We consider the design of forecasting competitions in which multiple forecasters make predictions about one or more independent events and compete for a single prize. We have two objectives: (1) to award the prize to the most accurate forecaster, and (2) to incentivize forecasters to report truthfully, so that forecasts are informative and forecasters need not spend any cognitive effort strategizing about reports. Proper scoring rules incentivize truthful reporting if all forecasters are paid according to their scores. However, incentives become distorted if only the best-scoring forecaster wins a prize, since forecasters can often increase their probability of having the highest score by reporting extreme beliefs. Even if forecasters do report truthfully, awarding the prize to the forecaster with highest score does not guarantee that high-accuracy forecasters are likely to win; in extreme cases, it can result in a perfect forecaster having zero probability of winning. In this paper, we introduce a truthful forecaster selection mechanism. We lower-bound the probability that our mechanism selects the most accurate forecaster, and give rates for how quickly this bound approaches 1 as the number of events grows. Our techniques can be generalized to the related problems of outputting a ranking over forecasters and hiring a forecaster with high accuracy on future events.

A prediction market is a useful means of aggregating information about a future event. To function, the market needs a trusted entity who will verify the true outcome in the end. Motivated by the recent introduction of decentralized prediction markets, we introduce a mechanism that allows for the outcome to be determined by the votes of a group of arbiters who may themselves hold stakes in the market. Despite the potential conflict of interest, we derive conditions under which we can incentivize arbiters to vote truthfully by using funds raised from market fees to implement a peer prediction mechanism. Finally, we investigate what parameter values could be used in a real-world implementation of our mechanism.

We investigate the limiting behavior of trader wealth and prices in a simple prediction market with a finite set of participants having heterogeneous beliefs. Traders bet repeatedly on the outcome of a binary event with fixed Bernoulli success probability. A class of strategies, including (fractional) Kelly betting and constant relative risk aversion (CRRA) are considered. We show that when traders are willing to risk only a small fraction of their wealth in any period, belief heterogeneity can persist indefinitely; if bets are large in proportion to wealth then only the most accurate belief type survives. The market price is more accurate in the long run when traders with less accurate beliefs also survive. That is, the survival of traders with heterogeneous beliefs, some less accurate than others, allows the market price to better reflect the objective probability of the event in the long run.

We consider the problem of belief aggregation: given a group of individual agents with probabilistic beliefs over a set of uncertain events, formulate a sensible consensus or aggregate probability distribution over these events. Researchers have proposed many aggregation methods, although on the question of which is best the general consensus is that there is no consensus. We develop a market-based approach to this problem, where agents bet on uncertain events by buying or selling securities contingent on their outcomes. Each agent acts in the market so as to maximize expected utility at given securities prices, limited in its activity only by its own risk aversion. The equilibrium prices of goods in this market represent aggregate beliefs. For agents with constant risk aversion, we demonstrate that the aggregate probability exhibits several desirable properties, and is related to independently motivated techniques. We argue that the market-based approach provides a plausible mechanism for belief aggregation in multiagent systems, as it directly addresses self-motivated agent incentives for participation and for truthfulness, and can provide a decision-theoretic foundation for the "expert weights" often employed in centralized pooling techniques.

I present a parallel algorithm for exact probabilistic inference in Bayesian networks. For polytree networks with n variables, the worst-case time complexity is O(log n) on a CREW PRAM (concurrent-read, exclusive-write parallel random-access machine) with n processors, for any constant number of evidence variables. For arbitrary networks, the time complexity is O(r^{3w}*log n) for n processors, or O(w*log n) for r^{3w}*n processors, where r is the maximum range of any variable, and w is the induced width (the maximum clique size), after moralizing and triangulating the network.

Graphical models based on conditional independence support concise encodings of the subjective belief of a single agent. A natural question is whether the consensus belief of a group of agents can be represented with equal parsimony. We prove, under relatively mild assumptions, that even if everyone agrees on a common graph topology, no method of combining beliefs can maintain that structure. Even weaker conditions rule out local aggregation within conditional probability tables. On a more positive note, we show that if probabilities are combined with the logarithmic opinion pool (LogOP), then commonly held Markov independencies are maintained. This suggests a straightforward procedure for constructing a consensus Markov network. We describe an algorithm for computing the LogOP with time complexity comparable to that of exact Bayesian inference.

To overcome the #P-hardness of computing/updating prices in logarithm market scoring rule-based (LMSR-based) combinatorial prediction markets, Chen et al. [5] recently used a simple Bayesian network to represent the prices of securities in combinatorial predictionmarkets for tournaments, and showed that two types of popular securities are structure preserving. In this paper, we significantly extend this idea by employing Bayesian networks in general combinatorial prediction markets. We reveal a very natural connection between LMSR-based combinatorial prediction markets and probabilistic belief aggregation,which leads to a complete characterization of all structure preserving securities for decomposable network structures. Notably, the main results by Chen et al. [5] are corollaries of our characterization. We then prove that in order for a very basic set of securities to be structure preserving, the graph of the Bayesian network must be decomposable. We also discuss some approximation techniques for securities that are not structure preserving.

In this article, we survey a number of mechanisms created to elicit predictions, many newly proposed within the last decade. We focus on the engineering questions: How do they work and why? What factors and goals are most important in their design? The primary goal of a prediction mechanism is to obtain and aggregate dispersed information, which often exists in tacit forms as beliefs, opinions, or judgements of agents. Coalescing information is a necessary first step for decision making in almost all domains. For example, consider seasonal influenza, a significant cause of illness and death around the world. Although it recurs every year, the geographic location, timing, magnitude, and duration of outbreaks vary widely. Many people possess relevant pieces of the full information puzzle, including doctors who meet patients, clinical microbiologists who perform respiratory culture tests, pharmacists who fill prescriptions, people who have the flu, and people who know people who have the flu.

In the context of binary Classification, we define disagreement as a measure of how often two independently-trained models differ in their Classification of unlabeled data.

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