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.
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