Peter Cotton

Softmax is the gradient of a perturbed maximum: independent Gumbel noise turns the multiclass maximum into log-sum-exp. General perturbed-optimizer expectations are typically evaluated by Monte Carlo, and the leading heteroscedastic image classifiers, whose output layer is exactly a low-rank Gaussian argmax, sample a temperature-softmax relaxation. For Gaussian perturbations with low-rank-plus-diagonal covariance, one shared conditional field yields every winner probability, the perturbed-max potential with a certified tail remainder, and, by a new reverse-mode formula, gradients with respect to means, loadings, and variances, at the asymptotic cost of a forward pass: deterministic quadrature at small factor rank, fixed-node Rao–Blackwellized factor integration in general. The dual regularizer's curvature is the effective resistance of a photo-finish graph, giving each input its own simplex geometry where softmax fixes one. The exact likelihood learns through decision-boundary rivals, and softmax is characterized as the unique link for which the Fenchel–Young loss and the likelihood coincide.

The operation of a novel open-source platform where mostly autonomous algorithms are tasked with predicting a large variety of streaming data is described. Reward and combination is achieved by means of a near-the-pin mechanism generalizing lottery countbacks to continuous space.

The Python package ice-skaters replaces each numeric stream entering a streaming machine-learning pipeline with two scalars produced by the stream’s own online distributional forecaster: the one-step predictive mean, which carries the level, and the standardized surprise, the probability integral transform of the arriving value pushed through the standard normal quantile, which carries the news and is bounded near |z| = 7 by construction. The package ships two estimators compatible with the river streaming-learning protocol: a transformer for input streams and a regressor wrapper that adds the target’s own pair while leaving the target raw. On river’s own regression datasets under river’s progressive-validation protocol, the wrapper alone beats the recommended StandardScaler pipeline on three of four datasets untouched and ten seeds out of ten under simulated feature contamination, while tying the fourth; in controlled simulation the full substitution beats raw features, a running z-score, a median/MAD winsorizer, and a Huberised loss thirty seeds out of thirty under every contamination type tested, at a small clean-data toll. We report the losing rows with the winning ones: distance-based learners do not benefit, entity-interleaved streams handicap a single forecaster per key, informative heavy tails should not be tamed, and the front-end costs roughly 900× a running scaler per tick. The theoretical contribution is a pathwise regret-transfer theorem with measured constants and no calibration assumption: conjugated through the forecaster’s predictive CDF, an online linear refinement of the surprises achieves log-loss within an explicitly computable O(d log T ) of every refinement in its class, on any data sequence whatsoever, and in particular can never lose more than O(d log T ) nats to the forecaster alone.

A claim about the behavior of a covariance or correlation method is a claim about a distribution over matrices, and the distribution is rarely stated. Papers validate estimators, factorizations, and portfolio constructions on whatever generator is at hand: a Wishart draw, a uniform LKJ sample, one empirical panel. We measure how much the choice matters. Across fifteen named, seeded ensembles at identical dimension and sample size, the Ledoit–Wolf shrinkage intensity ranges from $0.07$ to $0.86$, and eleven audited claims from eleven fields, from hierarchical risk parity to the Mantel test, each flip verdict as the ensemble changes, for stated structural reasons. The results assemble into a matrix of claims against measures whose outlined cells mark the structure each claim was demonstrated on or intended for. The randomcov library names the measures and seeds the draws, so a published claim can be re-run under all fifteen and its verdict reported per measure. A joint leaderboard of the scikit-learn and precise covariance estimators has five different winners across the fifteen ensembles. Eleven audits are completed here; a docket of candidates from astronomy, genomics, neuroscience, and chemometrics, including provably ensemble-invariant claims that serve as null controls, is assembled for the rest.

Hierarchical Risk Parity and its Schur-complementary generalization build a portfolio by recursive bisection, consulting only the covariance within each current block and discarding cross-block information the sample already contains. We repair this without re-optimizing: calibrate latent Thurstone abilities that reproduce the hierarchical weights under the dependence the recursion actually used, then re-race those same scores under the fuller dependence on hand. Tested on a known-truth synthetic market and $12,000$ real S\&P 500 and REIT trials, the repair reliably reduces out-of-sample tail risk, with most of the gain coming from the race's own reallocation rather than from correlation recovery per se. Whether genuine tail dependence adds further value beyond correlation is mixed on real data; separated from that estimation question, a confidence-weighted tail view gives a provably bounded-regret way to act on a tail belief however it was formed, mirroring the scope of Black–Litterman's own return views.

Hierarchical risk parity is defended on the regime where the sample covariance cannot be inverted. We test that defence on its own terms, with the population covariance known by construction so realized risk is exact, and find no interval of sample size where the method is the right answer. Below a tenth of an observation per asset it is beaten by inverse-variance weighting, which inverts nothing either and consults no tree. Above parity it is beaten by any regularised optimizer, in 78 percent of five hundred randomly drawn dependence structures, and by long-only minimum variance on the raw sample covariance in 76 percent, so the no-short-sale constraint alone closes the gap. We also show that no shrinkage class small enough to be interpretable can reproduce the method, by a counting argument confirmed at three universe sizes, and that nested clustered optimization on the same clusters tracks the shrinkage frontier while hierarchical risk parity does not, which locates the cost in the inverse-variance split rather than in the hierarchy.

