One comment. To conserve the probability as measure you must redefine the definition of confidence interval (see, for example, http://xxx.lanl.gov/abs/1209.6545) as new prior is used.

]]>In this reference in appendix you will find paradox. Any prior except uniform breaks the conservation of probability.

]]>I still do not get it: the probability measure has been modified when using another prior, but it still remains a probability measure.

]]>I keep in mind the construction of credible intervals.

For example, let t be parameter, X – observed data and

we try to find the 90% credible interval.

Case A: pi(t) – flat prior (let be 1).

0.9 = int_a^b { P(X|t) dt }.

Case B: pi(t) has dependence from t, i.e.

0.9 = int_c^d { P(X|t) pi(t) dt }.

Factually, in the case B for each value of P(X|t) is given

weight pi(t), i.e. we change scale of the axis t and, correspondingly,

we incorporate this changing to upper and lower bounds of integration.

a and b go to c and d in accordance with function pi(t).

Of course, this numerical method is correct for resolving the most of

task, but we loose the probabilistic sense of this inference.

How to use the posterior distribution in frame of probabilistic paradigma ?

Sergey Bityukov

]]>I am afraid I do not get your question…

]]>One question more in defense of notation with confidence distribution.

The using of any prior in Bayesian approach (except uniform, that is not

a prior) is only the deformation of abscissa (in case of one dimension),

i.e., factually, reweighting of the probability. What sense of the

probability with weight ?

Thank you for quick answer and for your comments.

In (5) we suppose that mu_b is known (const). In principle, it is possible to use the distribution of mu_b (for example by Monte Carlo), but we not considered this case.

]]>Thank you for pointing out this reference. I just read through it and I am still unconvinced of the appeal of the approach, I am afraid! Indeed, there is no result in your paper that proceeds from the confidence distribution perspective: a confidence interval on the only parameter in the model, μ_{s}, could be derived another way, as it has been in your references [31,32,33].

In addition, I find that the construction of those densities on the parameters are fraught with measure-theoretic danger: in particular, your derivation (5) and the subsequent density on μ_{s} do not seem right. Indeed, in (5) the weights p_{0} and p_{1} depend on the parameters, which means that the weighted sum of the densities is not necessarily integrable to 1… (it does in the case *ŝ=1*!), but also that the subsequent density on μ_{s} depends on μ_{b}.

We try to construct several examples of the confidence distributions with conserving

of probability. It points to possibility for Monte Carlo constructions of confidence

distributions. Confidence distributions, in our opinion, is very useful notion for

combining results. I give the link to our paper in AIP Proceedings (MaxEnt’2010)

http://proceedings.aip.org/resource/2/apcpcs/1305/1/346_1

Sincerely yours,

Dr. Sergey Bityukov

Institute for high energy physics, Protvino,

Moscow region, Russia