Bayesian Regression Markets

TL;DR

The paper tackles the challenge of incentivizing data sharing for regression tasks by introducing a Bayesian regression market that accounts for parameter uncertainty through posterior predictive inferences. It extends prior work by enabling uncertainty-aware valuations, using Shapley-value-based revenue allocation, and exploring both likelihood-based and information-based designs (KL-based) to mitigate financial risk. The authors prove universal and asymptotic market properties under different designs and demonstrate, via simulations and a real-world solar irradiance case study, that KL-based information-valued approaches reduce risk and stabilize payments, especially in small-sample or nonstationary settings. The framework offers practical implications for data markets by enabling robust, uncertainty-aware compensation mechanisms that align incentives for data owners and buyers in decentralized analytics tasks.

Abstract

Although machine learning tasks are highly sensitive to the quality of input data, relevant datasets can often be challenging for firms to acquire, especially when held privately by a variety of owners. For instance, if these owners are competitors in a downstream market, they may be reluctant to share information. Focusing on supervised learning for regression tasks, we develop a regression market to provide a monetary incentive for data sharing. Our mechanism adopts a Bayesian framework, allowing us to consider a more general class of regression tasks. We present a thorough exploration of the market properties, and show that similar proposals in literature expose the market agents to sizeable financial risks, which can be mitigated in our setup.

Figures (11)

• Figure 1: Schematic illustration of existing frameworks for data sharing with multiple buyers and sellers, where each figure depicts a building block consisting of a single interaction. The blue, red and green arrows indicate computational, information and monetary transactions between the buyer and the seller, respectively.
• Figure 2: Overview of in-sample market platform operations at time $t$, with agent $a_1$ as the central agent and the total number of support agents $M = \vert \mathcal{A}_{-c} \vert$. The time index $t$ is omitted for brevity. Recall that the blue, red and green arrows indicate computational, information and monetary transactions, respectively.
• Figure 3: In-sample market with increasing batch size. The dashed lines in (a) highlight the true coefficients. The histogram in (b) shows the in-sample NLL distribution. The bars in (c) are the cumulative revenues given the value of each datapoint provided.
• Figure 4: Empirical average of the percentage improvement in the NLL ratio for BLR relative to MLE, plotted as a function of sample size.
• Figure 5: Empirical average of expected Shapley values for each market design, plotted as a function of sample size. Solid and dashed lines correspond to features $x_{2, t}$ and $x_{3, t}$, respectively.

Theorems & Definitions (15)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 5
• Theorem 6
• Remark 7
• Corollary 8
• Remark 9
• Corollary 10