### Alessio Benavoli, Marco Zaffalon

## A discussion on learning and prior ignorance for sets of priors in the one-parameter exponential family

### Abstract

For a conjugate likelihood-priors model in the one-parameter exponential family of distributions, we show that, by letting the parameters of the conjugate exponential prior vary in suitable sets, it is possible to define a set of conjugate priors M which guarantees prior near-ignorance without producing vacuous inferences.
This result is obtained following both a behavioural and a sensitivity analysis interpretation of prior near-ignorance.
We also discuss the problem of the incompatibility of learning and prior near-ignorance for sets of priors in the one-parameter exponential family of distributions in the case of imperfect observations.
In particular, we prove that learning and prior near-ignorance are compatible under an imperfect observation mechanism if and only if the support of the priors in M is the whole real axis.

### Keywords

Prior near-ignorance, set of distributions, exponential family of distributions, imperfect observations

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### Authors’ addresses

**Alessio Benavoli**

Galleria 2

6928 Manno

**Marco Zaffalon**

Galleria 2

CH-6928 Manno

Switzerland

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