a new paradigm for improper priors

Gunnar Taraldsen and co-authors have arXived a short note on using improper priors from a new perspective. Generalising an earlier 2016 paper in JSPI on the same topic. Which both relate to a concept introduced by Rényi (who himself attributes the idea to Kolmogorov). Namely that random variables measures are to be associated with arbitrary measures [not necessarily σ-finite measures, the later defining σ-finite random variables], rather than those with total mass one. Which allows for an alternate notion of conditional probability in the case of σ-finite random variables, with the perk that this conditional probability distribution is itself of mass 1 (a.e.).  Which we know happens when moving from prior to proper posterior.

I remain puzzled by the 2016 paper though as I do not follow the meaning of a random variable associated with an infinite mass probability measure. If the point is limited to construct posterior probability distributions associated with improper priors, there is little value in doing so. The argument in the 2016 paper is however that one can then define a conditional distribution in marginalisation paradoxes à la Stone, Dawid and Zidek (1973) where the marginal does not exist. Solving with this formalism the said marginalisation paradoxes as conditional distributions are only defined for σ-finite random variables. Which gives a fairly different conclusion from either Stone, Dawid and Zidek (1973) [with whom I agree, namely that there is no paradox because there is no “joint” distribution] or Jaynes (1973) [with whom I less agree!, in that the use of an invariant measure to make the discrepancy go away is not a particularly strong argument in favour of this measure]. The 2016 paper also draws an interesting connection with the study by Jim Hobert and George Casella (in Jim’s thesis) of [null recurrent or transient] Gibbs samplers with no joint [proper] distribution. Which in some situations can produce proper subchains, a phenomenon later exhibited by Alan Gelfand and Sujit Sahu (and Xiao-Li Meng as well if I correctly remember!). But I see no advantage in following this formalism, as it does not impact whether the chain is transient or null recurrent, or anything connected with its implementation. Plus a link to the approximation of improper priors by sequences of proper ones by Bioche and Druihlet I discussed a while ago.

fiducial inference

In connection with my recent tale of the many ε’s, I received from Gunnar Taraldsen [from Tronheim, Norge] a paper [jointly written with Bo Lindqvist and just appeared on-line in JSPI] on conditional fiducial models.

“The role of the prior and the statistical model in Bayesian analysis is replaced by the use of the fiducial model x=R(θ,ε) in fiducial inference. The fiducial is obtained in this case without a prior distribution for the parameter.”

Reading this paper after addressing the X validated question made me understood better the fundamental wrongness of fiducial analysis! If I may herein object to Fisher himself… Indeed, when writing x=R(θ,ε), as the representation of the [observed] random variable x as a deterministic transform of a parameter θ and of an [unobserved] random factor ε, the two random variables x and ε are based on the same random preimage ω, i.e., x=x(ω) and ε=ε(ω). Observing x hence sets a massive constraint on the preimage ω and on the conditional distribution of ε=ε(ω). When the fiducial inference incorporates another level of randomness via an independent random variable ε’ and inverts x=R(θ,ε’) into θ=θ(x,ε’), assuming there is only one solution to the inversion, it modifies the nature of the underlying σ-algebra into something that is incompatible with the original model. Because of this sudden duplication of the random variates. While the inversion of this equation x=R(θ,ε’) gives an idea of the possible values of θ when ε varies according to its [prior] distribution, it does not account for the connection between x and ε. And does not turn the original parameter into a random variable with an implicit prior distribution.

As to conditional fiducial distributions, they are defined by inversion of x=R(θ,ε), under a certain constraint on θ, like C(θ)=0, which immediately raises a Pavlovian reaction in me, namely that since the curve C(θ)=0 has measure zero under the original fiducial distribution, how can this conditional solution be uniquely or at all defined. Or to avoid the Borel paradox mentioned in the paper. If I get the meaning of the authors in this section, the resulting fiducial distribution will actually depend on the choice of σ-algebra governing the projection.

“A further advantage of the fiducial approach in the case of a simple fiducial model is that independent samples are produced directly from independent sampling from [the fiducial distribution]. Bayesian simulations most often come as dependent samples from a Markov chain.”

