Topics — What a work is about. Nucleation holds whatever reading the owner files here, whether or not an Anthology of the SOTA topic could also hold it; instruction for ML practice belongs to the anthology (ADR-048). The words are seeded from the clusters the reading-time triage of 2026-09-25 found, and grow by decision.
Mathematical statistics — the theory of statistical inference read for itself — comparison and sufficiency of experiments (Blackwell, Le Cam deficiency), statistical decision theory, the sufficiency theorems (Fisher–Neyman, Basu, Bahadur) and bounds on estimation. Bayesian modelling is probabilistic-modeling (ADR-031).
43 of 1352 THEORY documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| THEORY-156 v3 | For experiments on a finite parameter set whose outcome spaces are standard Borel, being at least as informative for every decision problem is the same as being able to simulate the other by a randomisation | Blackwell (1953), LIT-781, Theorem 8 with Theorems 1–3: for experiments with finitely many states and decision problems with bounded losses, α attains every loss vector β attains, in every problem, exactly when a Markov kernel turns α’s observation into β’s under every state. The direction from simulation to informativeness is Blackwell (1951); the finite converse is Sherman and Stein (1951); the general converse is proved in 1953. It is a qualitative order and says nothing about how much worse an incomparable experiment is. The record re-derived the converse for standard Borel outcome spaces (note of 2026-10-29); arbitrary spaces, which the paper claims through a quoted 1951 result, are QUESTION-321. | Active |
| THEORY-157 | The classical theorems on sufficient statistics need only the copy/discard structure of Markov kernels: Basu’s theorem holds in every Markov category, and Fisher–Neyman and Bahadur need only strict positivity, not the existence of conditional distributions | Fritz (2020), LIT-772, read in NOTE-588, Theorems 14.5, 15.8 and 16.3 with the definitions of Sections 13–15. Sufficiency, completeness and ancillarity are stated with copy, discard, composition and almost-sure equality alone. Basu’s theorem follows with no further axiom; Fisher–Neyman and Bahadur use strict positivity, which Stoch has although it lacks regular conditionals. The match with the classical theorems is shown in FinStoch (Fisher–Neyman) and through bounded completeness in Stoch (Basu), not for measure-theoretic Fisher–Neyman. | Proposed |
| THEORY-161 v2 | In Zhao et al.’s (2025) and Chai et al.’s (2023) semantic rate–distortion frameworks, meaning is a latent variable with a known joint law with the observation, so their limits are indirect source-coding limits; Zhao et al.’s posterior-matching semantic distortion is, with KL divergence, exactly the information bottleneck, and with total variation a bound on the expected lost value of every bounded-loss decision | From the readings of Zhao et al. (2025), LIT-793, Chai et al. (2023), LIT-812, and the survey of Xin, Fan and Letaief (2024), LIT-823. These frameworks fix meaning as a hidden S with a known p(S, X); their theorems are indirect rate–distortion results that do not depend on what S is. Zhao et al.’s posterior distortion reduces, for KL, to I(S;X) − I(S;Y), making the function the information bottleneck (LIT-338), and for TV bounds the extra Bayes risk of every bounded loss. The two identities are this record’s derivations, not the papers’. | Active |
| THEORY-176 v2 | A task-loss stitching penalty compares representations in one decision problem, through a fixed top network and a restricted map class, so it is not Blackwell’s order: representations that are functions of each other can stitch differently, and a negative penalty shows the fitted map did not reproduce the layers it replaced | Bansal, Nakkiran & Barak (2021), LIT-850, define the stitching penalty L_ℓ(r; A) − L(A), with L_ℓ(r; A) the least task loss of A’s top layers fed r through a map from a simple family S, and use it to call one representation better than another. Read against Blackwell (1953), LIT-781, and THEORY-156, that ordering is one coordinate of the comparison of experiments with the decision rules cut down to A’s head composed with S. It is not invariant under Blackwell equivalence, which the paper intends, and it meets Blackwell’s order only where the fitted map reproduces the replaced layers exactly. The relations are this record’s derivations from the paper’s definitions, not the paper’s. | Active |
