1.1 What joint prices add

Two individual event prices leave their joint probability undetermined. Consider contracts paying \$1 when Team A or Team B wins its next game, and zero otherwise. The games may share conditions that affect both results. Suppose their prices are p_i=.65 and p_j=.40, as in Example 3.3. Under independence, a contract paying only when both teams win has price p_ip_j=.26. Such a joint contract is a parlay; its component events are its legs. A price of .31 or .21 is also compatible with those individual prices. The first puts more probability on joint success than independence does, and the second puts less.

Write \pi_{ij} for the parlay price. The four outcome probabilities are then \pi_{ij}, p_i-\pi_{ij}, p_j-\pi_{ij}, and 1-p_i-p_j+\pi_{ij}. They correspond respectively to both wins, only Team A winning, only Team B winning, and neither winning. Their nonnegativity restricts the parlay price to the interval [.05,.40] in this example. Thus one joint price completes the pair table, while either individual price alone leaves the dependence unresolved.

These probabilities first describe prices. Write Q for a coherent pricing law: one probability distribution that reproduces all the quoted contract prices. Write P for the physical outcome law, the distribution that generates resolved events. Encode a win by \omega_i=+1 and a loss by \omega_i=-1. The quoted marginal prices are p_i=Q(\omega_i=+1) and p_j=Q(\omega_j=+1). Market scoring rules elicit beliefs under their preference assumptions [2]. Equality Q=P requires a separate calibration assumption and evidence from resolved outcomes [26]. Each result applies to its named law. The symbol \mathbb{P} denotes that specified law, using Q for quote inversion and P for outcome estimation.

Resolved outcomes supply another route to the same four-cell table. An aligned round records both results from one specified occurrence of the pair, such as the two games in one week. Repeated rounds estimate how often each joint result occurs. Both legs can resolve without a traded parlay, and every round supplies one observation regardless of ticket count. The fixed-law estimates below require independent rounds with a common pair law. Trade signals and settled observations therefore have different sampling requirements.

The distinction between pair dependence and network interaction is equally necessary. Two results can depend on a shared third event even when their model has no direct interaction. The pair table measures their resulting association. Recovering the direct interaction requires additional observations or an independently justified restriction on the network. The paper makes these requirements explicit before drawing any network conclusion.

ParlayMarket [3] learns a shared Ising pricing law from trade targets. The question here is what specified probabilities and resolved observations determine, and with what uncertainty. The classical two-by-two table already supplies the exact pair inverse and its sampling interpretation [29, 30, 4]. Its use within a larger event model requires the compatibility, embedding, and observation distinctions developed below.

1.2 A joint model with pair interactions

For several events, a model must assign consistent probabilities to all joint outcomes. Rolling correlations, factor models, copulas, and Bayesian networks provide different descriptions of dependence. Here we use a model whose log probability contains individual-event terms and pair-interaction terms. For M binary events e_1, \ldots, e_M with outcomes \omega_i \in \{-1, +1\}, it has joint distribution \mathbb{P}(\omega_1, \ldots, \omega_M) \;=\; \frac{1}{Z} \exp\!\left(\sum_i h_i \omega_i + \sum_{i<j} J_{ij}\omega_i\omega_j\right), \tag{1} Here Z normalizes the probabilities to sum to one. The field h_i sets an event’s conditional bias, and the coupling J_{ij} measures its interaction with another event. An event index is also called a site, and its signed outcome a spin. This is the Ising model [5], also expressed by Besag’s auto-logistic model [22] and the Boltzmann machine [23]. It is an exponential family: its log probabilities are linear in fixed outcome statistics, apart from normalization. Three properties explain its use here:

1. The Ising density (1) is the pairwise maximum-entropy distribution on \{-1, +1\}^M with specified marginals and pairwise correlations; among distributions matching those moments it assumes the least additional structure.

2. The parameters (h_i, J_{ij}) have direct conditional interpretations: 2h_i + 2\sum_{j \neq i} J_{ij}\omega_j is the conditional log-odds of e_i given the other outcomes, and 4J_{ij} is the conditional log odds ratio of the pair (e_i, e_j) given the rest. Neither is a marginal quantity; Remark 2.2 states why h_i is not the marginal half-logit.

3. The pair (h, J) determines a consistent joint distribution over all 2^M outcomes, enabling the pricing of parlays and other joint-outcome contracts.

The pairwise maximum-entropy form omits higher-order log-linear interactions. It cannot represent a three-event law whose genuine joint interaction remains invisible in every pair table. Its two-event family nevertheless represents every positive pair law. Copulas, vine copulas, factor models, and directed graphical models impose different dependence assumptions. We use identifiability to mean that the stated observations determine one parameter value within the stated model. The results establish that property and quantify estimation error under the Ising parameterisation.

1.3 From pair probabilities to inference

The pair inverse starts with four probabilities and extracts their signed logarithmic contrasts. Its network interpretation then depends on what was observed beyond the pair. Sampling uncertainty enters when the probabilities themselves are estimated from resolved rounds. The following results address these successive questions.

(1)

Identifiability (Section 3). For M = 2, the triple (p_i, p_j, \pi_{ij}) of two marginals and the joint-outcome probability determines (h_i, h_j, J_{ij}) uniquely through an explicit closed form, and the admissible region of parlay prices is exactly the interior of the Fréchet-Hoeffding region (Theorem 3.1).

(2)

Embedding bias (Section 4). Applied to a pair inside a network of M > 2 coupled events, the closed form reads the direct coupling plus a two-hop correction through shared neighbours and a second correction carried by the correlations among those neighbours. Theorem 4.1 gives both terms and an unconditional fourth-order remainder bound, and an exactly solvable three-site chain confirms the two-hop constant and the remainder’s order.

