On Non-Bayesian Social Learning

We study a model of information aggregation and social learning recently proposed by Jadbabaie, Sandroni, and Tahbaz-Salehi, in which individual agents try to learn a correct state of the world by iteratively updating their beliefs using private observations and beliefs of their neighbors. No individual agent's private signal might be informative enough to reveal the unknown state. As a result, agents share their beliefs with others in their social neighborhood to learn from each other. At every time step each agent receives a private signal, and computes a Bayesian posterior as an intermediate belief. The intermediate belief is then averaged with the belief of neighbors to form the individual's belief at next time step. We find a set of minimal sufficient conditions under which the agents will learn the unknown state and reach consensus on their beliefs without any assumption on the private signal structure. The key enabler is a result that shows that using this update, agents will eventually forecast the indefinite future correctly

Consistency of Feature Markov Processes

We are studying long term sequence prediction (forecasting). We approach this by investigating criteria for choosing a compact useful state representation. The state is supposed to summarize useful information from the history. We want a method that is asymptotically consistent in the sense it will provably eventually only choose between alternatives that satisfy an optimality property related to the used criterion. We extend our work to the case where there is side information that one can take advantage of and, furthermore, we briefly discuss the active setting where an agent takes actions to achieve desirable outcomes.Comment: 16 LaTeX page

Algorithmic Identification of Probabilities

TThe problem is to identify a probability associated with a set of natural numbers, given an infinite data sequence of elements from the set. If the given sequence is drawn i.i.d. and the probability mass function involved (the target) belongs to a computably enumerable (c.e.) or co-computably enumerable (co-c.e.) set of computable probability mass functions, then there is an algorithm to almost surely identify the target in the limit. The technical tool is the strong law of large numbers. If the set is finite and the elements of the sequence are dependent while the sequence is typical in the sense of Martin-L\"of for at least one measure belonging to a c.e. or co-c.e. set of computable measures, then there is an algorithm to identify in the limit a computable measure for which the sequence is typical (there may be more than one such measure). The technical tool is the theory of Kolmogorov complexity. We give the algorithms and consider the associated predictions.Comment: 19 pages LaTeX.Corrected errors and rewrote the entire paper. arXiv admin note: text overlap with arXiv:1208.500

Non-Bayesian Social Learning, Second Version

We develop a dynamic model of opinion formation in social networks. Relevant information is spread throughout the network in such a way that no agent has enough data to learn a payoff-relevant parameter. Individuals engage in communication with their neighbors in order to learn from their experiences. However, instead of incorporating the views of their neighbors in a fully Bayesian manner, agents use a simple updating rule which linearly combines their personal experience and the views of their neighbors (even though the neighbors’ views may be quite inaccurate). This non-Bayesian learning rule is motivated by the formidable complexity required to fully implement Bayesian updating in networks. We show that, under mild assumptions, repeated interactions lead agents to successfully aggregate information and to learn the true underlying state of the world. This result holds in spite of the apparent naıvite of agents’ updating rule, the agents’ need for information from sources (i.e., other agents) the existence of which they may not be aware of, the possibility that the most persuasive agents in the network are precisely those least informed and with worst prior views, and the assumption that no agent can tell whether their own views or their neighbors’ views are more accurate.Social networks, learning, information aggregation

Successive Standardization of Rectangular Arrays

In this note we illustrate and develop further with mathematics and examples, the work on successive standardization (or normalization) that is studied earlier by the same authors in Olshen and Rajaratnam (2010) and Olshen and Rajaratnam (2011). Thus, we deal with successive iterations applied to rectangular arrays of numbers, where to avoid technical difficulties an array has at least three rows and at least three columns. Without loss, an iteration begins with operations on columns: first subtract the mean of each column; then divide by its standard deviation. The iteration continues with the same two operations done successively for rows. These four operations applied in sequence completes one iteration. One then iterates again, and again, and again,.... In Olshen and Rajaratnam (2010) it was argued that if arrays are made up of real numbers, then the set for which convergence of these successive iterations fails has Lebesgue measure 0. The limiting array has row and column means 0, row and column standard deviations 1. A basic result on convergence given in Olshen and Rajaratnam (2010) is true, though the argument in Olshen and Rajaratnam (2010) is faulty. The result is stated in the form of a theorem here, and the argument for the theorem is correct. Moreover, many graphics given in Olshen and Rajaratnam (2010) suggest that but for a set of entries of any array with Lebesgue measure 0, convergence is very rapid, eventually exponentially fast in the number of iterations. Because we learned this set of rules from Bradley Efron, we call it "Efron's algorithm". More importantly, the rapidity of convergence is illustrated by numerical examples