Practical robust estimators for the imprecise Dirichlet model

Marcus Hutter
2009 International Journal of Approximate Reasoning  
Walley's imprecise Dirichlet model (IDM) for categorical i.i.d. data extends the classical Dirichlet model to a set of priors. It overcomes several fundamental problems which other approaches to uncertainty suffer from. Yet, to be useful in practice, one needs efficient ways for computing the imprecise = robust sets or intervals. The main objective of this work is to derive exact, conservative, and approximate, robust and credible interval estimates under the IDM for a large class of
more » ... estimators, including the entropy and mutual information. URL: http://www.hutter1.net 1 Also called objective or aleatory probabilities. 2 We denote vectors by x :¼ ðx 1 ; . . . ; x d Þ for x 2 fn; t; u; p; . . .g, and i ranges from 1 to d unless otherwise stated. See also Appendix B. 3 Also called second order or subjective or belief or epistemic probabilities. 4 Strictly speaking, D should be the open simplex [10], since pðpÞ is improper for t on the boundary of D. For simplicity we assume that, if necessary, considered functions of t can and are continuously extended to the boundary of D, so that, for instance, minima and maxima exist. All considerations can straightforwardly, but cumbersomely, be rewritten in terms of an open simplex. Note that open/closed D result in open/closed robust intervals, the difference being numerically/practically irrelevant. 5 But see [7] for a proper Bayesian reconciliation of these principles.
doi:10.1016/j.ijar.2008.03.020 fatcat:nixglzqumzd4ffpmpu4op7p76u