(3)

Estimation (Section 6). The two-site model is saturated: the closed form on empirical joint-outcome counts is the maximum-likelihood estimator. Theorem 6.3 computes its Fisher information and identifies its efficient variance as Woolf’s log-odds-ratio variance; Theorem 6.4 gives a finite-sample concentration bound whose constant is computed from the two-site moments.

(4)

Robustness (Section 8). Theorem 8.1 propagates an L^\infty perturbation of the quoted triple to the inferred parameters: noise of size \eta moves each of the three inferred parameters by at most 2\eta/\kappa, where \kappa is the minimum of all four cell probabilities throughout the admitted uncertainty region.

Section 3.2 tests whether all quoted moments admit one joint law. Section 5 gives constructive direct recovery and proves why selected moments cannot certify a missing edge. Section 6.3 constructs count-based uncertainty sets, including zero-count samples. Section 7 separates sampling error from changes in the outcome law. Its certified arithmetic in Section 7.1 preserves interval inclusion through numerical evaluation.

Global compatibility and sufficient-information recovery determine which joint model the observations support. The confidence sets retain this distinction when observations are finite or their law changes. Computation results control approximation within a specified law. Section 10 evaluates these different quantities in synthetic examples.

2.1 Outcomes and conditional interactions

Consider M binary event types e_1, \ldots, e_M with outcomes \omega_i \in \{-1, +1\}; \omega_i = +1 denotes e_i occurring. A settled round is one joint realisation of the type vector: round r resolves (\omega_1^{(r)}, \ldots, \omega_M^{(r)}) as a single draw from the joint distribution below, and the sites index the types, not the individual resolutions. The two markets of Section 1 are types in this sense, with one round for each aligned pair of games. A type that resolves once and never again supplies exactly one round; Section 11, limitation (L9), records that one round supplies no useful precision guarantee for the estimates of Section 6.

The model assigns each outcome a real score before converting scores to probabilities. The negative of that score is called the Hamiltonian. Its sign convention agrees with the statistical-physics literature.

The coupling sign describes conditional association when all other outcomes are fixed. In a larger network, marginal association can also reflect paths through other sites, as Section 4 shows. The joint distribution is the Boltzmann density \mathbb{P}(\omega_1, \ldots, \omega_M) \;=\; \frac{1}{Z} \exp\!\bigl(-H(\omega)\bigr) \;=\; \frac{1}{Z} \exp\!\left(\sum_i h_i \omega_i + \sum_{i<j} J_{ij}\omega_i\omega_j\right), \tag{3} with partition function Z = \sum_{\omega \in \{-1,+1\}^M} \exp(-H(\omega)).

A marginal averages over the other outcomes; a conditional probability holds them fixed. For 0<p<1, the odds are p/(1-p) and their logarithm is \mathop{\mathrm{logit}}(p)=\log(p/(1-p)). The next calculation explains why converting a marginal price to log-odds does not usually recover its field.

3.1 Closed-form inversion and exact identifiability

A parlay trade on pair (i, j) pays \$1 if \omega_i = +1 and \omega_j = +1 jointly, zero otherwise; an admissible quote supplies the joint pricing probability \pi_{ij} \;=\; \mathbb{P}(\omega_i = +1, \omega_j = +1). The triple (p_i, p_j, \pi_{ij}) determines all four cell probabilities. Taking logarithms converts the Ising weights into a linear system. Signed sums cancel the normalizing constant and isolate the two fields and the coupling. The proof calls these sign patterns the Walsh basis; each basis function is a product of selected spins. Positivity is essential because the inverse takes the logarithm of every cell.

Applied to a pair (i, j) inside a network of M > 2 coupled events, the closed form (7) evaluated on the true pair marginal returns the effective pair coupling J^{\mathrm{eff}}_{ij} = J^*_{ij} + \Delta_{ij}, with J^* the network interaction matrix and \Delta_{ij} second order in the couplings; Section 4 computes \Delta_{ij}.

3.2 Global compatibility of quoted moments

Several valid pair tables can contradict one another when assigned to the same events. A moment is the expectation of a specified outcome statistic. For spins, singleton and pair moments are respectively averages of x_i and x_ix_j. The pair moment is a raw expectation. Covariance subtracts the product of singleton means, and Pearson correlation further divides by the two standard deviations. Let T(x) collect the available statistics at a full outcome x\in\{-1,+1\}^M. Let m be their quote-implied moment vector, using \mathbb{E}[x_i]=2p_i-1 and \mathbb{E}[x_ix_j]=4\pi_{ij}-2p_i-2p_j+1. The unknowns below are the probabilities of all full outcomes. Searching over these probabilities tests joint compatibility and, when necessary, measures the smallest adjustment to the quoted moments.

For three spins with all singleton means zero, set every pair moment to -.9. Each pair has positive cells (.025,.475,.475,.025). But every configuration satisfies x_1x_2+x_1x_3+x_2x_3\geq-1, while the requested sum is -2.7. The exact projection residual in these raw moment coordinates is 17/30 in maximum norm. Every correction must increase the correlation sum by 1.7, so one coordinate changes by at least 17/30. Uniform mass 1/6 on the six configurations other than +++, --- attains the bound, with each correlation -1/3. This projection is a boundary law. Mixing it with uniform mass gives a specified positive approximation and a larger recorded residual.

Exact enumeration makes the finite-state feasibility problem explicit. Floating-point solvers report their tolerance and their probability witness. An outer relaxation can certify inconsistency if even the relaxation is infeasible. Feasibility of a relaxation alone leaves joint compatibility unresolved. A nonnegative normalized probability witness establishes compatibility to its checked residual, regardless of how it was found.

