Facoltà di scienze economiche

## Learning under prior ignorance

### Thèse de doctorat : Università della Svizzera italiana, 2006 ; 2006ECO004.

It is well known that a state of prior ignorance is not compatible with learning, at least in a coherent theory of (epistemic) uncertainty. What is less widely known, is that there is another state of beliefs, called near-ignorance, that resembles ignorance very closely by satisfying some principles that can arguably be regarded as necessary in a state of ignorance, and that allows learning to... Plus

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Summary
It is well known that a state of prior ignorance is not compatible with learning, at least in a coherent theory of (epistemic) uncertainty. What is less widely known, is that there is another state of beliefs, called near-ignorance, that resembles ignorance very closely by satisfying some principles that can arguably be regarded as necessary in a state of ignorance, and that allows learning to take place. What this thesis does is to provide new and substantial evidence that also near-ignorance cannot be really regarded as a way out of the problem of starting statistical inference in conditions of very weak beliefs. The key to this result is focusing on a setting characterized by a variable of interest that is latent. We argue that such a setting is by far the most common case in practice. In the first part of the thesis we provide, for the case of categorical latent variables (and general manifest variables) a condition that, if satisfied, prevents learning to take place under prior near-ignorance. This condition is shown to be easily satisfied even in the most common statistical problems. We regard these results as a strong form of evidence against the possibility to adopt a condition of prior near-ignorance in real statistical problems. In the second part of the thesis we propose a slightly modified framework that allows learning to take place under a very weak specification of prior knowledge. The proposed approach is a first preliminary attempt to reconcile latent variables with a very weak specification of prior knowledge.