A New Axiomatization for Likelihood Gambles

This paper studies a new and more general axiomatization than one presented in [6] for preference on likelihood gambles. Likelihood gambles describe actions in a situation where a decision maker knows multiple probabilistic models and a random sample generated from one of those models but does not know prior probability of models. This new axiom system is inspired by Jensen’s axiomatization of probabilistic gambles. Our approach provides a new perspective to the role of data in decision making under ambiguity. 1 Likelihood gambles Likelihood gambles introduced in [5, 6] describe actions in situation of model ambiguity characterized by (1) there are multiple probabilistic models; (2) there is data providing likelihoods for the models and (3) there is no prior probability about the models. Formally, we consider a general problem described by a tuple (X,Y,Θ,A,x). X,Y are variables describing a phenomenon of interest. X is experiment variable whose values can be observed through experiments or data gathering (e.g. lab test results, clinical observations). Y is utility variable whose values determine the utility of actions (e.g. stages of disease, relative size of the tumor). Θ is the set of models that encode the knowledge about the phenomenon. To be precise, Θ is a set of indices and knowledge is encoded in probability functions Prθ(X,Y ) for θ ∈ Θ. A is the set of alternative actions (e.g. surgery, radiation therapy, chemotherapy) that are functions from utility variable Y to the unit interval [0, 1] representing utility. Fi∗I thank Bharat Rao for encouragement and support and UAI-2006 referees for their constructive comments. One should note that the use of utility rather than nally, evidence/data/observation gathered on experiment variable is X = x. A fundamental question to be answered is which among the alternative actions is the best choice given the information. We introduce the concept of likelihood gambles and derive a pricing formula that will allow their comparison. Note that given a model θ ∈ Θ and observation x, distribution on utility variable Y is Prθ(y|x). According to the classical Bayesian decision theory actions a ∈ A are values by their expected utility