5.1 Full observations and a known tree

Holding every other outcome fixed removes the effect of averaging over hidden configurations. The conditional four-cell contrast can then isolate a direct interaction. On a known tree, there is only one path between any two sites, so consistent edge tables also determine the full law. Here a tree is a connected interaction graph with no cycles; E(T) is its edge set and d_i its vertex degrees.

Complete sufficient moments also identify the unrestricted pairwise model by strict convexity of its log partition function. Exact likelihood fitting uses those moments. Conditional logistic likelihood uses aligned full-vector observations and estimates the same direct interactions under its model [22, 15]. Interaction screening provides a sparse inverse-Ising comparison with its own sampling assumptions [16]. The conditional-table formula is a population identification proof. Sparse or tree structure can reduce the data and computation required for its estimation.

5.2 Selected pairs and missing interactions

Suppose quotes are available only for a subset \mathcal{P}\subseteq\binom{[M]}{2} of pairs, where [M]=\{1,\ldots,M\}. The support of J is the set of pairs with nonzero direct interaction. Knowing that this support lies in \mathcal P is additional information about the model. The same selected quotes can identify a model under this restriction and fail to identify an unrestricted network. The next proof makes the difference precise through the number and independence of the observed moments.

Failure to identify the full model leaves a more specific question. Could selected moments at least prove that an unobserved edge must exist? For interior feasible moments, their maximum-entropy explanation always uses only the selected interactions. The following result therefore rules out that conclusion from those observations alone.

A concrete witness uses three zero-field spins. In one law, J_{02}=.4, J_{12}=.6, and J_{01}=0. The second law has only J_{01}=\operatorname{arctanh}(\tanh(.4)\tanh(.6))=.2069563793. Both laws have singleton means zero and the same (0,1) pair law. Both have full support. A detector that uses only those observations returns Nonidentifiable. Global quote inconsistency is a different finding. Proposition 3.4 tests compatibility of observations with any joint law.

5.3 Acquiring additional observations

An observation budget must specify which independent rounds it purchases. Accuracy for a selected pair and recovery of network support are different objectives. The following question concerns how simultaneous pair estimation changes the required sample size.

For a simpler allocation problem, suppose each pair’s variance coefficient is already known. The next proposition gives the best continuous allocation for a weighted sum of asymptotic variances. Here the index e denotes a pair, and w_e below is its specified decision weight.

For coupling estimation, a_e=w_e/(16)\sum_s1/p_{e,s} is the known asymptotic variance coefficient. With a=(1,4,9) and N=600, the allocation (100,200,300) has objective .06. Equal allocation has objective .07. These counts are separately purchased independent rounds. A full-vector round supplies several pair observations simultaneously and needs a different acquisition-cost model.

5.4 When pair interactions are insufficient

Some joint laws require interactions among three or more events. A complete set of conjunction probabilities determines such a law by inclusion–exclusion. The next transform generalizes the four-cell reconstruction and makes its information requirement explicit. The indicator X_i equals one when event i occurs and zero otherwise.

The complete table needs every required lower-order moment as well as the highest-order conjunction. A triple quote alone does not provide that table.

6.1 The saturated maximum-likelihood estimator

The closed-form inversion (7) takes the pair law as given. Resolved rounds instead supply counts in its four cells. The maximum-likelihood estimator (MLE) selects the parameter that makes those recorded counts most probable. The model fits every positive four-cell table, so it is called saturated. Consequently, when all counts are positive, the empirical frequencies themselves maximize the likelihood.

This is the classical 2\times2 log-linear exponential family [29][30, §9.2].

The statistics (\omega_i,\omega_j,\omega_i\omega_j) are sufficient: their sums retain all dependence of the likelihood on the parameter. Woolf’s log-odds-ratio variance describes its large-sample precision [4]. Section 6.2 also gives a finite-sample bound. Both results require the sampling and target conditions stated next.

The estimand. The pair marginal of the M-site distribution is itself a two-site Ising law: the family is saturated on the four cells, so exactly one parameter triple fits it. Write \theta^{\mathrm{eff}} = (h_i^{\mathrm{eff}}, h_j^{\mathrm{eff}}, J^{\mathrm{eff}}_{ij}) for that triple. At M = 2 it is (h_i, h_j, J_{ij}); at M > 2, Theorem 4.1 gives J^{\mathrm{eff}}_{ij} = J^*_{ij} + \Delta_{ij} with J^* the network interaction matrix. Every theorem of this section and Section 6.2 is a statement about \theta^{\mathrm{eff}}, the parameter of the law that generates the observations: the concentration below is around J^{\mathrm{eff}}_{ij}, and the embedding bias \Delta_{ij} does not shrink with N_p. The expansion in Section 4 requires environmental information beyond this pair law. Direct interactions are recovered through the additional observations or support information of Section 5.1.

The Fisher information measures local likelihood curvature and determines the estimator’s asymptotic covariance. Estimating the fields together with the coupling reduces the information available for that coupling. The Schur complement below removes this field contribution from the corresponding matrix block. The proof derives the variance both from this matrix and directly from the four empirical cell frequencies. An output labelled Boundary records an absent empirical cell and retains its uncertainty set, as Section 6.3 specifies.

6.2 Non-asymptotic concentration

Theorem 6.3(ii) describes a limit as the number of rounds grows. A finite-sample guarantee must also control the probability of an inaccurate estimate at the actual sample size. The proof first bounds errors in the three empirical moments using their bounded values in \{-1,+1\}. It then propagates those errors through the smooth inverse map. This requires a local region that stays away from every zero-cell boundary, which explains the radius in the theorem.