| THEORY-215 | A Hawkes process with a Lipschitz rate function and signed kernels has a unique stationary regime, which suitably started versions approach, when the Lipschitz-weighted absolute kernel mass is below one (a spectral radius below one, for a network) or when the rate is bounded | Brémaud and Massoulié (1996), LIT-1035, read in NOTE-728: Theorems 1, 2, 7 and 8, proved. Inhibition is allowed because the conditions use |h|. The conditions are sufficient, not necessary; the kernels are fixed in time; uniqueness is among solutions with finite mean intensity. In the linear excitatory case the condition is the branching ratio below one, but in the nonlinear signed case the spectral-radius quantity is a bound, not a branching ratio, and a bounded rate needs no bound on the kernels at all. Nothing is proved about cascade sizes or about what happens past the threshold. | Active |
| THEORY-280 v2 | The bisimulation metric between two finite Markov chains with observed states and a known cost bounded by 1 can be estimated without a transition model from independent samples of each chain’s discounted state-pair occupancy, to accuracy ε with probability 1 − δ from a number of samples polynomial in the state-space sizes, 1/(1−γ), log(1/δ) and 1/ε² | Calo, Jonsson, Neu, Schwartz & Segovia-Aguas (2025), LIT-948, read in NOTE-763. Bicausal occupancy couplings are characterised by constraints that use only each chain’s sampled occupancy measure over consecutive state pairs. A stochastic primal-dual method on them gets an ε-accurate estimate with probability 1−δ from O((|X||Y|(|X|+|Y|) + (|X|+|Y|)log(1/δ))/((1−γ)²ε²)) independent samples per chain. The states must be observed and enumerable, the cost known, and the samples drawn with restarts. A single running stream and latent states are outside it. | Active |
| THEORY-308 v2 | The neural Donsker–Varadhan estimator of mutual information is strongly consistent as an extremum estimator over compact families of critics, and, for critics bounded by M and L-Lipschitz in their d parameters, its uniform deviation from the population bound needs a number of samples set by d, M, L and the parameter radius, not by the data dimension | Belghazi et al. (2018), LIT-256, §3.3 and App. 8.2.2–8.2.3, read in NOTE-828. On a compact domain with Lebesgue densities, the supremum of the Donsker–Varadhan bound over some compact network family is within any η of the mutual information, and the empirical supremum converges to the population one almost surely. Uniform deviation ε needs Õ(d log d/ε²) samples for d parameters, under bounded, Lipschitz critics. The guarantee is for the exact empirical maximizer, not for gradient training. The finite-sample estimate is not a bound (Poole et al., LIT-246), and the proof has two gaps a reading found. | Active |
| THEORY-318 v2 | A bound k on the max-information between a dataset and an analyst’s chosen statistic multiplies by at most 2^k the probability that the statistic overfits that dataset, and an ε-differentially private mechanism on datasets of n points keeps k at most log₂e·εn, whatever the distribution on datasets | Dwork, Feldman, Hardt, Pitassi, Reingold & Roth (2015), LIT-964, Supplementary Material only, read in NOTE-848 (Skimmed: the main text was not reached). Theorem 2 and Lemma 5 are proved there and were checked: P[overfit] ≤ 2^{I∞}·max P[overfit], and ε-DP gives I∞ ≤ log₂e·εn. Thresholdout’s guarantee (Theorem 11), n = O(ln(m/β)/τ²·min{B, √(B ln(m/β))/τ}), is stated with its proof elsewhere. Statistical queries on a discrete universe only, and a procedure that prevents inflation, not a correction for a search already run. | Active |
| THEORY-391 v3 | Recovery of a k-sparse vector by ℓ₁ minimization from n random measurements in dimension N switches, as N grows with δ = n/N and ρ = k/n fixed, from near-certain success to near-certain failure across a curve ρ(δ) that, for Gaussian matrices, the authors’ earlier face-counting theorems fix; in 2,948,000 optimizations at N = 200, 400 and 1600, eight non-Gaussian ensembles approached that curve in a way consistent with gaps shrinking as N^(−1/2), an exponent fitted on the same data, while at finite N the location depended on the ensemble | Donoho and Tanner (2009), LIT-1160, read in NOTE-933 from arXiv 0906.2530v1. The Gaussian transition is a theorem of the authors’ earlier papers, restated here. The universality is an observation from 2,948,000 optimizations at N = 200, 400 and 1600: strict finite-N universality is rejected, and agreement as N → ∞ is not rejected. Sparse and Hadamard matrices sit about one transition width low at small N. The title states only what was measured. Whether the agreement is a limit theorem is QUESTION-168. | Active |
| THEORY-534 | The optimality of maximum likelihood was never one theorem: each historical proof holds under conditions its author left implicit, and of the classical counterexamples the practically important one is a parameter count growing with the sample, while Hodges-type superefficiency is confined to a set of measure zero | Stigler (2007), LIT-044, read in full in NOTE-074. Fisher’s 1922 proof rests on sufficiency, his 1925 proof on exact normality, and his private 1930 proof on multinomial data and smooth degree-0 estimators. Rigorous treatments (Wald 1943, Cramér 1946) came at the cost of unwieldy conditions. Neyman–Scott, where the MLE of σ² tends to σ²/2, is the failure Stigler weighs as practical; Hodges’s superefficiency is possible only on a Lebesgue-null set (Le Cam 1953). It is a history: the theorem statements are reconstructions, and it says nothing about modern overparameterized models beyond one hedged warning. | Active |