We give analytical conditions under which the Schur portfolio with the best out-of-sample variance lies inside the bridge, our hierarchical-to-minimum-variance family, with provable examples on both sides. Cross-block estimation error separates the optimum from both ends under stated endpoint conditions, and an exchangeable four-asset family is solved exactly, its optimizer decreasing from full coupling to the middle. The mechanism persists under whole-matrix noise, with a sharp second-order sign criterion at full coupling, and the median-optimal coupling climbs with sample size. Every portfolio is the minimum-variance portfolio of some positive definite matrix, so the dial moves implied beliefs.

Hierarchical equal risk contribution cuts a dendrogram into clusters, holds the inverse-variance portfolio inside each cluster, and budgets across clusters by inverse cluster variance. We show that its budget rule is already the one the global minimum-variance portfolio uses, and that what it lacks is conditioning in two places: each cluster on the other clusters, and each asset on its cluster mates. Damping the two Schur complements by γ and η gives a square with the method at one corner and the global optimum at the opposite corner. The inner dial has a closed form that interpolates the naive inverse-variance quantities and Stevens’ regression quantities, keeps every weight positive up to an explicit frontier, and on an equicorrelated cluster traces the straight segment from inverse variance to the cluster’s minimum-variance portfolio. Under estimation error the best point can be strictly inside the square, and we give the two-dial local theorem at the minimum-variance corner, a sign criterion for the inner dial with both signs realized, and exact examples, including one whose optimum is η = 3/5 with gaps 1/45 and 1/20 to the two ends.

Thurstone portfolio polishing calibrates latent abilities that reproduce a benchmark under a reference dependence law and re-races them under a richer one, with a scalar ϕ dialling the estimated correlation in. Choosing the reference to be the block-diagonal part of the estimate makes that blend exactly the estimate with its cross-cluster entries scaled by ϕ, and for a two-block partition the conditional covariance of that matrix is the damped Schur complement at γ = ϕ2 . The tilt dial is therefore the Schur dial under a square root. What is new is the pair of endpoints. Every bridge catalogued so far ends at Σ−1 u, a linear solve that can short any asset without bound; this one ends at a race, which lies on the simplex, splits a duplicated position rather than doubling it, and stays bounded. The taper is positive semidefinite for every ϕ, so unlike an unmatched pricing space the√path has no pole, and the closed-form reliability of the cross-block estimate transfers as ϕ⋆ = γ ⋆ , which sets the dial from data rather than by tuning.

A two-tier allocation holds a portfolio inside each cluster and then decides how much capital each cluster gets. The second decision needs a k × k matrix, and every construction in this family so far builds that matrix from the same cluster portfolios it holds. It need not. Separating what is held from what it is priced against gives a family with two spaces rather than one, and the far end of the bridge is exact for every choice of the pricing space: the k conditioned directions already sum to Σ−1 u, so any non-degenerate pairing recovers it. Choosing one representative asset per cluster, as hierarchical core-orbital allocation does, therefore cannot affect the far end. The interior is a different matter. Once the two spaces differ the k × k matrix is no longer symmetric, and it can pass through singularity inside the dial: we give a positive-definite 6 × 6 example where it does, and on estimated covariances it does so in about a quarter of markets. Pricing against the cluster portfolios keeps the matrix a Gram matrix of Σ and removes the possibility.

We introduce the Schur pseudo-likelihood, a one-parameter family that damps the cross-block coupling of the Gaussian likelihood through its Schur complements, for scoring and regularizing covariance and correlation estimates in high dimensions. The Gaussian log-likelihood is the standard criterion, but when the dimension rivals the sample size it is governed by the smallest, least-identifiable eigenvalues of the estimate, and as a criterion for ranking or selecting estimates it becomes unreliable; damping the coupling restores it. The optimal damping has a closed form—the reliability of the coupling, a James–Stein shrinkage. On real crypto returns, choosing a shrinkage estimate by the Schur pseudo-likelihood rather than by the full likelihood yields several-fold lower out-of-sample portfolio variance once the dimension rivals the sample size.

Traditional covariance shrinkage estimates pull an empirical covariance matrix Σ towards a lower information anchor (often λI), reducing estimation error at the cost of suppressing correlation. This note proposes the opposite: geodesic covariance inflation—a controlled movement of Σ towards a perfectly-correlated limit. Intended not as a consistent estimator but merely a device for use in the context of portfolio construction, we propose that instead of discounting correlation, we exaggerate it in a geometrically natural way, yielding a family of positive–definite matrices indexed by an inflation parameter γ ∈ [0, 1].

