Aggregation under bias: Rényi divergence aggregation and its implementation via machine learning markets

Abstract

Trading in information markets, such as machine learning markets, has been shown to be an effective approach for aggregating the beliefs of different agents. In a machine learning context, aggregation commonly uses forms of linear opinion pools, or logarithmic (log) opinion pools. It is interesting to relate information market aggregation to the machine learning setting. In this paper we introduce a spectrum of compositional methods, Rényi divergence aggregators, that interpolate between log opinion pools and linear opinion pools. We show that these compositional methods are maximum entropy distributions for aggregating information from agents subject to individual biases, with the Rényi divergence parameter dependent on the bias. In the limit of no bias this reduces to the optimal limit of log opinion pools. We demonstrate this relationship practically on both simulated and real datasets. We then return to information markets and show that Rényi divergence aggregators are directly implemented by machine learning markets with isoelastic utilities, and so can result from autonomous self interested decision making by individuals contributing different predictors. The risk averseness of the isoelastic utility directly relates to the Rényi divergence parameter, and hence encodes how much an agent believes (s)he may be subject to an individual bias that could affect the trading outcome: if an agent believes (s)he might be acting on significantly biased information, a more risk averse isoelastic utility is warranted.