| THEORY-641 v2 | Under a well-specified, identifiable parametric Markov model of the target given the joint past, with ergodic lagged states, transfer entropy is the limit of the normalised log-likelihood ratio for the null of no dependence on the source’s past, so it is Granger-sense predictive improvement; for finite-state chains the plug-in estimator is exactly that statistic, and under the null and the usual regularity it is asymptotically χ² with the difference in parameter counts as degrees of freedom | Barnett & Bossomaier (2012), LIT-048, read in full in NOTE-053. The consistency result is a one-line consequence of Props. 1–2, and Prop. 2 is argued informally. The χ² null is imported from Wald (1943), whose regularity conditions are not checked. The non-central χ² for T > 0 is at best a local-alternative approximation. The finite-state identity and the Gaussian equivalence (half the Granger statistic) are exact. | Active |
| THEORY-704 | In a stationary multivariate time series whose components include every relevant process, with interventions that are isolated exogenous changes and no instantaneous causation, a missing Granger-causal edge from a to b means intervening on a has no effect on b at the next step, non-ancestry means no effect at any horizon, and a set whose observed past blocks every back-door path into b’s ancestors identifies the effect from observational data, as a multi-step linear-predictor coefficient in the Gaussian case | Eichler & Didelez (2007), LIT-1164, read in full in NOTE-941. The back-door results are proved; the edge-to-effect statements in the conclusion are said to follow by the same reasoning; the front-door proof is deferred. All conditions are sufficient, and Assumption 2.3 is untestable from observational data. Two worked effects were checked. | Active |
| THEORY-705 v2 | For a target and two sources with finite alphabets, if unique information is to depend only on the two source–target pair marginals, then the minimum of the conditional mutual information over all joints with those marginals is an upper bound on every non-negative decomposition’s unique information, and the shared and synergistic information it implies are lower bounds; this unique information vanishes exactly when the other source is Blackwell-sufficient, and the measures are non-negative, satisfy the bivariate partial-information axioms and the identity property, and do not extend to three sources on the partial-information lattice | Bertschinger et al. (2014), LIT-1162, read in full in NOTE-937. The bounds, the Blackwell characterisation and the properties are proved. The motivating property is argued, not derived. Example 30 and the Table 1 rows reproduce, except two entries that conflict with the paper’s own Proposition 18. Under this measure, the record’s transfer-entropy examples split exactly as under I_min. | Active |
| THEORY-709 | Mutual information between continuous variables can be estimated from k-nearest-neighbour statistics, with the joint k-th-neighbour distance in the maximum norm fixing the scale at which marginal neighbours are counted, so that the biases of the three entropy estimates largely cancel; the one approximation is a locally constant density, the systematic error scales with k/N, and exactness for independent variables is observed in every tested case but not proved | Kraskov, Stögbauer & Grassberger (2004), LIT-1163, read in NOTE-938 (formal sections in full, numerics at lower resolution). The estimators are derived up to one approximation and one neglected correction; the rest is numerical. Checked by simulation on correlated Gaussians. | Active |
| THEORY-717 | For experiments with finitely many states and finitely many outcomes, being at least as informative in every decision problem is the same as being reproducible from the other by a Markov matrix | Blackwell (1953), LIT-781, Theorems 4–6, read in NOTE-595: the Sherman– Stein theorem, with a new minimax proof. With n states and finite outcome sets, write experiments as n × N Markov matrices P and Q. Every risk vector Q attains in every decision problem is attainable with P exactly when Q = PM for some Markov matrix M. The arbitrary-outcome extension is THEORY-156, still Proposed. A qualitative order: most pairs are incomparable, and it says nothing about how far apart they are. | Active |