This side argument in “favour” of the fiducial approach is most curious as it brings into the picture computational aspects that do not have any reason to be there. (The core of the paper is concerned with the unicity of the fiducial distribution in some univariate settings. Not with computational issues.)

all those ε’s…

A revealing [and interesting] question on X validated about ε’s… The question was about the apparent contradiction in writing Normal random variates as the sum of their mean and of a random noise ε in the context of the bivariate Normal variate (x,y), since using the marginal x conditional decomposition led to two different sets of ε’s. Which did not seem to agree. I replied about these ε’s having to live in different σ-algebras, but this reminded me of some paradoxes found in fiducial analysis through this incautious manipulation of ε’s…

what makes variables randoms [book review]

When the goal of a book is to make measure theoretic probability available to applied researchers for conducting their research, I cannot but applaud! Peter Veazie’s goal of writing “a brief text that provides a basic conceptual introduction to measure theory” (p.4) is hence most commendable. Before reading What makes variables random, I was uncertain how this could be achieved with a limited calculus background, given the difficulties met by our third year maths students. After reading the book, I am even less certain this is feasible!

“…it is the data generating process that makes the variables random and not the data.”

Chapter 2 is about basic notions of set theory. Chapter 3 defines measurable sets and measurable functions and integrals against a given measure μ as

which I find particularly unnatural compared with the definition through simple functions (esp. because it does not tell how to handle 0x∞). The ensuing discussion shows the limitation of the exercise in that the definition is only explained for finite sets (since the notion of a partition achieving the supremum on page 29 is otherwise meaningless). A generic problem with the book, in that most examples in the probability section relate to discrete settings (see the discussion of the power set p.66). I also did not see a justification as to why measurable functions enjoy well-defined integrals in the above sense. All in all, to see less than ten pages allocated to measure theory per se is rather staggering! For instance,

does not appear to be defined at all.

“…the mathematical probability theory underlying our analyses is just mathematics…”

Chapter 4 moves to probability measures. It distinguishes between objective (or frequentist) and subjective measures, which is of course open to diverse interpretations. And the definition of a conditional measure is the traditional one, conditional on a set rather than on a σ-algebra. Surprisingly as this is in my opinion one major reason for using measures in probability theory. And avoids unpleasant issues such as Bertrand’s paradox. While random variables are defined in the standard sense of real valued measurable functions, I did not see a definition of a continuous random variables or of the Lebesgue measure. And there are only a few lines (p.48) about the notion of expectation, which is so central to measure-theoretic probability as to provide a way of entry into measure theory! Progressing further, the σ-algebra induced by a random variable is defined as a partition (p.52), a particularly obscure notion for continuous rv’s. When the conditional density of one random variable given the realisation of another is finally introduced (p.63), as an expectation reconciling with the set-wise definition of conditional probabilities, it is in a fairly convoluted way that I fear will scare newcomers out of their wit. Since it relies on a sequence of nested sets with positive measure, implying an underlying topology and the like, which somewhat shows the impossibility of the overall task…

“In the Bayesian analysis, the likelihood provides meaning to the posterior.”

Statistics is hurriedly introduced in a short section at the end of Chapter 4, assuming the notion of likelihood is already known by the readers. But nitpicking (p.65) at the representation of the terms in the log-likelihood as depending on an unspecified parameter value θ [not to be confused with the data-generating value of θ, which does not appear clearly in this section]. Section that manages to include arcane remarks distinguishing maximum likelihood estimation from Bayesian analysis, all this within a page! (Nowhere is the Bayesian perspective clearly defined.)

“We should no more perform an analysis clustered by state than we would cluster by age, income, or other random variable.”

The last part of the book is about probabilistic models, drawing a distinction between data generating process models and data models (p.89), by which the author means the hypothesised probabilistic model versus the empirical or bootstrap distribution. An interesting way to relate to the main thread, except that the convergence of the data distribution to the data generating process model cannot be established at this level. And hence that the very nature of bootstrap may be lost on the reader. A second and final chapter covers some common or vexing problems and the author’s approach to them. Revolving around standard errors, fixed and random effects. The distinction between standard deviation (“a mathematical property of a probability distribution”) and standard error (“representation of variation due to a data generating process”) that is followed for several pages seems to boil down to a possible (and likely) model mis-specification. The chapter also contains an extensive discussion of notations, like indexes (or indicators), which seems a strange focus esp. at this location in the book. Over 15 pages! (Furthermore, I find quite confusing that a set of indices is denoted there by the double barred I, usually employed for the indicator function.)

“…the reader will probably observe the conspicuous absence of a time-honoured topic in calculus courses, the “Riemann integral”… Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance.” Jean Dieudonné, Foundations of Modern Analysis

In conclusion, I do not see the point of this book, from its insistence on measure theory that never concretises for lack of mathematical material to an absence of convincing examples as to why this is useful for the applied researcher, to the intended audience which is expected to already quite a lot about probability and statistics, to a final meandering around linear models that seems at odds with the remainder of What makes variables random, without providing an answer to this question. Or to the more relevant one of why Lebesgue integration is preferable to Riemann integration. (Not that there does not exist convincing replies to this question!)

commentaries in financial econometrics

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)