We develop Thurstone portfolio polishing: a way to take an existing allocation on the simplex—a capitalization index, an equal-weight book, any benchmark—and re-tilt it under a view on dependence rather than returns. Reading the weights as the winning probabilities of a race among assets (a Thurstonian model of choice), we back out the latent abilities that reproduce them with a fast exact inverse, then re-evaluate the race under a richer dependence law. The marquee instance is a tail-sensitive analogue of Black–Litterman: where Black–Litterman tilts a benchmark by a view on expected returns through a linear inverse of a covariance, we tilt by a view on lower-tail co-movement through a fast exact nonlinear inverse of a correlated race, inverting no covariance and solving no optimization. Driven by a tail-dependent simulation the polish shades down exactly the names that crash together beyond what their correlation can encode—a crash-proofing overlay we verify both in a known-truth lab and on the Dow. The construction's organizing property is redundancy (clone) consistency: a near-duplicate asset splits a position rather than doubling it—the cure for the red-bus/blue-bus failure of any Luce-type rule, and the same near-singularity that simultaneously makes the polish robust to ill-conditioning (it samples from a correlation rather than inverting one), low in turnover (its weights are locally Lipschitz in the correlation), and a smooth de-duplicator of correlated holdings. The polish is feasible on the simplex without optimization, reproduces its benchmark by construction, admits an arbitrary performance simulation so dependence beyond second moments can be matched, and solves an implied mean–(convex-regularizer) program whose penalty the simulation fixes; capitalization weighting is its degenerate, correlation-blind member. We prove the supporting properties (feasibility, benchmark reproduction, redundancy and tail consistency, the implied objective, and a Lipschitz turnover bound) and implement the method in the allocation package, built on thurstone.

Let G(K) = supm (m − K)F̄ (m) be the optimized value of an enquiry with breakeven strike K, against a best competing quote with survival curve F̄ . For every positive, strictly decreasing enquiry-value curve, the coordinate Φ = − log(G/G⋆ ) converts buy– sell imbalance into an exact translation of two log-value coordinates. The translation is induced by state-independent shifts of the break-even strikes exactly for G = κ + Be−hK on the active strike interval; state-independent shifts of the submitted quotes, or global rationalizability by a finite-mean markup distribution, force κ = 0. For general G the canonical sidewise gauge need not satisfy the edge compatibility that lets both strike ladders derive from one inventory potential; we quantify the defect. On a finite inventory lattice, under nonsingularity of the anchor linearization, any smooth deformation path yields a perturbation hierarchy whose every coefficient uses one linear operator, with distributional perturbations entering through an envelope formula and local rationalizability conditions. Reflection symmetry makes the skew odd and the convexity response even in the log-odds of the flow. All formulas and asymptotic orders are checked numerically.

Intuitively, accuracy in microstructure prediction at a granular level must relate to profitability for a market participant, but this is not trivial to formalize. Here we provide an approximation using a stylized steadystate model for over-the-counter trading modeled as a sequence of sealed bid auctions. A simple picture emerges due to a mildly surprising feature of this model: one does not need to know the fair price, only where others are bidding.

We provide analytic results for the optimal control problem faced by a market maker who can only obtain and dispose of inventory via a sequence of sealed-bid auctions. Under the assumption that the best competing response is exponentially distributed around a commonly discerned fair market price we examine properties of the market maker’s optimal behavior. We show that simple adjustments to skew and width accommodate customer arrival imbalance. We derive a straightforward relationship between the market marker’s fill probability and direct holding costs. A simple formula for optimal bidding in terms of (non-myopic) inventory cost is presented. We present the results as a perturbation of an improvement to a “linear skew, constant width” (CWLS) market making heuristic.

Rescaling a probability vector when an item is withdrawn or masked out is one of the great unconscious modelling acts. When the support shrinks because something was learned about a fixed experiment, rescaling is the conditional law. When somebody instead intervenes on what may be chosen, the old vector does not determine the new one, and rescaling amounts to assuming Gumbel noise in an additive random-utility model. A Gaussian is an equally canonical, equally tuning-free choice. We race this Gaussian renormalization against linear renormalization on thirty-nine held-out comparisons, re-litigating the constant-ratio rule versus Thurstone's Case V. Gaussian renormalization has the lower log loss in thirty rows and leads in fourteen sources against one. Differences are small: a median of 0.004 nats.

A k-subset of items will be chosen from n according to values taken by n auxiliary variables X1 , . . . , Xn interpreted as performances in a contest. Item i is chosen if Xi ≤ X (k) where X (k) is the k’th order statistic. A numerical algorithm is presented for computing many k-combination choice probabilities quickly, for small k but potentially large n ≫ 1, 000, 000. No assumption is made on the 1-margin distributions of the Xi , and the analytical convenience survives the introduction of dependence via a factor model also. The computation of rank probabilities for k items is a corollary. The algorithm is provided in the winning package, on PyPI.

A subset of k items will be chosen from n according to values taken by n variables X1 , . . . , Xn interpreted as performances in a contest. Item i is chosen if Xi ≤ X (k) where X (k) is the k’th order statistic. A numerical algorithm is presented for computing many k-combination choice probabilities quickly, for small k but large n ≫ k in the millions. Rank probabilities for k can also be computed. Some users of the winning package may wish to calibrate models for latent Xi from partial information such as winning or losing probabilities. Others may prefer to supply arbitrary performance distributions. The analytical convenience of this method also survives the introduction of dependence in the Xi via a low-dimensional Copula.