For completeness, write a_s=(s_i,s_j,s_i s_j), c_s=s_i s_j, and p_{\min}=\min_s p_s(\mu^*). Since p_s(\mu)=(1+a_s\cdot\mu)/4, \nabla J(\mu)=\frac1{16}\sum_s\frac{c_s a_s}{p_s(\mu)},\qquad \|\nabla^2J(\mu)\|_{\mathrm{op}}\leq\frac{3}{16k_\rho^2},\quad k_\rho=p_{\min}-\frac{\sqrt3\rho}{4}>0. The Hessian bound follows from \|a_sa_s^\top\|_{\mathrm{op}}=3. Integrating it gives the row-norm bound \|\nabla J(\mu^*)\|_2+3\rho/(16k_\rho^2). At the example’s cells (.45,.20,.20,.15), \rho=.02 gives k_\rho=.1413397 and a bound of 1.431917.

6.3 Confidence sets with finite and zero counts

The preceding concentration constant depends on the unknown population law. A cell confidence set can instead be constructed directly from the observed counts. It contains the true four-cell vector with a declared probability and remains defined when a cell has no observations. The construction bounds each cell count by a binomial interval, then intersects the four intervals with the probability simplex. That simplex is the set of nonnegative cell vectors whose coordinates sum to one.

For 0<k_s<n, the endpoints are beta quantiles with parameters (k_s,n-k_s+1) and (k_s+1,n-k_s). Set L_s=0 at k_s=0 and U_s=1 at k_s=n. An absent empirical cell returns Boundary and the confidence set. The finite unregularized MLE does not exist there. An optional estimate (k_s+\alpha)/(n+4\alpha), with declared \alpha>0, returns Regularized. Its finite inverse is distinct from the unregularized MLE and retains the confidence set. The vector (0,.2,.3,.5) is a boundary law with moments (-.6,-.4,0).

6.4 Independent rounds, pooling, and decisions

A data record counts independent resolved rounds for each estimand. It records the horizon, selection rule, missing resolutions, and any shared shocks. Exchangeability alone does not establish independence. Pooling requires a common pair law and independent units, or a replacement dependence-aware guarantee. Five independent strata with the same pair law can contribute 520 rounds each to a 2600-round estimate. Repeated tickets on one result contribute one round.

The illustrative concentration bound requires 75{,}065 rounds at \epsilon=\delta=.05, and 2{,}507{,}478 at .01. At one round per week, 75{,}000 rounds represent 1442.3 years using 52 weeks per year. Practical decisions therefore use the available sample size and its reported uncertainty. The exact cell construction supports modest samples without asserting a narrow interval. Section 10 reports synthetic coverage and shrinkage error at 104, 520, 2600, and 5000 rounds. A positive decision witness uses counts (2250,1000,1000,750). Its 95\% coupling enclosure is [.06428,.19745], which supports a positive effective coupling. Empirical utility requires prospective resolutions, a declared decision loss, and comparison at the actual data horizon.

7.1 Certified evaluation of the confidence interval

Near a cell boundary, numerical rounding can decide whether a parameter interval is finite. An outward enclosure rounds the lower endpoint down and the upper endpoint up, so the exact interval remains inside it. Exact rational inputs permit these enclosures through every arithmetic operation in Proposition 7.2. The required logarithms and square roots reduce to a positive series and an integer square root. The first two lemmas justify these numerical bounds; the corollary applies them to the statistical interval without reducing its coverage.

The integer b specifies component precision. The final interval reports its actual rational endpoints and width. Close to the boundary, test \eta_W^+<\kappa before using the interior error bound. If that test fails, retain the cell uncertainty set and its potentially infinite parameter interval. Outward decimal display uses integer floor for a lower endpoint and integer ceiling for an upper endpoint after multiplication by 10^d. For example, -1/3 displays as [-.3334,-.3333] at four decimal places. The calculation accepts rational inputs, exact cell counts, and an immutable catalogue. A resource limit returns an explicit failure. These arithmetic guarantees preserve the specified confidence statement. Sampling validity and drift evidence remain its statistical premises.

9.1 Marginals and exact computation for small M

The marginal probability of event e_k occurring is \mathbb{P}(\omega_k = +1) \;=\; \frac{\sum_{\omega: \omega_k = +1} \exp(-H(\omega))}{Z} \;=\; \frac{Z_k^+}{Z}, \tag{21} where Z_k^+ sums over configurations with \omega_k = +1.

Direct enumeration computes the partition function and all marginals in O(M^2 2^M) operations. The feasible size depends on the measured resource budget. Tree inference is exact by sum-product, while general junction-tree cost depends exponentially on graph width [36].

9.2 Mean-field approximation for larger M

Mean field approximates the joint law with a product distribution. Its magnetisation at site i is the mean spin, so a value m_i corresponds to event probability (1+m_i)/2. The approximation chooses each site’s mean consistently with the average influence of its neighbours. Its variational objective minimizes Kullback–Leibler divergence from the product distribution to the specified Ising law. This divergence measures the expected logarithmic probability ratio under the first law. Algorithm selection depends on graph structure, error requirements, and computation cost.

The row sum of coupling magnitudes bounds how much one mean-field update can amplify an input error. When its maximum is below one, repeated updates contract that error and determine a unique fixed point. The theorem separates this convergence guarantee from the variational bound on the normalizing constant.

Convergence to the fixed point leaves the approximation error relative to the true law unresolved. The next bound controls that error in singleton means. Its proof compares the exact conditional expectation with the mean-field update by a second-order Taylor estimate.

