A nonparametric predictive alternative to the Imprecise Dirichlet Model: the case of a known number of categories

Nonparametric Predictive Inference (NPI) is a general methodology to learn from data in the absence of prior knowledge and without adding unjustified assumptions. This paper develops NPI for multinomial data where the total number of possible categories for the data is known. We present the general upper and lower probabilities and several of their properties. We also comment on differences between this NPI approach and corresponding inferences based on Walley's Imprecise Dirichlet Model

Theory of agent-based market models with controlled levels of greed and anxiety

We use generating functional analysis to study minority-game type market models with generalized strategy valuation updates that control the psychology of agents' actions. The agents' choice between trend following and contrarian trading, and their vigor in each, depends on the overall state of the market. Even in `fake history' models, the theory now involves an effective overall bid process (coupled to the effective agent process) which can exhibit profound remanence effects and new phase transitions. For some models the bid process can be solved directly, others require Maxwell-construction type approximations.Comment: 30 pages, 10 figure

Market response to external events and interventions in spherical minority games

We solve the dynamics of large spherical Minority Games (MG) in the presence of non-negligible time dependent external contributions to the overall market bid. The latter represent the actions of market regulators, or other major natural or political events that impact on the market. In contrast to non-spherical MGs, the spherical formulation allows one to derive closed dynamical order parameter equations in explicit form and work out the market's response to such events fully analytically. We focus on a comparison between the response to stationary versus oscillating market interventions, and reveal profound and partially unexpected differences in terms of transition lines and the volatility.Comment: 14 pages LaTeX, 5 (composite) postscript figures, submitted to Journal of Physics

Multiplpe Choice Minority Game With Different Publicly Known Histories

In the standard Minority Game, players use historical minority choices as the sole public information to pick one out of the two alternatives. However, publishing historical minority choices is not the only way to present global system information to players when more than two alternatives are available. Thus, it is instructive to study the dynamics and cooperative behaviors of this extended game as a function of the global information provided. We numerically find that although the system dynamics depends on the kind of public information given to the players, the degree of cooperation follows the same trend as that of the standard Minority Game. We also explain most of our findings by the crowd-anticrowd theory.Comment: Extensively revised, to appear in New J Phys, 7 pages with 4 figure

Nonparametric predictive inference and interval probability

This paper presents the unique position of A(n)-based nonparametric predictive inference within the theory of interval probability. It provides a completely new understanding, leading to powerful new results and a well-founded justification of such inferences by proving strong internal consistency results

DYNAMICAL SOLUTION OF A MODEL WITHOUT ENERGY BARRIERS

In this note we study the dynamics of a model recently introduced by one of us, that displays glassy phenomena in absence of energy barriers. Using an adiabatic hypothesis we derive an equation for the evolution of the energy as a function of time that describes extremely well the glassy behaviour observed in Monte Carlo simulations.Comment: 11 pages, LaTeX, 3 uuencoded figure

Statistical Mechanics of Dilute Batch Minority Games with Random External Information

We study the dynamics and statics of a dilute batch minority game with random external information. We focus on the case in which the number of connections per agent is infinite in the thermodynamic limit. The dynamical scenario of ergodicity breaking in this model is different from the phase transition in the standard minority game and is characterised by the onset of long-term memory at finite integrated response. We demonstrate that finite memory appears at the AT-line obtained from the corresponding replica calculation, and compare the behaviour of the dilute model with the minority game with market impact correction, which is known to exhibit similar features.Comment: 22 pages, 6 figures, text modified, references updated and added, figure added, typos correcte

Dynamical Solution of the On-Line Minority Game

We solve the dynamics of the on-line minority game, with general types of decision noise, using generating functional techniques a la De Dominicis and the temporal regularization procedure of Bedeaux et al. The result is a macroscopic dynamical theory in the form of closed equations for correlation- and response functions defined via an effective continuous-time single-trader process, which are exact in both the ergodic and in the non-ergodic regime of the minority game. Our solution also explains why, although one cannot formally truncate the Kramers-Moyal expansion of the process after the Fokker-Planck term, upon doing so one still finds the correct solution, that the previously proposed diffusion matrices for the Fokker-Planck term are incomplete, and how previously proposed approximations of the market volatility can be traced back to ergodicity assumptions.Comment: 25 pages LaTeX, no figure

On Bernoulli Experiments with Imprecise Prior Probabilities

This paper deals with a situation where the prior distribution in a Bayesian treatment of a Bernoulli experiment is not known precisely. Imprecision for Bernoulli experiments is discussed for the case where the prior densities are defined in the form of intervals of measures, and a simple model with conjugate imprecise prior densities is used for ease of calculation in updating. Attention is focused on imprecision with regard to predictive probabilities. The main aim of the paper is an analysis of imprecision, emphasizing the important relationship between information and imprecision. On the basis of this relationship, a rule for updating the set of prior distributions is proposed that is different from the theory advocated by Walley and suggests a new area for research. Imprecision within the chosen model is compared with intuition, and, although a complete lack of prior information cannot be represented perfectly, it can be approximated well