Using prompts such as: “my favorite state in the US is [MASK]”, and “my favorite Western state in the U.S. is [MASK]” we infer that Thurston models are a better match to the revealed preferences of large language models than the application of Luce’s Choice Axiom. There is some irony in this finding given that Softmax functions, responsible for the token probabilities we interpret as preference, suggest Independence of Irrelevant Alternatives.

Rating systems almost universally assume that contest performance is ability plus normal or logistic noise. We present a setting where the assumption is visibly false and measurably costly: Formula 1, where for decades the most likely outcome of starting a race was not finishing it. A lattice-based Thurstonian rater that accepts arbitrary noise densities outperforms TrueSkill herbrich2007, Elo elo1978, Glicko-2 glickman2012 and OpenSkill weng2011 over 1,158 races, and replacing its normal noise with a two-component density, a Gaussian pace term plus a separated block of slow mass representing retirement, improves every metric. Estimating the block's mass from the trailing retirement rate improves it further. Symmetric heavy tails and skew do not help, so the gain is attributable to the block, not to non-normality in general.

The probability that a competitor wins a noisy race—the Thurstonian choice probability, equivalently the gradient of an expected-maximum potential—is a rule for attributing credit among correlated contributors. It is a probability share ($\sum_i w_i = 1$), symmetric, and redundancy-aware (contributors that move together share a single contributor's credit rather than double-counting); it is differentiable in the contributors' abilities, it updates online, and it costs $O(Mn)$ in a single Monte-Carlo pass with no coalition enumeration. It is not a Shapley value and does not try to be: Shapley measures coalitional cooperation—a contributor's average marginal value as coalitions form—whereas the winning probability measures selection relevance, how often a contributor is the single best in the field at hand. The two answer different questions, and a contributor that is never individually best yet improves every blend it joins earns Shapley credit but little race credit; the race does not capture such coalitional value. (There is a clean mathematical link—the winning-probability field integrates along the diagonal to an Aumann–Shapley value of an ability-scaling game—which we record but do not lean on.) We show that the headline redundancy property—an even split of credit among near-duplicates—is a property of the calibrated equal-ability race rather than of raw scores, and we demonstrate the rule on forecast combination and feature attribution. This develops the credit reading of the construction in the companion paper on Thurstone portfolios.

Consider N channels firing at random times, with rates driven by a hidden state they all share. Conditionally on that state each channel is Poisson, and an observer records which channels fire and in what order. When the driver mixes fast relative to the firing rates, the probability that channel i fires first approaches the law of Luce’s Choice Axiom in the stationary mean rates. We give the first correction in the timescale ratio, governed by the time-integrated covariance K of the rates, and every claim below holds at that order. Recording the first two arrivals recovers all N 2 −2N observable combinations of K, matching what blocking channels and rerunning the process would give, while the winner alone recovers none of them. The diagonal of K is unobservable, absorbed into the rates, so only correlated modulation leaves a trace. And K is symmetric when the driver satisfies detailed balance, so its observable asymmetry certifies a driver held out of equilibrium, visible with three channels and never with two.

Many systems reveal themselves only through races: N latent processes compete, and an observer records which one finished first. The natural counterfactual question — what happens if competitor i is removed? — turns out to be exactly nonidentified from winner frequencies alone: the redistribution of the removed competitor’s probability is governed by a runner-up kernel M on which winner data places no constraint whatsoever. We give the sharp partial-identification set, and a data hierarchy that resolves it: observing the runnerup identifies every single-deletion counterfactual, and top-(k+1) ranking prefixes identify every deletion of up to k competitors. A Markov substitution resolvent turns the runner-up kernel into predictions for arbitrary blocked sets. We validate the hierarchy on real firstpassage physics — Brownian narrow escape, where a truncation identity makes one reflecting trajectory answer every blocked-set counterfactual exactly — finding that winner-only models degrade with deletion depth while the runner-up kernel predicts at the sampling-noise floor, and that the substitution process is empirically Markov. Finally, we show the kernel is not merely estimable but computable: Green-function hitting splits of the domain predict M to within sampling noise with no trajectory data, and the geometric kernel outperforms the empirically measured one as an input to counterfactual prediction. Two practical warnings emerge: the stationary encounter chain differs materially from the first-transition kernel that counterfactuals require, and common time changes make proportional-hazards (softmax) redistribution exact — so deviations from it measure differential structure only.

Lichess maintains a separate Glicko-2 rating for each time control, which is stratification: condition on the category, estimate within it, and accept the sparse-cell variance that follows. We compare it with a partially pooled alternative on 59{,}399 decisive games from one month of the site's open database, in which a player's ability is a level estimated from all of their games plus shrunk bullet and blitz offsets. With both systems tuned on the same validation split, the pooled rating is better by 0.008 to 0.012 nats of held-out log loss across three independent splits, every interval excluding zero; the estimator accounts for -0.0080 of that and the factor structure for -0.0033, and two thirds of the estimator contribution is the privilege of a batch fit rather than a better method. Once Glicko-2's recency parameter is tuned rather than left at its default, stratification is worth nothing to either side. Opening family is unidentifiable because players choose openings, opponent-strength curvature is real but almost entirely shared, and modern Lichess no longer poses the question, with 0.3% of players mixing time controls against 52% in 2013.

