In this paper we introduce an approach and algorithms for model mixing in large prediction problems with correlated predictors. We focus on the choice of predictors in linear models, and mix over possible subsets of candidate predictors. Our approach is based on expressing the space of models in terms of an orthogonalization of the design matrix. Advantages are both statistical and computational. Statistically, orthogonalization often leads to a reduction in the number of competing models by eliminating correlations. Computationally, large model spaces cannot be enumerated; recent approaches are based on sampling models with high posterior probability via Markov chains. Based on orthogonalization of the space of candidate predictors, we can approximate the posterior probabilities of models by products of predictor-specific terms. This leads to an importance sampling function for sampling directly from the joint distribution over the model space, without resorting to Markov chains. Comp...

The term Soft Computing (SC) represents the combination of emerging problem-solving technologies such as Fuzzy Logic (FL), Probabilistic Reasoning (PR), Neural Networks (NNs), and Genetic Algorithms (GAs). Each of these technologies provide us with complementary reasoning and searching methods to solve complex, real-world problems. After a brief description of each of these technologies, we will analyze some of their most useful combinations, such as the use of FL to control GAs and NNs parameters; the application of GAs to evolve NNs (topologies or weights) or to tune FL controllers; and the implementation of FL controllers as NNs tuned by backpropagation-type algorithms.

Currently, there is renewed interest in the problem, raised by Shafer in 1985, of updating probabilities when observations are incomplete (or set-valued). This is a fundamental problem in general, and of particular interest for Bayesian networks. Recently, Gr unwald and Halpern have shown that commonly used updating strategies fail in this case, except under very special assumptions. In this paper we propose a new method for updating probabilities with incomplete observations. Our approach is deliberately conservative: we make no assumptions about the so-called incompleteness mechanism that associates complete with incomplete observations. We model our ignorance about this mechanism by a vacuous lower prevision, a tool from the theory of imprecise probabilities, and we use only coherence arguments to turn prior into posterior (updated) probabilities. In general, this new approach to updating produces lower and upper posterior probabilities and previsions (expectations), as well as partially determinate decisions. This is a logical consequence of the existing ignorance about the incompleteness mechanism. As an example, we use the new updating method to properly address the apparent paradox in the `Monty Hall' puzzle. More importantly, we apply it to the problem of classification of new evidence in probabilistic expert systems, where it leads to a new, so-called conservative updating rule.

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The 30th anniversary of The Journal of Portfolio Management is a milestone in the rich intellectual history of modern finance, firmly establishing the relevance of quantitative models and scientific inquiry in the practice of financial management. One of the most enduring ideas from this intellectual history is the Efficient Markets Hypothesis (EMH), a deceptively simple notion that has become a lightning rod for its disciples and the proponents of behavioral economics and finance. In its purest form, the EMH obviates active portfolio management, calling into question the very motivation for portfolio research. It is only fitting that we revisit this groundbreaking idea after three very successful decades of this Journal. In this article, I review the current state of the controversy surrounding the EMH and propose a new perspective that reconciles the two opposing schools of thought. The proposed reconciliation, which I call the Adaptive Markets Hypothesis (AMH), is based on an evolutionary approach to economic interactions, as well as some recent research in the cognitive neurosciences that has been transforming and revitalizing the intersection of psychology and economics. Although some of these ideas have not yet been fully articulated within a rigorous quantitative framework, long time students of the EMH and seasoned practitioners will no doubt recognize immediately the possibilities generated by this new perspective. Only time will tell whether its potential will be fulfilled. I begin with a brief review of the classic version of the EMH, and then summarize the most significant criticisms leveled against it by psychologists and behavioral economists. I argue that the sources of this controversy can

A predictor is asked to rank eventualities according to their plausibility, based on past cases. We assume that she can form a ranking given any memory that consists of finitely many past cases. Mild consistency requirements on these rankings imply that they have a numerical representation via a matrix assigning numbers to eventuality-case pairs, as follows. Given a memory, each eventuality is ranked according to the sum of the numbers in its row, over cases in memory. The number attached to an eventuality-case pair can be interpreted as the degree of support that the past case lends to the plausibility of the eventuality. Special instances of this result may be viewed as axiomatizing kernel methods for estimation of densities and for classification problems. Interpreting the same result for rankings of theories or hypotheses, rather than of specific eventualities, it is shown that one may ascribe to the predictor subjective conditional probabilities of cases given theories, such that her rankings of theories agree with rankings by the likelihood functions.

De Finetti's treatise on the theory of probability begins with the provocative statement PROBABILITY DOES NOT EXIST, meaning that probability does not exist in an objective sense. Rather, probability exists only subjectively within the minds of individuals. De Finetti defined subjective probabilities in terms of the rates at which individuals are willing to bet money on events, even though, in principle, such betting rates could depend on statedependent marginal utility for money as well as on beliefs. Most later authors, from Savage onward, have attempted to disentangle beliefs from values by introducing hypothetical bets whose payoffs are abstract consequences that are assumed to have state-independent utility. In this paper, I argue that de Finetti was right all along: PROBABILITY, considered as a numerical measure of pure belief uncontaminated by attitudes toward money, does not exist. Rather, what exist are de Finetti's "previsions," or betting rates for money, otherwise known in the literature as "risk neutral probabilities." But the fact that previsions are not measures of pure belief turns out not to be problematic for statistical inference, decision analysis, or economic modeling.

This paper develops an axiomatic theory of decision making under uncertainty that dispenses with the state space. The results are subjective expected utility models with unique, action-dependent, subjective probabilities, and a utility function defined over wealth-effect pairs that is unique up to positive linear transformation.