The problem considered in Denœux’s

The problem considered in Denœux's paper [1] is very important because, in many practical situations, we have a parameterspace Θ and the only available information is provided by an observation X ∈ X, which defines a likelihood on Θ,through a conditional probability distribution p(x|θ ). If there is a prior distribution on Θ, then the problem can be solvedusing Bayesian statistics, but in many situations this prior distribution is not available. In fact, in the paper [2] of this issue,we consider a similar problem, which is studied within the theory of imprecise probabilities. In this paper a sound treatmentis considered using the theory of belief functions. I agree in that profile likelihood (i.e., the consonant belief functionwith a contour function equal to the relative likelihood) is a distinguished solution to the problem, but I have some doubtsabout the unicity of the solution. The argument provided in the paper is that this plausibility is the least informative onein BX , where this set is given by all the plausibility functions that are compatible with Bayes' rule. This compatibility issatisfied when the combination of the plausibility with any prior probability in X using Dempster's rule is equal to theposterior of the prior given the likelihood. However, in Section 2.4 it is discussed the incompatibility with Dempster's rulewhen two conditionally independent pieces of information are available, each one of them providing a likelihood. It is saidthat to transform each one of them into a plausibility function and then combine the results using Dempster's rule is notthe same as to consider them as a joint information defining the product likelihood, and compute the plausibility associatedwith this product likelihood. The only possible explanation for this incompatibility is that it might be that different kindsof evidence require different combination mechanisms. If this argument is accepted in this case, why not to accept it forprior and likelihood information? In fact these are very specific types of information which could have specific proceduresto be combined. Then, if compatibility with Bayesian inference is transformed into a weaker requirement in which the

The Problem considered in Denœux's Paper [1] is very important because, in many practical Situations, we have a parameter
Space theta and the only available information is provided by an Observation X ∈ X, which defines a likelihood on theta,
Through a conditional probability. distribution p (x | θ). If there is a prior Distribution on theta, then the Problem Can be Solved
using Bayesian Statistics, but many in this prior Situations Distribution is not available. In Fact, in the Paper [2] of this Issue,
we consider a similar Problem, which is studied Within the Theory of imprecise probabilities. In this Paper a Sound Treatment
is considered using the Theory of belief functions. I Agree in that likelihood Profile (IE, the consonant belief function
with a Contour function Equal to the Relative likelihood) is a distinguished Solution to the Problem, but I have doubts Some
of the Unicity About the Solution. The argument that this is provided in the Paper Plausibility is the Least informative one
in BX, where this is SET Plausibility Given by all the functions that are Compatible with Bayes' Rule. Compatibility this is
satisfied when the combination of the Plausibility with any prior probability in X using Dempster's Rule is Equal to the
posterior of the prior Given the likelihood. However, it is discussed in Section 2.4 the incompatibility with Dempster's Rule
when Two pieces of information are available conditionally Independent, each one of them providing a likelihood. It is said
that each one of them to transform Into a function and then Plausibility Combine the results using Dempster's Rule is not
the Same as to consider them as a Joint Defining the product information likelihood, and Compute the Plausibility associated
with this product likelihood. The only possible explanation for this incompatibility is that it Might be that different kinds
of different combination Require Evidence Mechanisms. If this argument is accepted in this Case, why not to accept it for
prior and likelihood information? In Fact these are very specific types of information which could have specific procedures
to be combined. Then, if compatibility with Bayesian inference is transformed into a weaker requirement in which the.

The problem considered in Den œ UX 's paper [] is 1 very, important because in many practical situations we have, a parameter
space. Θ and the only available information is provided by an observation X ∈ X which defines, a likelihood, on Θ
through a conditional. Probability distribution P (x | θ). If there is a prior distribution on Θ then the, problem can be solved
using, Bayesian statisticsBut in many situations this prior distribution is not available. In fact in the, paper [] of 2 this issue
we, consider a. Similar problem which is, studied within the theory of imprecise probabilities. In this paper a sound treatment
is considered. Using the theory of belief functions. I agree in that profile likelihood (i.e, the consonant belief function
.With a contour function equal to the relative likelihood) is a distinguished solution to, the problem but I have some doubts
about. The unicity of the solution. The argument provided in the paper is that this plausibility is the least informative one
in. BX where this, set is given by all the plausibility functions that are compatible with Bayes rule. This compatibility. ' Is
.Satisfied when the combination of the plausibility with any prior probability in X using Dempster 's rule is equal to the
posterior. Of the prior given the likelihood. However in Section, 2.4 it is discussed the incompatibility with Dempster s rule
when. ' Two conditionally independent pieces of information, are available each one of them providing a likelihood. It is said
.That to transform each one of them into a plausibility function and then combine the results using Dempster 's rule is not
the. Same as to consider them as a joint information defining the product likelihood and compute, the plausibility associated
with. This product likelihood. The only possible explanation for this incompatibility is that it might be that different kinds
.Of evidence require different combination mechanisms. If this argument is accepted in this case why not, to accept it for
prior. And likelihood information? In fact these are very specific types of information which could have specific procedures
to. Be combined. Then if compatibility, with Bayesian inference is transformed into a weaker requirement in which the


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