The Python package humpday provides twenty-three derivative-free optimizers behind one uniform contract: minimise a black-box function on the unit cube under a hard evaluation budget. Every algorithm is implemented in pure Python with no required dependencies (a numpy backend is used transparently when present) and again in zero-dependency JavaScript, with parity tests holding the two in agreement, so the same optimizer runs on a server or in a browser. A recommender selects an algorithm from the problem’s dimension, budget, and measured evaluation cost, using rankings precomputed on the package’s own benchmark: eighty-one objectives ported from real applications, each paired with an interactive browser demonstration. Scoring applies seeded smooth bijections of the cube that relocate every optimum, so no method, human or machine, can score by memorising solutions; the suite remains valid when the candidates are written by language models. We describe the design and two findings the benchmark has produced: optimizer rankings on synthetic test functions correlate only moderately with rankings on the real-world suite (Kendall τ between 0.42 and 0.49 across budgets), and the suite supported the generation and out-of-sample validation of Alloy, a machine-designed optimizer that now ships in the package.

Practitioners choose a derivative-free optimizer either by consulting a benchmark leaderboard or by reasoning about the problem. We test both. Using a memorisation-proof suite of real-world objectives, each relocated by a seeded cube-to-cube diffeomorphism, we rank a panel of $22$ optimizers and ask how well that ranking is predicted by the same panel's ranking on synthetic analytic benchmark functions, and by a language model asked to reason, problem by problem, about which optimizer should win. Both predictors are essentially uninformative: the rank correlation with real-world performance is statistically indistinguishable from zero in either case. A language-model selector that picks an optimizer per problem in fact does worse than a constant policy of always using one robust direct-search method. The two failures share a mechanism. Both the benchmark and the model over-trust model-based trust-region methods (Powell's NEWUOA and BOBYQA), which dominate smooth analytic functions yet rank near the bottom on the real suite. This is a draft, and several confirmatory runs are still in progress (Section sec:caveats).

We discover rather than hand-design a derivative-free optimizer by evolving a parameterised DE/ES-hybrid template against a memorisation-proof suite of real-world objectives. A genetic algorithm searching $14$ behavioural knobs reaches a normalised regret of $0.110$ versus a Nelder–Mead/Differential-Evolution/CMA-ES panel (mean rank $1.47$ of $4$), against $0.191$ for the original $12$-knob template. The gain comes from one added mechanism — a cheap, R2-gated separable-quadratic trust-region jump — which we validate three ways: an A/B re-evolution, a single-gene toggle ablation (disabling it more than doubles regret), and independent re-discovery by the production search, which drives the jump's firing probability to $\!1$. The discovered optimizer is a hedged hybrid, found by grounding fitness in real (disguised) evaluation rather than a synthetic proxy.

Place K established algorithms at the vertices of a simplex. A point is a vector of mixing weights, and a language model turns the point into a working program: the heaviest vertex supplies the architecture and the others graft their ideas into it, in the stated proportions (Figure 2 shows the literal prompt). Scoring the program on a benchmark makes the simplex a continuous space of algorithm designs that ordinary derivative-free optimizers can search. The construction’s first product is Alloy, generated at the equal-weights point: on twentynine problems never used in any selection step it has the best mean rank at every budget from 60 to 480 evaluations and beats six competitors including CMA-ES pairwise (p < 10−10 ); it ships in the humpday package. Letting weights go negative extends the recipes beyond the simplex; of six semantics for a minus sign, tested under a paired-ablation protocol, one survives out of sample, with a rule attached: negative weights repair weak hosts and spoil strong ones. Failures are reported throughout: the continuous search never beat the obvious point, single generations vary severalfold, and in-sample scores mislead reliably. Two further worked examples test the construction’s reach: cache-eviction policies, where static blends fail and adaptive shares rediscover the design of CAR, and distributional forecasting, whose champion dominates synthetic suites, loses on real data, and ships anyway, recast as a transform inside the forecaster it could not beat.

It is shown that an algorithm given two chances to improve its position on the path of an exponentiated Ornstein-Uhlenbeck (OU) process should not choose its final position between the first two locations. It is sometimes easy, therefore, to diagnose failure of an algorithm to learn the optimal policy. The proof introduces the notion of rapidity on an OU bridge, and complements similar results in managerial science that are used as metaphors for complex but ill-defined business and strategy problems.

Derivative-free optimizers spend most of their budget on one-dimensional line searches. We use the terminal-placement rule of the companion theory paper as a line-search inner loop: with its scale and correlation length estimated online, it flees, probes, or commits in one to three evaluations. On the HumpDay physics benchmark it is a competitive inner loop on rough and high-dimensional objectives at tight budgets, where its early exits buy more search directions than a classical line search. A projection argument then lifts the same rule off the line: after two observations under any radial Gaussian-process prior, the entire next-point decision lives in a plane, in any ambient dimension. A fixed off-line placement does not beat a line search across the catalog, but choosing it by expected improvement under a fitted Gaussian process does, and increasingly so with dimension: the value of the acquired point, not its position, is what turns the exact reduction into a win.

