align}}+\lambda_{\text{area}}\mathcal L_{\text{area}}, $$ with $$ \mathcal L_{\text{align}} = \mathbb E_{(x,y)\sim \tilde p} \left[ -\log p_\theta(y\mid x)+D_{KL}(p_\theta(z\mid x)\,\|\,q_\phi(z\mid y)) \right]. $$ This model uses STN-based patch extraction, diagonal covariances in practice, and \(K=5\) in all experiments. On MultiCIFAR10, the reported detection mAP values were WSDDN \(49.47\) and CoMix \(75.83\); on MultiMNIST, WSDDN \(88.27\) and CoMix \(87.04\) [2206.06404]. A different usage arises in the analysis of shifted Gaussians through the King function. For dimensionless radial coordinate \(r=v/\varsigma\) and shift parameter \(k=2u/\varsigma\), the King kernel is $$ \mathcal K_l(r;k,\varsigma) = \frac{1}{\pi^{3/2}\varsigma^3(2l+1)}\,e^{-(k/2)^2}e^{-r^2}i_l(kr), $$ where \(i_l\) is the modified spherical Bessel function. The paper establishes the King differential equation, a Gaussian-weighted self-adjoint operator, a unitary equivalence to the free radial Schrödinger operator, and a spectral representation on the imaginary branch. On the real branch, $$ \mathscr K_l^{\mathbb R}=\{\Psi_{l,k}(r)=e^{-r^2}i_l(kr)\mid k\in \mathbb R_{>0}\} $$ is not orthogonal in $$ H_{\rm rad}=L^2\big((0,\infty); r^2 e^{-r^2}\,dr\big), $$ but its linear span is dense. This yields discrete approximations of the form $$ f(r)\approx \sum_{s=1}^S c_s\,\mathcal K_l(r;k_s,\varsigma), $$ with least-squares fitting in \(H_{\rm rad}\) [2606.12455]. Against these usages, the general theory of discrete mixture representations provides the broadest abstraction: $$ P_\theta=\sum_{i\in E} w_\theta(i)\,Q_i. $$ That theory includes classical noncentral \(\chi^2\) mixtures, barycentric convexity, minimality and non-minimality phenomena, information inequalities, Rao–Blackwellization, and NPMLE for mixing probabilities, but explicitly notes that “King Mixture Representations” is not a standard term there [2206.11094]. Taken together, these works indicate that the phrase does not name a single unified theory across fields. Its most specific and technically developed meaning is the exact finite-mixture reparametrization of proper species sampling processes compatible with Kingman’s paintbox, while other occurrences denote either mixture-based compositional embeddings or approximation schemes built from King functions. Source: https://www.emergentmind.com/topics/king-mixture-representations