The susceptibility below is the derivative of the mean-field means with respect to the fields. It measures how the approximation responds to a small field change. Here \|\cdot\|_{\mathrm{op}} is the Euclidean operator norm, and \lambda_{\max} denotes the largest eigenvalue. The formula concerns this approximation; exact environmental covariances use Proposition 4.3.

The iteration bound controls distance to the mean-field fixed point. Proposition 9.3 controls singleton approximation error within the specified Ising law. Estimation accuracy, physical calibration, and trading profit require separate evidence.

For example, take two sites with zero fields and coupling J_{12}=0.5. Mean field converges immediately from zero and returns both singleton probabilities correctly. Its product joint probability is 0.25. The exact joint probability is (1+\tanh(0.5))/4=0.3655292893. Thus exact singleton predictions can accompany a joint pricing error of 0.1155292893.

10.1 Identification, uncertainty, and allocation

The three-site chain has zero fields and interactions J_{02}=.4, J_{12}=.6, J_{01}=0. Its effective endpoint coupling is .2069563793. The alternative isolated-edge law matches the selected observation interface within 1.2\times10^{-16}. The smallest cell across both laws is .03588, while their full tables differ by up to .11462. Thus the negative witness uses two positive laws.

Exact full-vector likelihood fitting recovers the population endpoint interaction within 1.4\times10^{-11}. Conditional logistic fitting recovers it within 1.2\times10^{-11}. On the same 5000 independently sampled full-vector rounds, both estimates are .0154254. The pair-only estimate on those rounds is .206945 and retains its effective-coupling label. The fixture supplies no parlay quote, so the quote interface returns no estimate while aligned outcomes remain usable.

For the pair law (.45,.20,.20,.15), the true effective coupling is .1308120. Each row below uses 2000 independent training replications. The regularized estimate uses \alpha=.5. Coverage refers to the conservative coupling enclosure in Proposition 6.8. The Wald interval is the point estimate plus or minus 1.96 estimated standard errors. In Table 1, RMSE means root mean squared estimation error across replications. Enclosure coverage is the fraction of replications whose cell-derived coupling enclosure contains the generating coupling. Median width describes that enclosure’s precision.

At 2600 rounds, the coupling enclosure certifies positive sign in .9635 of replications. At 104 rounds, that fraction is .001. The regularized cell predictor’s expected log loss on an independent test draw, integrated exactly under the generating law, falls from 1.30217 to 1.28797 across the first and last rows. The generating law’s entropy is 1.28767. These quantities measure predictive precision under the synthetic law. A real data audit must separately establish its sample size, sampling contract, and decision utility.

The zero-count fixture (0,20,30,50) returns Boundary and a lower-unbounded coupling interval. Its declared \alpha=.5 estimate remains finite. The eight-cell saturated reconstruction has maximum cell error below 10^{-15}. Known-variance allocation returns (100,200,300) and equal Karush–Kuhn–Tucker (KKT) multiplier values for the three coordinates. These multipliers express the equality-constrained optimum of Proposition 5.4. The incompatible three-spin quotes return the analytical maximum-norm residual 17/30.

10.2 An informative window under full-law drift

Set T=32768 and generate independent categorical outcomes with laws p_s=(a_s,b_s,b_s,a_s),\qquad a_s=.4-2\times10^{-6}(T-s),\qquad b_s=.1+2\times10^{-6}(T-s). All cells are positive. The current coupling is J(p_T)=\tfrac12\log4=.693147\ldots. The cell drift bound is L=2\times10^{-6} per round. Declare four windows ending at T, with lengths 512, 2048, 8192, and 32768. Allocate \delta_W=.05/4 to each and use D_W=L(N-1)/2. Python’s random.Random with seed 20260905 supplies one uniform variate per round, assigned in cell order (++,+-,-+,--).

Selecting the narrowest computed coupling enclosure chooses 8192 rounds and retains a positive lower endpoint. All four enclosures contain the current coupling in this realization. The longest window reduces sampling error but increases pooling bias, which its drift term covers. This realization demonstrates an informative output. Proposition 7.2 supplies the coverage statement under the stated law, rather than a coverage frequency from one draw. The interval estimates the current effective coupling. Direct-edge identification and physical calibration retain their separate observation requirements.

10.3 Environmental and computational checks

Small cycle environments have fields linearly spaced from -.2 to .2 and equal edge coupling J. Exact enumeration gives the following covariance row sums and their influence bounds.

Enumeration checks these examples. The coupling argument in Proposition 4.3 establishes the general constant.

A six-site path uses the field vector and successive edge couplings h=(.15,-.1,.2,-.15,.1,-.05),\qquad J_E=(.22,-.18,.20,.16,-.21). The model determines all reference inputs. The reconstruction uses its exact singleton and tree-edge marginals. Each resulting law prices the same six singleton and fifteen pair statistics. Exact enumeration and known-tree reconstruction agree within 3\times10^{-16}. The independent law with exact singleton marginals has maximum pair-moment error .209898. Mean field has maximum singleton-mean error .013955 and pair-moment error .211517. Its fixed-point iteration takes 25 steps to a maximum coordinate change below 10^{-12}. Here r=.4, so the contraction guarantee holds despite the pair error. The 64-state reference makes these errors directly observable. It does not establish a universal dimension cutoff or speed ranking.

10.4 Market mechanisms and loss

An accurate joint distribution does not by itself determine a market maker’s monetary loss. That loss also depends on how requests change prices, what shares are issued, and what premiums are collected. This subsection compares those execution choices on common synthetic inputs. An automated market maker is the trading rule that quotes prices and issues the requested contracts. We use binary indicators X_A,X_B\in\{0,1\} here, with joint cell order 00,01,10,11. This is the same outcome space as the spin representation, with the zero outcome corresponding to spin -1. A primitive is one executed sub-trade within a request.