| THEORY-743 | For finite X and Y, every information-bottleneck optimum at every β gives one encoder to all inputs sharing p(y|x), so it is a stochastic coarsening of the minimal sufficient partition (the causal states of memoryless transduction); as β → ∞ the optima converge to that partition, but in general no finite β attains it, except when Y is a function of X | Shalizi and Crutchfield (LIT-1240, read in NOTE-976) assert that the information bottleneck recovers the causal states as β → ∞. The record proves it, against stochastic encoders as well as partitions, and adds that every finite-β optimum factors through those states. So the IB hierarchy always nests under its top. It says nothing about whether optima at two finite β nest. Finite alphabets only. | Active |
| THEORY-778 | In the Random Hierarchy Model with v symbols per level, m rules per symbol and s-tuples, the covariance between two tokens whose lowest common ancestor is ℓ levels up has zero mean over grammars and root-mean-square size about ((1 − m/v^(s−1))/(v³ m^(2ℓ−1)))^(1/2), falling by a factor m per level, while its estimate from P sequences carries noise of about (v²P)^(−1/2), so the two meet at P_ℓ ≈ v m^(2ℓ−1)/(1 − m/v^(s−1)); all to leading order in large v and m | Cagnetta and Wyart (2024), LIT-883, App. D–F, read in NOTE-687; re-derived to leading order by the promotion audit and checked against exact covariances of sampled grammars. A statement about the statistics of the generative model only: it says nothing about what a trained network learns, or when, and nothing about natural language. | Active |
| THEORY-812 | As dimension grows with the undersampling and sparsity ratios fixed, the ℓ₁ phase transition for recovering sparse vectors from random measurements sits at the same curve for a broad class of random-matrix ensembles, those with a kind of orthogonality or weak independence, as for Gaussian matrices, while differing at finite size | The general reading that THEORY-391 held before the promotion audit narrowed it to what 2,948,000 optimizations at three problem sizes showed. Donoho and Tanner (2009), LIT-1160, read in NOTE-933, state it as the hypothesis of “weak universality” and pose the class of ensembles it holds for as an open problem. Nothing about non-Gaussian ensembles is proved. Filed under ADR-047 v4 so the wide form stays a thesis evidence can confirm or reject. | Proposed |
| THEORY-894 | A flow of pairwise comparisons on a graph splits orthogonally and uniquely into a gradient part, the l2-best global ranking, and a cyclic part, which splits into curl (local) and harmonic (global) inconsistency; a finite game’s flow of unilateral payoff differences has no curl part, so every finite game is uniquely a potential game plus a harmonic game plus a nonstrategic remainder, and potential games are exactly those with no harmonic part; the decomposition is cardinal | Jiang, Lim, Yao & Ye (2011), LIT-1439, Theorems 4.7, 4.8 and 5.1: the Hodge/Helmholtz decomposition of edge flows, and HodgeRank’s least-squares ranking with its curl and harmonic residuals. Candogan, Menache, Ozdaglar & Parrilo (2011), LIT-1442, Theorems 4.1 and 5.1: every finite game is P ⊕ H ⊕ N, and potential games are P ⊕ N. Both read from arXiv; the proofs were followed. Settled as mathematics. Whether its shares measure a negotiation’s potential depends on cardinal assumptions that the record’s orderings do not supply. | Active |
| THEORY-951 | For independent identically distributed observations, and neglecting how far the log-likelihood-ratio walk overshoots its boundaries, Wald’s sequential probability ratio test accepts the null hypothesis with probability (A^h − 1)/(A^h − B^h), where h ≠ 0 solves E e^{hz} = 1, and takes on average [(1 − γ) log B + γ log A]/E z observations; no sequential test with the same error probabilities needs fewer on average under either hypothesis than Wald’s lower bounds, which the test attains when the overshoot is zero, and for a normal mean it saves 47–63% of the fixed-sample test’s observations at error rates from .01 to .05; the overshoot-free formulas are exact for Brownian motion with drift, which the paper never mentions | Wald (1945), LIT-1523, read in NOTE-1165. The SPRT’s operating characteristic and mean sample size follow from two identities of the stopped walk, and its optimality from Jensen’s inequality, all exact when the walk stops on the boundary. Optimality with overshoot is only conjectured in the paper. The drift-diffusion limit that the record takes from Bogacz et al. (LIT-569, THEORY-068) is not stated by Wald. It is the case in which his approximations become exact. | Active |
| THEORY-952 | In Wald’s sequential probability ratio test, which samples until the likelihood ratio of two simple hypotheses leaves an interval (B, A), the realized error probabilities satisfy α/(1 − β) ≤ 1/A and β/(1 − α) ≤ B whenever the test terminates with probability one, with or without independent observations; so the boundaries A = (1 − β)/α and B = β/(1 − α), set with no distribution theory, give errors α′ ≤ α/(1 − β), β′ ≤ β/(1 − α) and α′ + β′ ≤ α + β, and they would be exact only if the ratio stopped on the boundary | Wald (1945), LIT-1523, read in NOTE-1165. The boundaries of a sequential likelihood-ratio test follow from the error rates by a two-line argument that needs only termination, so the test needs no sampling distribution. The approximate boundaries are conservative in sum. Either error alone may rise, but only by a factor 1/(1 − β) or 1/(1 − α) at most. Nothing here says how many observations the test takes or that it is optimal: that is THEORY-951, and it needs i.i.d. observations. | Active |
