Distributed Experimental Design: Bayes-optimal Fusion of Local Designs

Abstract: We develop a decision-theoretic framework for distributed Bayesian experimental design in which local agents evaluate candidate experiments using expected information gain and transmit their local design decisions to a fusion center. Unlike centralized Bayesian design, where all likelihood components and information-gain values are available to a single planner, the fusion center in the distributed setting chooses a global experiment from compressed local recommendations. We derive the Bayes-optimal fusion rule, which selects the experiment with largest conditional expected centralized information gain given the observed local design decisions. This rule is analogous in spirit to optimal fusion rules in distributed detection, but differs fundamentally because the underlying utility is expected information gain and the resulting loss is information-gain regret rather than classification error. We also establish information-loss bounds and identify conditions under which the decision-only fusion rule is asymptotically equivalent to the centralized design. Numerical experiments show that Bayes-optimal fusion closely approximates the centralized oracle, whereas majority voting can be highly suboptimal when a minority of sites carry disproportionate information.

• The paper develops a Bayes-optimal fusion rule that combines binary local design recommendations using site reliability, prior probabilities, and asymmetric information-loss costs.
• The paper proves that expected information regret decreases exponentially with the number of informative sites, approaching centralized design under conditional independence and sufficient separation.
• The paper’s simulations show near-oracle performance from one-bit messages, with mean regret of 0.010 nats versus 0.166 for majority voting in a heterogeneous Gaussian model, while naive voting worsens as sites increase.

Problem setting and motivation

This paper formulates distributed Bayesian experimental design as a decision-fusion problem. Local agents evaluate candidate experiments using expected information gain (EIG) and transmit only their binary design recommendations to a fusion center, which must select a single system-wide experiment. The central observation is that the experiment preferred by a majority of local sites need not be the one with the largest centralized EIG: in heterogeneous systems, a minority of highly informative sites can dominate global information content while being numerically outvoted.

The setting is motivated by physically distributed experiments (multicenter clinical trials, sensor networks), communication constraints that permit only small messages, privacy or regulatory restrictions on sharing likelihoods, and computational benefits of parallel local design. The authors position the problem at the intersection of Bayesian experimental design, distributed decision making, and information-theoretic utility maximization, arguing it is not reducible to any of these standard formulations. The key distinction from classical distributed detection is that the fused quantity is an experimental recommendation rather than a hypothesis decision, and the loss is information-gain regret rather than classification error.

The model assumes $M$ sites, a common finite experiment set (in the analysis, $\mathcal{E}=\{0,1\}$), conditional independence of site observations given $\theta$ and the chosen designs, and a common-treatment restriction $\xi_1 = \cdots = \xi_M = \delta(D)$. Each site computes its local EIG $U_m(\xi_m)$, transmits $D_m = \arg\max_{\xi_m} U_m(\xi_m)$, and the fusion center applies a rule $\delta: \{0,1\}^M \to \{0,1\}$. The loss incurred by choosing $a$ when the centralized optimum is $b$ is $L(a,b) = U(b) - U(a)$.

Bayes-optimal fusion rule

The first main result characterizes the optimal fusion rule. Writing $\mathcal{E}=\{0,1\}$0 for the centralized information-optimal design with prior probabilities $\mathcal{E}=\{0,1\}$1, and assuming the recommendations are conditionally independent given $\mathcal{E}=\{0,1\}$2 with $\mathcal{E}=\{0,1\}$3 and $\mathcal{E}=\{0,1\}$4, the Bayes-optimal rule selects design 1 if and only if

$\mathcal{E}=\{0,1\}$5

Structurally this mirrors the Chair–Varshney fusion rule, but the threshold carries an experimental-design interpretation: the ratio $\mathcal{E}=\{0,1\}$6 shifts the decision toward whichever incorrect choice would cause the larger EIG loss. When $\mathcal{E}=\{0,1\}$7, the fusion center demands stronger evidence before selecting design 1; under symmetric losses the rule reduces to a reliability-weighted likelihood-ratio test on the recommendations. In the homogeneous case ($\mathcal{E}=\{0,1\}$8, $\mathcal{E}=\{0,1\}$9), the statistic depends only on the vote count $\theta$0 through a weighted-majority threshold, which coincides with ordinary majority voting only in special symmetric cases.

Information-loss bound and asymptotic equivalence

The second main result bounds the expected information-gain regret. Since regret is bounded by $\theta$1, Chernoff/Chernoff-information arguments applied to the log-likelihood-ratio statistic $\theta$2 yield

$\theta$3

where $\theta$4 is the local Chernoff coefficient. Under the identically distributed condition $\theta$5, the error probability and hence the expected regret decay as $\theta$6, where $\theta$7 is the Chernoff information between $\theta$8 and $\theta$9. The condition $\xi_1 = \cdots = \xi_M = \delta(D)$0 has a direct reading: a site is more likely to recommend experiment 1 when experiment 1 is globally optimal. The result cleanly separates two effects—the information geometry of the experiment through $\xi_1 = \cdots = \xi_M = \delta(D)$1 and the reliability of local recommendations through $\xi_1 = \cdots = \xi_M = \delta(D)$2. Notably, even weakly informative recommendations suffice for exact recovery of the centralized design with probability tending to one; no quantitative characterization of how weak "weakly informative" can be before performance degrades materially is provided beyond the exponential rate itself.