The posterior probability that alternative i is best, pi = Pr(θi = maxj θj | D), drives stopping rules, budget allocation, derandomized Thompson sampling, and safety constraints, and the reinforcement-learning and Bayesian optimization literatures describe it as intractable: recent work variationally approximates it from marginal means and variances alone. We observe that in two standard settings the posterior covariance is exactly of factor form Σ = W W ′ + diag(d) at every step: simulation replicates under common random numbers, and per-arm Gaussian observations against a factor prior. In that case the whole vector p is a one-dimensional integral per factor node, computable for all n alternatives at once in time linear in n by machinery previously developed for pricing correlated contests, and certified here against Monte Carlo. In a stopping problem under common random numbers, rules that stop on marginal-only estimates of p hold their nominal error by overspending, between 1.5 and 3.4 times the replicates of the exact rule, which stops earliest in every correlation regime at the nominal error level. In a correlated Gaussian bandit, sampling from the exact vector reproduces Thompson sampling’s regret to within replication noise, as the occupancy identity requires, while the variational policy pays ten percent more regret and sits at measured total-variation distance 0.13 to 0.16 from the object it stands in for.

In a budgeted Brownian race, a controller watches a cloud of Brownian paths, pays per unit time for every path kept alive, may irreversibly prune, and spends a finite path-time budget to maximize the expected terminal maximum. We solve the mean-field version: a Poissonized population under per-path controls, with the budget enforced in expectation. The expected maximum has an exact layer-cake form whose gradient is a pivotal value, and for a fixed price on path-time the linearized best response is an obstacle problem whose keep sets are upper tails: the wavefront policy, which retains a path while its propagated future pivotal value exceeds its carrying cost. A fully corrective Frank–Wolfe method over wavefront policies, with dual bisection on the price, solves the budget problem and certifies its answer, bounding suboptimality by the Frank–Wolfe gap plus a complementary slackness term. In the committed example the adaptive policy improves the expected maximum by 19% over the best single screening date and by 30% over static thinning at the same expected budget; the finite-n race with a pathwise budget is left to future work.

A prompt that fails a handful of scored attempts is rated impossible by the standard pass-at-k extrapolation 1 − (1 − p̂)k , and rated so at every k. On nine thousand prompts with eight scored samples each, that plug-in step — the same one the coverage diagnostic of Malladi et al. (2026) applies as f16 (p) = 1 − (1 − p)16 — costs seven-fold in held-out log loss and understates aggregate pass@8 by thirteen points. Replacing the plug-in evaluation with the posterior expectation E[1 − (1 − p)k | s] under a two-parameter difficulty prior, fitted once across prompts by marginal maximum likelihood, removes most of the damage: log loss 0.48 against 3.37, the extrapolated curve within two points of truth. Probit-normal and beta-binomial priors tie to three decimals, so nothing hangs on the family. The bias shrinks as the per-prompt sample count grows, and the data cover one model on one task suite.

The Clay Mathematics Institute posts one million dollars on each Millennium problem. We ask what solving one is worth to the person attacking it, and what the purse contributes to the effort. A Brownian attempt run between a target and an abandonment floor buys success probability p at expected cost ζ²p/(1−p), where ζ² is the cost of diffusing across the remaining distance, and this price list recovers the reward from the cost of an attempt, the number of attackers and the observed rate at which problems fall. The level of the reward is not identified by one solution in twenty-six years: the point estimate is $96M, and the hazard interval carries it from $18M to $3.8B. The surplus over the threshold at which anyone would start is identified, at about twice the cost of one attempt, and the purse is 55% of it at every point of the interval.

Thompson sampling is easy to execute: sample once from a posterior and choose the action with the largest sampled value. Its action probabilities can nevertheless be difficult to recover. For a finite action set, the logging propensity of action $i$ is the posterior winner probability \[ p_i = \!( U_i = \max_j U_j H, x ), \] where $U = (U_1, , U_N)$ is the jointly sampled utility vector. These propensities are required by inverse-propensity and doubly robust off-policy estimators. A familiar one-dimensional product formula is exact when the sampled action values are independent. It is generally invalid for a shared posterior parameter draw, because univariate Gaussian marginals do not determine winner probabilities. We study correlated Gaussian Thompson sampling under the factor-plus-diagonal representation \[ U_i = \mu_i + v_i^ f + D_i\,\epsilon_i, f N(0, I_k), \epsilon_i iid N(0,1), \] with $D_i > 0$. Conditional on the factors, the actions are independent. This yields a shared product-CDF field from which all $N$ winner probabilities can be evaluated together in $O(QN(L+k))$ arithmetic, where $Q$ is the number of factor nodes and $L$ is the size of a one-dimensional utility quadrature. We prove that Gaussian marginals alone are insufficient, give the correct Gaussian orthant representation, establish continuity to the singular pure-factor boundary, and bound the error induced by a factor approximation through Gaussian total variation. We also describe the Jacobian of the propensity map as a weighted ``photo-finish'' graph Laplacian, providing all-action sensitivity to posterior-mean bonuses. For off-policy evaluation, we derive exact conditional bias formulas when an approximate propensity vector is substituted into inverse-propensity or doubly robust estimators. The formulas show why small absolute errors on rare actions can create large counterfactual bias. The paper closes with a falsifiable computational and statistical experimental program. The intended claim is not that factor-Gaussian evaluation replaces direct posterior sampling in every Thompson-sampling model, but that it makes correlated all-action propensities explicit in an important structured regime where independence-based calculations are wrong and brute-force Monte Carlo can be costly.