| THEORY-954 | When the information a design yields depends on the unknown parameter, as in nonlinear models or nonlinear functions of linear ones, Bayesian design criteria are normal approximations to expected utility that average a local criterion over the prior, such as the expected log det of the Fisher information, and locally optimal design is the special case of a point-mass prior; a locally optimal design can leave the quantity of interest inestimable, as for the turning point of a quadratic regression on [−1, 1] with best guess ½, where the locally c-optimal design puts half its mass at 0 and half at 1 and identifies only θ₁ + θ₂ | Chaloner and Verdinelli (1995), LIT-1525, read in NOTE-1161. Once the information depends on θ, no design is optimal without saying which θ it is optimal for, and the prior is how that is said. The approximations’ accuracy is “not much known” in general. The claim that optimal Bayesian designs gain support points as the prior disperses rests on examples, not a theorem, and is not part of this account. | Active |
| THEORY-956 | In the normal linear model with known error variance and a conjugate normal prior of precision R/σ², the design that maximizes the expected Shannon information of the posterior is the one maximizing det(nM(η) + R), Bayes D-optimality, and the design that minimizes expected quadratic loss with weight matrix A is the one minimizing tr A(nM(η) + R)^-1, Bayes A-optimality; both criteria depend on the design and not on the data, so in this model a sequential design gains nothing over the best fixed one | Chaloner and Verdinelli (1995), LIT-1525, read in NOTE-1161. The alphabetical criteria are utilities in disguise, and the utility fixes the criterion. This holds exactly only in the normal linear model with known variance. With unknown variance only Bayes A keeps its simple form, and in nonlinear models the criteria are approximations (THEORY-954). E- and G-optimality have no utility behind them. | Active |
| THEORY-972 | In the author–topic model the likelihood of a corpus depends on the topics Φ and the authors’ topic distributions Θ only through the document–word matrix ΦΘA, A being fixed by the observed authorship, so LDA (each document its own author) and the topic-free author model (each author one topic) are special cases; and the likelihood leaves Φ and Θ unidentified not only up to the permutations of topics the paper names but along a continuum of changes of topic basis whenever every topic gives every word positive probability | Rosen-Zvi, Chemudugunta, Griffiths, Smyth and Steyvers (2010), LIT-1546, read in NOTE-1180. The likelihood sees members’ topic profiles only through the word distributions they jointly produce, and a small change of topic basis that keeps every distribution a distribution produces exactly the same corpus probabilities. Only the Dirichlet priors then choose among the profiles. It says nothing about the posterior’s concentration, and nothing about conditions under which Φ and Θ are identified up to permutation. | Active |
| THEORY-985 | The beta process BP(c, B0) is the de Finetti mixing measure of the Indian buffet process: given n objects, the next has each existing feature with probability equal to the number holding it over c + n and a Poisson(cγ/(c + n)) number of new ones, so n objects hold Poisson(γ Σ_{i<n} c/(c + i)) distinct features, which is the IBP’s αH_n at c = 1 and γ = α, and c alone sets how much objects share | Thibaux and Jordan (2007), LIT-1537, read in NOTE-1171. The beta process plays for the IBP the role the Dirichlet process plays for the Chinese restaurant process, and its concentration c separates how much objects share from the mass γ that sets features per object. The conjugacy it rests on is cited, not proved, in the paper. It does not give dependence between features or over time. | Active |
| THEORY-993 | In the mixed membership stochastic blockmodel the probability of an edge from p to q, given the memberships, is (1 − ρ)π_pᵀBπ_q, so the graph’s likelihood given the memberships is unchanged by π ↦ Rπ, B ↦ R⁻ᵀBR⁻¹ for any invertible R that keeps the memberships on the simplex and B in [0, 1], a continuum around any solution with B’s rates strictly between 0 and 1; and the sparsity parameter ρ enters only through (1 − ρ)B, so it is fixed by nothing but the bound B ≤ 1, and the paper’s printed estimator of it is the fraction of absent edges whatever the fit | Airoldi, Blei, Fienberg and Xing (2008), LIT-1553, read in NOTE-1179. The paper says the blockmodel and the memberships do not compete, because one is global and asymmetric and the other local and symmetric. Given the memberships they do: a change of block basis moves every node’s memberships toward the centre and sharpens B, with the same edge probabilities. What can still separate them is the Dirichlet family assumed for memberships, not the graph. The confounding of ρ with B’s scale is exact even in the marginal likelihood. | Active |
