A behavioural justification for using sensitivity analysis in imprecise multinomial processes

Abstract

We consider two fundamentally different ways of defining an imprecise-probabilistic multinomial process. On the one hand, we have the so-called sensitivity analysis approach. The corresponding imprecise-probabilistic joint model is defined as the lower envelope of joint models of precise multinomial processes, each of which has a different marginal model taken from some closed and convex set of candidate marginal models. For finite subsets of variables in this process, this corresponds to using Walley's so-called type-2 product of identically distributed variables. On the other hand, we consider a behavioural approach, defining an imprecise multinomial process by imposing exchangeability and either forward irrelevance or epistemic independence. Our main result is that both approaches lead to the exact same imprecise multinomial process. This fairly technical result has an important philosophical consequence as well: it provides the sensitivity analysis approach and the related type-2 product with a behavioural justification. We compare our justification favourably with a previous attempt by Cozman and explain how it fits into the more general problem of providing a behavioural justification for so-called strongly independent models, or equivalently, for using lower envelopes of stochastically independent models.