ParlayMarket uses trade-target gradients to update a shared Ising law [3]. Its learning and expected-loss bounds impose additional local curvature, gradient-noise, and sensitivity conditions. The Automated Parlay Market Maker (APMM) maintains separate LMSR books for subsets of events [37]. Each book records its own contract inventory and prices. APMM propagates inventory updates from smaller subsets into their containing books. Its mixed YES/NO contract universe is larger than ParlayMarket’s basic positive-conjunction universe. APMM’s loss calculation includes charged premiums, issued holdings, and premium-free propagation. Neither mechanism’s learning conditions establish physical calibration of observed prices.

A price update determines a monetary loss only after execution specifies the shares issued and the premium collected. The following experiment records both quantities through settlement. Each arrival supplies one Bernoulli target for A or B. The executed contracts are A, its complement, B, and its complement, with unit settlement payouts. Both mechanisms natively accept this common interface. An APMM pair request supplies a complete joint target and causes several sub-trades. A scalar ParlayMarket request supplies different information. The first comparison executes singleton requests and reports pair-book prices as diagnostics. A second comparison below supplies complete pair information through explicit adapters.

An arrival is one trader request with a stated probability target and purchase budget. The maker receives a premium when it issues shares and pays their specified unit amount when the outcome settles. All runs use liquidity b=10, initial maker cash K=100, a budget of 100 per arrival, and zero fees. Every mechanism receives the same prior, target sequence, and declared settlement law within each fixture. Cash covers every possible issued payout after each accepted request. A request exceeding its trader budget or this cash constraint leaves prices, cash, and holdings unchanged.

For a singleton quote q and target \tau, define the share difference and minimum buy-only bundle by d=b(\mathop{\mathrm{logit}}\tau-\mathop{\mathrm{logit}}q),\qquad h_1=\max(d,0),\qquad h_0=\max(-d,0). The trader receives h_0,h_1 shares and pays c=b\log\bigl((1-q)e^{h_0/b}+qe^{h_1/b}\bigr). This cost difference is specified in ParlayMarket [3, Appendix E.4] and APMM [37, Eq. 8]. Adding equal shares to both outcomes adds the same amount to premium and payout, so the minimum choice preserves net loss. Let T contain the 0/1 singleton and pair products of ParlayMarket’s exponential family. Let \phi denote the coefficients of this 0/1 exponential family. In the update below, P_\phi denotes the replay’s pricing law, rather than the physical outcome law P used earlier. The ParlayMarket replay (PM replay) takes one step of size \eta=.2 using the pre-trade law: \phi'=\phi-\eta\frac{q-\tau}{q(1-q)}\operatorname{Cov}_{P_\phi}(T(X),X_i). Exact enumeration supplies the expectation in its trade-target update [3, Eq. 13 and Appendix E.4]. A purchase gauge chooses how many equal shares to add to both outcomes without changing their net payoff difference. The minimum gauge adds none beyond the required nonnegative bundle. The PM replay implements Appendix E.4 with exact inference, this gauge, and unpaid internal shadow resets. A shadow reset changes the internal pricing state without a corresponding external trade. These resets issue zero external shares. Its reported losses refer to this declared replay realization.

APMM adds the real bundle to the selected singleton block. The resulting change propagates into containing books, with zero extra premium and zero extra issued holdings [37, Sections 3.2–3.3]. The exact joint-LMSR reference applies the same real bundle to its four outcome inventories and prices from their normalized exponential weights. For every mechanism, let D_t(x) be all real shares payable at outcome x through arrival t. With collected premiums c_s and fees f_s, define L_t(x)=D_t(x)-\sum_{s\leq t}(c_s+f_s),\qquad R=\max_{t,x}\max(L_t(x),0). The terminal maker cash is K-L_T(x), and R is the reserve required by that particular path. All four outcomes are enumerated. The expected loss uses the explicitly supplied synthetic law.

The independent fixture starts at the uniform law and settles under (.06,.14,.24,.56), ordered 00,01,10,11. It repeats the arrivals (A,.8),(B,.7) thirty times. Every method reduces maximum singleton error from .3 to at most .034747. The correlated fixture starts at (.4,.1,.1,.4) and settles under (.16,.04,.16,.64). It receives (A,.8),(B,.68) once. The repeated-information fixture gives that same pair of targets four times, with the same prior and settlement law. Table 4 summarizes complete ledgers for these paths.

After the correlated fixture’s first arrival, APMM’s joint table is (.16,.04,.16,.64), while its untouched singleton B book quotes .5. The resulting book disagreement is .18. After the second arrival, both singleton books equal their supplied targets, but the joint’s maximum marginal disagreement is .138697. Repeated targets preserve that disagreement. ParlayMarket’s quotes come from one law throughout, while its gradient step continues to leave a target discrepancy. Repeated information therefore produces additional real trades and loss in this declared execution rule. These finite results identify effects of the update and execution choices. They supply no parameter-independent ranking or general loss bound for either mechanism.

A separate APMM fixture executes a full pair target (.16,.04,.16,.64) from the uniform prior. Its three sub-trades run bottom-up through A, B, and the pair. Each premium uses the current book after lower-order propagation, then charges only its own block change. Their premiums are 9.162907, 4.462871, and 12.237754. The total issued payouts are (21.400662,7.537718,21.400662,35.263605). Expected loss is 3.854895, and maximum loss is 9.400073. A budget sufficient only for the first sub-trade rejects the full request without changing any block. This checks complete pair accounting within APMM. The following construction matches the information supplied to the two interfaces.