| THEORY-999 v2 | In the finite beta–Bernoulli feature model with prevalences Beta(α/K, 1), the expected number of occupied features among N objects is K(1 − (α/K)B(α/K, N + 1)), strictly below the Indian buffet process’s αH_N for every finite K and tending to it as K grows, and each object holds Kα/(K + α) features on average rather than α | Griffiths and Ghahramani (2011), LIT-1556, read in NOTE-1172, derive the IBP as the K → ∞ limit of πk ~ Beta(α/K, 1), zik ~ Bernoulli(πk), and give K+ ~ Poisson(αH_N) for the limit only. At finite K the count falls short, by 27% at N = 12, α = 3, K = 16. It says nothing about stick-breaking truncations, which approach the IBP much faster. | Active |
| THEORY-1006 | In the dependent Indian buffet process each item’s feature indicator is a Gaussian process thresholded at the quantile of a stick length that does not change with the covariate, so its prevalence is the same at every covariate value and its persistence is set separately by the covariance; the whole feature matrix at a covariate value is an Indian buffet process only when there is no shared feature-level process, and with one the items are correlated and fewer features are occupied | Williamson, Orbanz and Ghahramani (2010), LIT-1548, read in NOTE-1175. Thresholding at a fixed quantile keeps each feature’s prevalence at its stick length at every covariate value, which is what a hand-written logistic transition does not do. The paper’s statement that Z(t) is an IBP at each t holds for its collapsed model; with a shared feature-level GP it holds item by item only. Persistence for a fixed correlation still falls with prevalence. | Active |
| THEORY-1010 | On ten corpora simulated from three-level hierarchical LDA itself (100 documents of 250 words, a vocabulary of 100, sparse topics with η = 0.005, γ = 1), the highest-probability state of the collapsed Gibbs sampler recovered the generating tree exactly in eight; in one of the other two a wrong tree had higher posterior probability than the true one | Blei, Griffiths and Jordan (2010), LIT-1542, §6.1 and Fig. 6, read in NOTE-1176. It shows that the sampler can invert hLDA on data hLDA generated, with sparse, well-separated topics. It does not show that the tree is identified in general, or that a tree found in real data was there to be found. One failure is a posterior that prefers the wrong tree, which no better sampler would fix. | Active |
| THEORY-1012 | Latent features made dependent through a layer of categories, each observation’s features being a function of how many of its categories carry each feature, stay finite on finite data when a feature carried by none of an observation’s categories is always off; under that condition a category can switch features on and never off, so features depend on one another by sharing a category, or, where an observation has a single category, by excluding one another, and no feature or category suppresses another | Doshi-Velez and Ghahramani (2009), LIT-1544, read in NOTE-1184. Z = f(CM) with f(0) = 0 gives correlated nonparametric feature models with O(log log N) expected features for the DP-IBP and IBP-IBP. The correlations are motifs, sets of features switched on together, plus, in the DP-IBP, exclusion between categories; suppression is outside them, and negative correlation is left as future work. The models’ predictive advantage over the IBP is mixed in the paper’s own table. | Active |
| THEORY-1015 | Groups of objects drawn from beta processes share features only if the measure they are drawn around is discrete; in the hierarchical beta process that measure is itself a beta process draw, and at each shared feature a group’s prevalence is Beta(c_j b, c_j(1 − b)) around the shared prevalence b, so a small c_j lets groups differ and a large one pulls them together | Thibaux and Jordan (2007), LIT-1537, read in NOTE-1171. The hierarchical beta process B ~ BP(c0, B0), Aj ~ BP(cj, B) gives several groups one unbounded feature library at group-specific prevalences. It treats each object as in one group, exchangeable within it, and has no dynamics; its one experiment, 58% against 50% on a single text split, is suggestive only. | Active |