Illustrative models

Two examples ground the theory. In the scalar Gaussian model with $\xi_1 = \cdots = \xi_M = \delta(D)$3 and site-specific precision contributions $\xi_1 = \cdots = \xi_M = \delta(D)$4, all quantities are closed-form: local EIG is $\xi_1 = \cdots = \xi_M = \delta(D)$5, centralized EIG is $\xi_1 = \cdots = \xi_M = \delta(D)$6, and the centralized optimum reduces to comparing total Fisher informations $\xi_1 = \cdots = \xi_M = \delta(D)$7 versus $\xi_1 = \cdots = \xi_M = \delta(D)$8. Even here, the optimal fusion rule is a reliability-weighted rule with an information-loss threshold, not majority voting.

The second model is a distributed binary-response experiment with logistic success probability $\xi_1 = \cdots = \xi_M = \delta(D)$9. Exact EIG is not closed-form because the logistic likelihood is non-conjugate to the Gaussian prior, so the paper uses a Laplace/Fisher-information approximation, giving $U_m(\xi_m)$0 with prior-averaged Bernoulli Fisher information. A useful structural insight emerges: informativeness requires both large design sensitivity $U_m(\xi_m)$1 and response probability away from 0 or 1, since the factor $U_m(\xi_m)$2 is maximized at $U_m(\xi_m)$3. The fusion theorem itself is distribution-free at the level of local decisions—only computation of $U_m(\xi_m)$4 and $U_m(\xi_m)$5 changes across models.

Numerical studies

Monte Carlo experiments compare four rules—the centralized oracle, Bayes fusion estimated from training data via conditional mean centralized EIG given the vote count $U_m(\xi_m)$6, majority voting, and random fusion—in a deliberately heterogeneous configuration where type-1 sites are less frequent but carry larger precision contributions.

For the scalar Gaussian model with $U_m(\xi_m)$7:

Rule	Mean EIG	Mean regret	90% regret	Oracle agreement
Centralized oracle	2.326	0.000	0.000	1.000
Bayes fusion	2.316	0.010	0.020	0.876
Majority vote	2.160	0.166	0.381	0.263
Random fusion	2.215	0.111	0.353	0.501

Bayes fusion loses only 0.010 nats of EIG relative to the oracle despite receiving only binary messages—an order-of-magnitude improvement over majority voting's 0.166 mean regret. As $U_m(\xi_m)$8 grows, Bayes-fusion regret falls from 0.029 ($U_m(\xi_m)$9) to 0.002 ($D_m = \arg\max_{\xi_m} U_m(\xi_m)$0) with oracle agreement rising to 0.954, consistent with the asymptotic equivalence theorem. Majority voting exhibits the opposite trend: regret grows to 0.186 and agreement collapses to 0.060, because adding more sites amplifies the structural bias toward the larger but less informative class. This is a strong and somewhat counterintuitive claim worth emphasizing: more data can make naive fusion strictly worse.

The binary-response model shows the same pattern. At $D_m = \arg\max_{\xi_m} U_m(\xi_m)$1, Bayes fusion achieves mean utility 1.8510 versus the oracle's 1.8570 (mean regret 0.0060, median and 90% regret both zero, 94.2% oracle agreement), while majority voting attains only 1.5226 mean utility with 13.2% agreement—worse than random fusion in utility terms. As $D_m = \arg\max_{\xi_m} U_m(\xi_m)$2 increases to 51, Bayes regret drops to 0.0010 with 98.6% agreement, whereas majority-vote regret rises to 0.3860 with agreement falling to 2.8%. The learned Bayes rule switches to experiment 1 at a vote count well below the majority threshold, reflecting the statistical association between experiment-1 recommendations and high-precision sites.

Two implications follow directly. Methodologically, distributed Bayesian design should be formulated as an information-gain fusion problem rather than a voting problem. Practically, communication-efficient design is feasible: near-centralized performance is achievable from one-bit messages whenever the centralized EIGs of competing experiments are sufficiently separated and the local recommendations are informative enough to recover the global ordering with high probability.

Limitations and open questions

Several restrictions bound the applicability of the results. The framework assumes a common finite experiment set implemented uniformly across all sites; allowing site-specific assignments would turn the problem into a different assignment problem over $D_m = \arg\max_{\xi_m} U_m(\xi_m)$3, which the paper explicitly does not address. The optimality and asymptotic results rely on conditional independence of local recommendations given $D_m = \arg\max_{\xi_m} U_m(\xi_m)$4 and on known or estimable reliabilities $D_m = \arg\max_{\xi_m} U_m(\xi_m)$5; correlated recommendations, arising for example from conditionally dependent experiments, fall outside the current theory. The binary-response analysis depends on a Fisher-information/Laplace approximation to EIG whose accuracy degrades for diffuse priors. All results concern discrete design spaces; the continuous-design case remains open. Finally, the numerical studies use synthetic heterogeneity structures; whether the estimated conditional-mean fusion rules retain their advantage under real-world multicenter variability is not established.

Conclusion

The paper develops a decision-theoretic framework in which compressed local design recommendations are fused by maximizing conditional expected centralized EIG, yielding a Chair–Varshney-like rule with an information-loss threshold. Finite-sample regret bounds with exponential decay at the Chernoff rate establish asymptotic equivalence between decision-only fusion and centralized design under separation and concentration conditions. Numerical evidence in Gaussian and binary-response models shows near-oracle performance from one-bit messages, while majority voting degrades—and can worsen with additional sites—under heterogeneous informativeness. Extending the framework to multiple candidate experiments, richer message types, sequential design, dependent experiments, and continuous design spaces remains open.