page_title: Info-gap decision theory
section_title: Bayesian analysis
hierarchical_section_title: Info-gap decision theory / Alternatives / Comparative analysis / Bayesian analysis
caption_attribution_description: Probability distribution of the revenue function realizations
context_page_description: Info-gap decision theory is a non-probabilistic decision theory that seeks to optimize robustness to failure – or opportuneness for windfall – under severe uncertainty, in particular applying sensitivity analysis of the stability radius type to perturbations in the value of a given estimate of the parameter of interest. It has some connections with Wald's maximin model; some authors distinguish them, others consider them instances of the same principle.
  It has been developed since the 1980s by Yakov Ben-Haim, and has found many applications and described as a theory for decision-making under "severe uncertainty". It has been criticized as unsuited for this purpose, and alternatives proposed, including such classical approaches as robust optimization.
context_section_description: As for Laplaces' principle of insufficient reason, in this context it is convenient to view it as an instance of Bayesian analysis.
  The essence of the Bayesian analysis is applying probabilities for different possible realizations of the uncertain parameters. In the case of Knightian (non-probabilistic) uncertainty, these probabilities represent the decision maker's "degree of belief" in a specific realization.
  In our example, suppose there are only five possible realizations of the uncertain revenue to allocation function. The decision maker believes that the estimated function is the most likely, and that the likelihood decreases as the difference from the estimate increases. Figure 11 exemplifies such a probability distribution.
  Now, for any allocation, one can construct a probability distribution of the revenue, based on his prior beliefs. The decision maker can then choose the allocation with the highest expected revenue, with the lowest probability for an unacceptable revenue, etc.
  The most problematic step of this analysis is the choice of the realizations probabilities. When there is an extensive and relevant past experience, an expert may use this experience to construct a probability distribution. But even with extensive past experience, when some parameters change, the expert may only be able to estimate that  is more likely than , but will not be able to reliably quantify this difference. Furthermore, when conditions change drastically, or when there is no past experience at all, it may prove to be difficult even estimating whether  is more likely than .
  Nevertheless, methodologically speaking, this difficulty is not as problematic as  basing the analysis of a problem subject to severe uncertainty on a single point estimate and its immediate neighborhood, as done by info-gap. And what is more, contrary to info-gap, this approach is global, rather than local.
  Still, it must be stressed that Bayesian analysis does not expressly concern itself with the question of robustness.
  Bayesian analysis raises the issue of learning from experience and adjusting probabilities accordingly. In other words, decision is not a one-stop process, but profits from a sequence of decisions and observations.
caption: Figure 11 – Probability distribution of the revenue function realizations
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