Decentralized perpetuals protocols have collectively reached billions of dollars of daily trading volume, yet are still not serious competitors on the basis of trading volume with centralized venues such as Binance. One of the main reasons for this is the high cost of capital for market makers and sophisticated traders in decentralized settings. Recently, numerous decentralized finance protocols have been used to improve borrowing costs for perpetual futures traders. These protocols have grown to over $2.5 billion dollars of assets while generating over $890 million in fees in 2024. We formalize this class of mechanisms utilized by protocols such as Jupiter, Hyperliquid, and GMX, which we term Perpetual Demand Lending Pools (PDLPs). We then formalize a general target weight mechanism that generalizes what GMX and Jupiter are using in practice. We explicitly describe pool arbitrage and expected payoffs for arbitrageurs and liquidity providers within these mechanisms. Using this framework, we show that under general conditions, PDLPs are easy to delta hedge, partially explaining the proliferation of live hedged PDLP strategies. Our results suggest directions to improve capital efficiency in PDLPs via dynamic parametrization.

Scoring rules, cost-function market makers, and opinion pools are taught as separate subjects with separate notation. All three are generated by one convex function of the forecast. A proper scoring rule is that function read at the report, a cost-function market maker is its Fenchel conjugate, and the linear and logarithmic opinion pools are the two Kullback-Leibler barycenters. Merging two market makers is infimal convolution, so liquidity adds. The mechanisms are given one input-output signature so that they can be chained, and several familiar statistical methods, Bayesian model averaging, least squares, the Kalman filter and split-conformal prediction, are then chains with one degree of freedom frozen. Every statement is worked on a single three-outcome example. --

The microprediction platform built distributional prediction supply chains, chaining pools so that one's output became the message or the settled outcome of the next. The games were built before their game theory was examined, and the question for each is whether truthful reporting was really the best move. A chain is proper where an exogenous outcome anchors every stage; a downstream stage can then be scored on its own residual or, for the log score, at the top level. By that test the games mostly hold up, one of them only by a lucky accident. Flaws aside, the setup has continuing pedagogical merit, as it illustrates the inherent limitation of conformal prediction. --

The choice between the logarithmic score and the continuous ranked probability score (CRPS) is usually argued on four grounds: locality, propriety, robustness, and interpretability. A fifth applies when the elicited object is a reusable predictive density and forecasts are chained, each stage refining the residual left by the last. The log-likelihood of a composed forecast is the base score plus the log-density of each residual refinement, so credit is additive and each stage's increment is itself the proper score of what that stage added; CRPS has no comparable per-stage identity. A population-level example shows the two scores preferring measurably different forecasts, and CRPS remains the right target when the decision is coarse or the tail unmodelable. Conformal prediction is the limiting case, a method that produces no density and is scored on a metric that does not require one. --

We study automated market makers for contracts that settle to bounded real payoffs, such as a saturated surprise relative to a published forecast, rather than to the indicator of an outcome. The cost function $C(q)=b\sum_d(qv_d/b)$, built from residual directions $v_d$ supplied by the market maker and its capital $b>0$, has a convex conjugate that is finite exactly on the zonotope $V[-1,1]^D$. The worst-case loss is therefore exactly $b\,D 2$, the price set is the interior of the zonotope, and the Hessian is a covariance quoted by vectors, revealed through cross-impact and depleted by inventory. Any cost that is a state function refunds itself over a round trip, so revenue must be path-dependent, and a per-unit fee $f\|\|_1$ set by each maker is a dead band in price space: a trader with belief $$ transacts nothing exactly when $ p(q)+[-f,f]^n$, and otherwise solves a lasso whose solution is sparse across contracts. Aggregating fee-bearing makers is an inf-convolution whose supply is the horizontal sum of band-limited supplies, so a consolidated limit order book, with a level at each maker's quote, depth equal to its curvature and exact zero fills for uncompetitive makers, follows as a theorem.

A model can be more capable than every competitor and nevertheless disappear from most economic production. We formalize this claim in a competitive economy of prediction tasks in which standardized units of compute (“tokens”) are itinerant: at zero switching cost they may move between tasks and may instantiate whichever available algorithm they choose. Mobility generates a no-arbitrage condition under which marginal profit-and-loss per token is equalized across all active uses of compute, and any technology offering a strictly smaller return on the same subcontractible task receives zero execution share. As the division of labor becomes progressively finer, execution is therefore selected by task-conditional marginal economic productivity per standardized unit of compute—not by generic capability. We exhibit an explicit equilibrium in which a generalist that is strictly more capable than every specialist on every task nonetheless has compute market share tending to zero, and we give sufficient conditions, in both directions, relating asymptotic execution share to membership of the active productivity frontier. Measurement follows from the same accounting: a market-scoring-rule prediction mechanism pays each entry its marginal contribution at the state it faced, and for a sole frontier supplier on one task at its competitive scale, average PnL per token equals the shadow price of compute divided by the elasticity of task value with respect to effective compute, so at a common shadow price it ranks task-level productivity exactly when that elasticity declines with crowding, as it does under log production; administered deployment or mixed scarcity regimes can invert the unconditioned quotient, and an arbitrary model-level lifetime quotient has no such representation without additional structure. The model supplies no canonical taskinvariant scalar quality: any aggregate PnL-per-token ranking embeds a task distribution and deployment weighting, measuring the economic niches a model occupies.