| THEORY-1019 | Under the Markov Indian buffet process the expected number of features on at the first time step is α, as in the Indian buffet process, but it rises over time toward α·E[ln((2 − b)/(1 − b))] with b ~ Beta(γ, δ), which is 9.04 at γ = 10 and δ = 1, so after the first step the rows are not distributed as an Indian buffet process | Van Gael, Teh and Ghahramani (2008), LIT-1577, read in NOTE-1199. A chain starting off is on at step t with probability a(1 − (b − a)^t)/(1 − b + a). As M → ∞ the a’s form a Poisson process of intensity α/a on (0, 1), so E[number on at t] = α ∫₀¹ E_b[(1 − (b − a)^t)/(1 − b + a)] da. That is α at t = 1 and tends to α·E[ln((2 − b)/(1 − b))]. The number of features ever on is still Poisson(αH_T), as the paper says. This concerns the expected count only, not the full row distribution beyond it. | Active |
| THEORY-1034 | In the Markov Indian buffet process each feature’s probability of staying on is drawn independently of its probability of switching on, so how long a feature persists once on does not depend on how rare it is; the process is defined for the time steps of one sequence, not for several objects over time | Van Gael, Teh and Ghahramani (2008), LIT-1577, read in NOTE-1199. Each column is a two-state chain from an off state with a_m = P(0 → 1) ~ Beta(α/M, 1) and b_m = P(1 → 1) ~ Beta(γ, δ), independent, and the stick-breaking construction keeps each b paired with its a as an independent draw. Rarity (small a) and persistence (b) are separate per-feature parameters. It does not say that prevalence is held fixed over time, which it is not (THEORY-1019), nor how to share the chains across several objects. | Active |
| THEORY-1037 | In the infinite hidden Markov model, drawing at each time step a slice variable uniformly below the probability of the transition taken leaves the posterior unchanged and leaves only finitely many state trajectories with positive probability, so a whole trajectory can be resampled exactly by forward filtering and backward sampling, with no truncation of the state space | Van Gael, Saatci, Teh and Ghahramani (2008), LIT-1622, read in NOTE-1204. u_t ~ Uniform(0, π_{s_{t−1}s_t}) is an auxiliary variable, so sampling u given s and then s given u leaves p(s | y) invariant; given u, the forward recursion sums only over transitions with π > u_t, of which there are finitely many per row. It says the sampler is exact, not how fast it mixes (THEORY-1044). | Active |
| THEORY-1041 | In the Indian buffet process the feature probabilities in decreasing order are products of independent Beta(α, 1) draws, so the k-th has mean (α/(α + 1))^k, and keeping only the first K of them gives an expected number of occupied features among N objects that approaches αH_N from below: 8.97 at K = 16, N = 12 and α = 3, against 9.31 in the limit and 6.83 for the finite model with sixteen independent prevalences | Teh, Görür and Ghahramani (2007), LIT-1627, read in NOTE-1210. μ_(k) = ∏{l≤k} ν_l with ν_l ~ Beta(α, 1). Truncated at K, the expected number of occupied features is Σ{k≤K} Σ_{j=1}^{N} (−1)^{j+1} C(N, j) (α/(α + j))^k, which sums to αH_N as K → ∞. At N = 12, α = 3: 6.80, 8.97, 9.27 and 9.31 at K = 8, 16, 24 and 32. It concerns the prior’s expected count, not a posterior, and the paper itself computes no truncation error. | Active |
| THEORY-1043 | The finite feature model in which each of K features has prevalence Beta(α/K, 1) has, as K grows, a limit on equivalence classes of binary matrices that ignore the order of columns, and the Indian buffet process, in which the i-th object takes each earlier feature with probability m/i and Poisson(α/i) new ones, produces the same distribution, so its objects are exchangeable and the number of features they hold is Poisson(αH_N) | Griffiths and Ghahramani (2005), LIT-1623, read in NOTE-1208, and their review (LIT-1556). P([Z]) → α^{K+}/∏_{h≥1} K_h! · exp{−αH_N} · ∏_k (N − m_k)!(m_k − 1)!/N! on lof-classes, and the buffet gives the same. It is a statement about the limit: at finite K the expected count of occupied features is below αH_N (THEORY-999), and an ordered truncation of the same limit is a different approximation (THEORY-1041). | Active |
| THEORY-1044 | For the infinite hidden Markov model the beam sampler reaches the posterior in fewer iterations than the single-site Gibbs sampler, and is less sensitive to initialization and hyperpriors, particularly when hidden states persist and emissions are uninformative, at a cost per iteration close to the Gibbs sampler’s | Van Gael, Saatci, Teh and Ghahramani (2008), LIT-1622, read in NOTE-1204. Two synthetic designs (an almost cyclic 4-state HMM, length 800, 20 runs; persistent 4-state HMMs with self-transitions to 0.999, length 4000, 60 starts), a well-log series and a text task. The beam sampler is exact by construction (THEORY-1037); how much faster it mixes is shown only in iterations and by eye. | Proposed |