The supplementary scripts market_ledgers.py and test_market_ledgers.py record the complete arrivals, premiums, fees, issued payouts, and prefix reserves. Fourteen checks cover cash conservation, entropy loss, gradient differentiation, propagation, pair routing, fees, and atomic rejection. The estimation results of this paper remain separate from these execution mechanisms.

11.1 Relation to prior Ising-based models of financial markets

The econophysics literature [7, 8, 9, 10, 11] applies Ising-type models to equity markets with J_{ij} modelling trader-imitation strength and \omega_i modelling a trader’s decision. The present model is different: spins are event outcomes (binary resolutions of real-world events), and J_{ij} is a pairwise interaction term in the joint distribution over outcomes, identified from the specified joint probabilities or estimated from aligned resolved outcomes. The trader-interaction reading of J is not claimed.

The Ising structure-learning literature provides complementary results. Ravikumar, Wainwright, and Lafferty [15] prove sample-complexity bounds for recovering the sparsity pattern of J from samples of the full spin vector via \ell_1-regularised logistic regression. Santhanam and Wainwright [24] give the matching information-theoretic lower bound n = \Omega(c^d \log p) for structure recovery, and Bresler [25] gives the first \mathrm{poly}(p)-time algorithm without the incoherence condition. Here n counts full-vector samples, p the sites, and d the maximum degree. The constant c in the cited lower bound depends on its interaction class. Vuffray et al. [16] give a sample-optimal interaction-screening estimator. All of these operate on samples of the full \omega-vector and target the full adjacency under sparsity regularisation. The parlay-MLE of Section 6.1 operates on samples of the pair-marginal of \omega and targets one effective pair coupling with unpenalized estimation; the sample-complexity regime is different (parametric \epsilon^{-2}\log(1/\delta) per pair rather than combinatorial scaling in the full-adjacency support). Combining the two, by using market-derived pair-level moment estimates as side information for the \ell_1 or interaction-screening estimator, is a natural direction.

On what market prices identify probabilistically: Manski [26] showed that under heterogeneous beliefs or risk aversion, a prediction-market price estimates a functional of the cross-trader belief distribution rather than the objective probability. A risk-neutral scoring rule represents the participating beliefs under its assumptions. Physical calibration requires independent resolved outcomes.

11.2 Relation to combinatorial prediction markets

Hanson’s combinatorial information markets posed the joint-outcome pricing problem [1]. Chen and Pennock [13] developed a utility framework for bounded-loss market makers on combinatorial outcome spaces, and Abernethy, Chen, and Wortman Vaughan [14] showed that automated market-making on combinatorial markets admits a convex-optimization formulation. A central design question in that literature is the choice of prior over joint outcomes; the standard LMSR cost function implicitly uses a uniform prior. The Ising parameterisation (3) provides an alternative, namely a prior with pairwise moment constraints, and the MLE of Theorem 6.3 gives a procedure for estimating its effective pair parameters from aligned resolved outcomes. Whether an LMSR with an Ising-parameter prior strictly dominates the uniform-prior LMSR on specific classes of correlated-event markets is an open question.

The Claim as Primitive defines the cost-function pricing interface that consumes these joint probabilities [32]. One-Way Coupling of Prediction Markets to Automated Market Makers defines the incentive-compatible channel that can move an automated-market-maker price from a prediction signal [33].

11.3 Open problem: dynamic identification from lead-lag data

The parlay-based estimator requires joint observations for each pair (i,j). Marginal price paths provide a different type of observation. The static Ising law specifies the distribution of resolved outcomes, but it does not specify a stochastic process for pre-resolution prices or returns. A Granger statistic measures the additional predictive contribution of one variable’s past, conditional on the included past variables. The contemporaneous R^2 below instead measures the fraction of variation explained by a regression within the same round.

The zero-field two-site law gives the exact contemporaneous normalization \begin{aligned} \rho_{ij}:=\mathop{\mathrm{Corr}}(\omega_i,\omega_j)&=\tanh J_{ij}, &J_{ij}&=\operatorname{arctanh}\rho_{ij},\\ R_{\mathrm{contemp}}^2&=\rho_{ij}^2=\tanh^2 J_{ij} =J_{ij}^2+O(J_{ij}^4). \end{aligned} \tag{24} Indeed, Z=4\cosh J_{ij}, symmetry gives zero means and unit variances, and \partial_{J_{ij}}\log Z=\mathbb{E}[\omega_i\omega_j]. The one-regressor contemporaneous identity R^2=\mathop{\mathrm{Corr}}^2 and the Taylor expansion then give (24). This identity is contemporaneous. It does not identify J_{ij} from a lagged Granger statistic. For example, let the vectors \omega_t be independent across time, with each vector drawn from the same two-site law and J_{ij}\neq 0. Equation (24) still holds within each round, while every population lagged-Granger partial-R^2 is zero.

A lead-lag estimator therefore requires an explicit transition law. One possible starting point is the synchronous kinetic Ising model \mathbb{P}(\omega_{t+1}\mid\omega_t) =\prod_j \frac{\exp\!\left\{\omega_{j,t+1}\left(a_j+\sum_i K_{ij}\omega_{i,t}\right)\right\}} {2\cosh\!\left(a_j+\sum_i K_{ij}\omega_{i,t}\right)}. \tag{25} If this transition law holds and the lag vector has full support, its conditional probabilities identify the directed coefficients K_{ij}. Consistent estimation also requires a stationary or explicitly modelled non-stationary sampling law. Relating K_{ij} to the static symmetric couplings J_{ij} requires an additional model. Deriving that relation for marginal-market price paths remains open.