We consider evidence in support of what we term the Weak Indispensable Markets Hypothesis (IMH) provided by the M6 Financial Forecasting Competition. The Weak IMH asserts in the presence of a well-established market, those who eschew prices as inputs for proximate predictive modeling tasks will under-perform out of sample. We also consider the Strong IMH which asserts that forecasting should be considered a market-inspired engineering endeavor in addition to a modeling task under the usual rubric of statistics or machine learning methods (put simply: if a market doesn’t exist to help you, make one!). The competition established that neither principle’s application is obvious to participants or organizers; it hinted at a hidden quality crisis in data science generally; and it suggests that broadening the usual concept of analytic pipelines to insert collective intelligence might be part of the remedy.

Conformal prediction builds prediction sets that cover the truth at a rate you choose, finite-sample and distribution-free, assuming only exchangeable data. That guarantee is marginal. de Finetti's theorem describes the exchangeability it rests on, and in the finite form the mixing measure may be signed kerns2006. A short lemma decomposes the slope of conformal's calibration-conditional coverage into a non-negative threshold term (the classical Beta-law fan) and a term carrying the sign of the de Finetti measure. Positive (extendable) mixtures make conformal conditionally adaptive; the signed corner of de-meaned, ranked, or compositional scores makes it anti-adaptive. The marginal guarantee is the same either way; only what it hides changes.

The companion paper shows that the log-score regret of a single-shape conformal predictor to the conditional oracle is the mutual information $I(R;X)$ between the residual and the input. Here we rederive that quantity from a betting mechanism rather than from the score. Treat the predictor as the crowd in a parimutuel pool on the residual: bettors put money in, and the pot is split among the winners in proportion to their stake on the realised outcome. In the continuous limit the payoff is the ratio of the bettor's density to the crowd's. An entrant who knows only the marginal breaks even, which is marginal coverage stated as wealth. An entrant who conditions on $X$ grows his bankroll at rate exactly $I(R;X)$. The gap is the rent. This is not a metaphor: the nearest-the-pin pool of the microprediction platform, and the continuous density version run in the MidOne contest, are this mechanism, and a conformal predictor is the entrant that prices the pool flat in $X$. We then give two ways to measure the rent on a fitted predictor: a static lower bound from distance covariance, and a sequential e-process whose growth rate estimates $I(R;X)$ and which is an anytime-valid test for conditional miscoverage.

Temporal dependence charges split conformal prediction two different prices, and they scale differently. The coverage price is a tax that vanishes with the calibration size: for a stationary $$-mixing score process the loss is at most $\min_\/(n+1)+2()\$ barberpananjady2026. The sharpness price is a rent that does not: the per-step log-score regret of the marginally calibrated predictive against the past-conditional oracle equals the entropy-rate gap $ H=H(S_0)-h$, the mutual information between the present score and its infinite past, a constant independent of $n$. For Gaussian score processes the rent exponentiates into width — oracle intervals are narrower by the factor $e^- H$ at matched coverage, which for an AR(1) score process with parameter $$ is $1-^2$. Dependence is therefore asymptotically free for the certificate and permanently valuable for the forecast.

We give the validity of split conformal prediction an exact contragredient form: the placement acceptance map is equivariant for the permutation action, and the Reynolds identity $ P^f,v= f,Pv$ reads its orbit average either as moving the evaluation functional or as averaging the induced acceptance vector over placements. A natural primal discrepancy construction under the same action is Steinitz balancing: ordering a zero-sum population so that every prefix sum stays bounded. The centered conformal placement residuals form such a population, and an explicit ordering keeps the running placement-acceptance average within $1/(2t)$ of the orbit level $k/N$, exact once all $N$ candidate test placements have been traversed; the vector-valued lemma balances several coverage levels along one traversal. On the function–measure pairing, the measure-side counterpart of calibration is balanced sampling and exact cubature. A transitive distributional symmetry, exchangeability in the standard setting, turns the deterministic orbit average into the coverage probability of the designated test placement.

We view conformal prediction as a repairable univariate transform to be included alongside other time-series operators in a forecasting grammar. By this device we evaluate the wisdom of stopping after the conformalizing step, something that is tacit in conformal prediction and departed from by only two of the 623 papers proposing conformal methods in our survey. Across 17,364 economic time-series, early stopping costs approximately 0.4 nats. We interpret this through the lens of a new information identity we term the conformal information gap, whereby it is possible to formalize why conformal calibration, at a fixed score and stratification, cannot exploit forecastability.