My comment(arie)s on the moment approach to Bayesian inference by Ron Gallant have appeared, along with other comment(arie)s:
Invited Article
Reflections on the Probability Space Induced by Moment Conditions with
Implications for Bayesian Inference
A. Ronald Gallant . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Commentaries
Dante Amengual and Enrique Sentana .. . . . . . . . . . 248
John Geweke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253
Jae-Young Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Oliver Linton and Ruochen Wu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261
Christian P. Robert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Christopher A. Sims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
Wei Wei and Asger Lunde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .278
Author Response
A. Ronald Gallant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .284
While commenting on commentaries is formally bound to induce an infinite loop [or l∞p], I remain puzzled by the main point of the paper, which is that setting a structural distribution on a moment function Z(x,θ) plus a prior p(θ) induces a distribution on the pair (x,θ) in a possibly weaker σ-algebra. (The two distributions may actually be incompatible.) Handling this framework requires checking that a posterior exists, which sounds rather unnatural (even though we also have to check properness of the posterior). And the meaning of such a posterior remains unclear, as for instance in this assertion that (4) above is a likelihood, when it does not define a density in x but on the object inside the exponential.
“…it is typically difficult to determine whether there exists a p(x|θ) such that the implied distribution of m(x,θ) is the one stated, and if not, what damage is done thereby” J. Geweke (p.254)