| THEORY-1046 | In infinite sparse factor analysis, where a binary mask says whether a source is active at a data point and an independent Gaussian value says how strongly, the evidence of that data point for the source being active, with its value integrated out, is σ̂·exp(μ̂²/(2σ̂²)) for the posterior mean μ̂ and standard deviation σ̂ of the value; when the residual has no component along the source’s loading the ratio is σ̂, below one, so the data favour an inactive source over an active one at zero by a factor that the prior on the value sets | Knowles and Ghahramani (2007), LIT-1626, read in NOTE-1209. With y = g x + ε, x ~ N(0, 1), ε ~ N(0, σ_ε² I): ∫ p(y | x) p(x) dx / p(y | x = 0) = σ̂ exp{μ̂²/(2σ̂²)}, σ̂² = σ_ε²/(σ_ε² + g′g), μ̂ = g′ε/(σ_ε² + g′g). It is the Gaussian case only; the Laplacian case is the paper’s (14), checked the same way. It concerns one data point and one source, with the loading known. | Active |
| THEORY-1096 | Blackwell’s comparison of experiments, its easy direction: an experiment is a family of distributions indexed by a common parameter, and if one experiment is a garbling of another by a Markov kernel, every risk attainable with the garbling in any decision problem with bounded losses is attainable with the original, on any outcome spaces | Split from THEORY-156 by the status review of 2026-10-27 (ADR-057). THEORY-156’s equivalence needs a second proof of the general converse and stays Proposed. Ten of its twelve dependents used only its definitions (an experiment, a garbling, the informativeness order), or only the easy direction, that a garbling never helps a decision-maker free to choose any rule. The easy direction holds on any outcome spaces, and its proof is two lines, re-derived below under ADR-047 v2’s ruling, which ADR-057 keeps. It is Theorem 3 of Blackwell (1953), LIT-781, read in NOTE-595, and Blackwell’s 1951 result. The finite converse is THEORY-717, Active. | Active |
| THEORY-1177 | For experiments on a finite parameter set, being at least as informative for every decision problem is the same as being able to simulate the other by a randomisation | The wide form THEORY-156 held before the twelfth status review narrowed it, kept as a Proposed reading by the ADR-047 kind-1 sweep of 2026-10-30: the source asserts it, and THEORY-156, Active, is its evidence. Blackwell (1953), LIT-781, claims the equivalence for any outcome space, through Theorem 2, quoted from his 1951 paper. The record’s re-derivation reaches only outcome spaces that are standard Borel, since its last step needs a regular conditional distribution. | Proposed |
| THEORY-1269 | In overparameterized linear regression, iterated self-training with fresh covariates, noiseless pseudo-labels and minimum-norm students applies a fresh random orthogonal projection to the estimate each round, so noise from the first fit decays geometrically while signal along a direction of variance s decays at the slower rate s/(s + τ); risk is U-shaped under anisotropy and monotone under isotropy, tends to the risk of predicting zero, and every student fits its teacher exactly on its own sample | Wu, Yang and Sun (2026), LIT-1884, Theorem 3.2, Corollaries 3.3–3.4 and B.1, and the projection identity of §3.2; read in NOTE-1471 and checked by Monte Carlo in the spiked model. Active for the spiked closed form; the general concentration result (Theorem 4.2) is the paper’s, proof not checked. It does not cover reused data, noise injected every round, or nonlinear students. | Active |
| THEORY-1312 | A phase-amplitude coupling measure can be significant when only one non-sinusoidal rhythm is present, because the rhythm’s harmonics carry amplitude that depends on its own phase; such coupling is confined to bands spanning the rhythm’s harmonics, so a measured coupling is evidence of one rhythm modulating another only after waveform shape is controlled | Donoghue, Schaworonkow and Voytek (2021), LIT-1999, §5 and Fig. 5D–E, read in NOTE-1529 from the PsyArXiv preprint, standing on Kramer et al. 2008 and Cole et al. 2017. Active: checked by a short simulation in which a sharp-peaked 10 Hz wave gave a modulation index of 0.021 between its phase and 25–60 Hz amplitude against 0.0001 for a cosine, and none in a single-harmonic or an 80–250 Hz band. It does not say any particular published PAC is an artefact, nor that broadband coupling far above the harmonics is. | Active |
| THEORY-1337 | Neural field recordings carry aperiodic 1/f-like activity with power at every frequency, whose exponent varies with age, state and task; so power in a band, correlated power changes across bands, and ratios of band power can arise from the aperiodic component with no oscillation changing, and 1/f structure is a ubiquitous background of field data rather than a mark of any particular organization | Donoghue, Schaworonkow and Voytek (2021), LIT-1999, §§1 and 3, read in NOTE-1529. Active: the spectral argument is elementary (an aperiodic signal has power at all frequencies, and a change in its exponent moves every band), and the simulations and cited cases (the ADHD theta/beta ratio) are in the text. It does not say aperiodic activity is noise without physiological meaning; the paper calls it “a physiologically informative feature”. | Active |