11.4 Limitations

The results are stated under explicit modelling assumptions:

(L1)

Pairwise interactions only. The Ising Hamiltonian (2) omits genuine higher-order log-linear interactions. Event triples with joint dependence but pairwise independence (e.g., bracket-conditional tournaments) are misspecified.

(L2)

Stationarity. Theorems 6.3 and 6.4 assume \theta^{\mathrm{eff}} is constant over the observation window. Section 7 estimates a current conditional pair law with separate sampling and full-law drift bounds. Establishing those drift bounds requires a declared model or independent simultaneous evidence.

(L3)

Symmetric J_{ij} = J_{ji}. The Ising parameterisation is undirected. Asymmetric causal relationships cannot be represented; a directed graphical-model alternative is more expressive and lacks the simple closed-form identifiability of Theorem 3.1.

(L4)

LMSR-to-Ising bridge. The inversion (7) assumes the quoted marginals and the quoted parlay price equal the corresponding Boltzmann probabilities. Section 8 propagates a declared quote-error set. Scoring-rule equilibrium alone supplies no physical calibration certificate. A full model of the price-formation process mapping the Boltzmann distribution to observed order-book prices is not developed here.

(L5)

Admissible-region boundary. When a parlay price lies near the Fréchet-Hoeffding boundary, the closed-form estimator is unstable and the MLE has large variance (Remark 8.2). Section 6.3 supplies boundary output and uncertainty sets.

(L6)

Mean-field convergence condition. Theorem 9.2 requires r<1. This condition can fail in dense networks with strong coupling; exact enumeration and structured inference remain available according to their resource and graph-width requirements.

(L7)

Survivorship bias. Quote availability and outcome availability are recorded separately. Selective listing or missing resolutions can alter either target population. Inclusion in outcome inference depends on aligned observations and the declared sampling law.

(L8)

Causal sufficiency. Two events that co-move through an unobserved common factor produce nonzero \hat J_{ij} with zero direct interaction. Effective coupling records the observed pair law. Direct-interaction interpretation requires the additional network model and observations in Section 5.

(L9)

One-shot pairs. Definition 6.2 counts settled rounds, not contracts, so a pair of event types that resolves once supplies N_p = 1 whatever the open interest, and Theorems 6.3 and 6.4 give no useful precision guarantee from it. Proposition 6.8 still returns a defined uncertainty set. The estimation results apply to pair types that recur, such as per-fixture, per-round or per-day markets. Pooling structurally similar pairs into one type to raise N_p requires the common-law and independence conditions of Assumption 6.1(R2). Those conditions are data-model assumptions.

12.1 Notation and regularity

Fix a pair (i, j) and suppress indices: \omega = (\omega_i, \omega_j), \theta = (h_i, h_j, J_{ij}) \in \Theta \subset \mathbb{R}^3. Let \mathbb{P}_\theta denote the two-site Ising distribution with density p_\theta(\omega) \;=\; \exp\!\bigl(h_i \omega_i + h_j \omega_j + J_{ij} \omega_i \omega_j - A(\theta)\bigr), \qquad A(\theta) = \ln\!\sum_\omega \exp(\cdot). The sufficient statistics are T(\omega) = (\omega_i, \omega_j, \omega_i \omega_j) \in \{-1, +1\}^3, and the model is a minimal regular exponential family on the 4-point support \{-1, +1\}^2 with three-dimensional natural parameter space \Theta = \mathbb{R}^3 (full-rank, since the three sufficient statistics are affinely independent on the support) [18, Ch. 1][19, Ch. 2]. Minimality, together with \mathbb{R}^3 being the full domain of finiteness of the log-partition function A, places the model in the canonical regular-exponential-family setting of [18, Thm. 3.6]. The mean-parameter map \theta \mapsto \mu(\theta) := \mathbb{E}_\theta T is consequently a C^\infty diffeomorphism from \mathbb{R}^3 onto the interior of the convex hull of T(\{-1, +1\}^2) \subset \mathbb{R}^3, with Jacobian d\mu / d\theta = I(\theta) = \mathop{\mathrm{Cov}}_\theta(T) \succ 0 everywhere in \Theta. Injectivity of \theta \mapsto \mathbb{P}_\theta on \Theta follows immediately.

Assumption 6.1 states the sampling and target conditions used throughout the fixed-law estimates.

Identifiability also follows from strict convexity of the log-partition function A on the interior of \Theta [18, Thm. 1.13]: distinct natural parameters give distinct mean parameters \mathbb{E}_\theta T.

12.2 Schur complement and the Woolf identity

Writing the cells as p_{s_i s_j} = \tfrac{1}{4}(1 + m_i^* s_i + m_j^* s_j + C_{ij}^* s_i s_j), expanding S = I_{33} - b^\top A^{-1} b in (m_i^*, m_j^*, C_{ij}^*), and clearing denominators verifies the identity S^{-1} \;=\; \frac{1}{16}\sum_{s_i, s_j \in \{\pm 1\}} \frac{1}{p_{s_i s_j}} used in Theorem 6.3(iii). The identity is also forced, with no computation, by the delta-method derivation in that proof: both expressions are the asymptotic variance of the same estimator.

Accompanying computations.

The archive market-ledgers-supplement.zip contains the market ledgers, pair adapter, synthetic experiments, and drift and certified-arithmetic fixtures. Its README.md gives reproduction commands, dependency versions, comparison rules, and the scope of each computation. The archive’s SHA-256 digest is

a8a6ea2c1212bbb5f70b32ff0d52f964eec86fddc893239967dc4332a478c489.