pure birth process

The Riddler has a rather simplistic riddle this week since it essentially asked for the expectation of a pure birth process (also known as the Yule process) at time t. Since the population size at time t has a geometric distribution with expectation

e^λt.

It however took me a while to recover this result on my own on Easter afternoon, as I went for the integrals rather than the distribution itself and the associated differential equations. Interestingly (in a local sense!), I first following the wrong path of looking at the average time to the first birth, 1/λ, then to the second, 2/λ, and so on. Wrong since of course expectations do not carry this way… For a unit rate,  λ=1, the average time to reach 10 births is about 3, while the average number of births over t=3 is essentially 20.

double yolk priors [a reply from the authors]

[Here is an email I received from Subhadeep Mukhopadhyay, one of the authors of the paper I discussed yesterday.}

Thank for discussing our work. Let me clarify the technical point that you raised:

– The difference between Leg_j(u)_j and T_j=Leg_j(G(θ)). One is orthonormal polyn of L₂[0,1] and the other one is L₂[G]. The second one is poly of rank-transform G(θ).

– As you correctly pointed out there is a danger in directly approximating the ratio. We work on it after taking the quantile transform: evaluate the ratio at g⁻¹(θ), which is the d(u;G,F) over unit interval. Now, this new transformed function is a proper density.

-Thus the ratio now becomes d(G(θ)) which can be expended into (NOT in Leg-basis) in , in eq (2.2), as it lives in the Hilbert space L₂(G)

– For your last point on Step 2 of our algo, we can also use the simple integrate command.

-Unlike traditional prior-data conflict here we attempted to answer three questions in one-shot: (i) How compatible is the pre-selected g with the given data? (ii) In the event of a conflict, can we also inform the user on the nature of misfit–finer structure that was a priori unanticipated? (iii) Finally, we would like to provide a simple, yet formal guideline for upgrading (repairing) the starting g.

Hopefully, this will clear the air. But thanks for reading the paper so carefully. Appreciate it.

double yolk priors

“To develop a “defendable and defensible” Bayesian learning model, we have to go beyond blindly ‘turning the crank’ based on a “go-as-you-like” [approximate guess] prior. A lackluster attitude towards prior modeling could lead to disastrous inference, impacting various fields from clinical drug development to presidential election forecasts. The real questions are: How can we uncover the blind spots of the conventional wisdom-based prior? How can we develop the science of prior model-building that combines both data and science [DS-prior] in a testable manner – a double-yolk Bayesian egg?”

I came through R bloggers on this presentation of a paper by Subhadeep Mukhopadhyay and Douglas Fletcher, Bayesian modelling via goodness of fit, that aims at solving all existing problems with classical Bayesian solutions, apparently! (With also apparently no awareness of David Spiegelhalter’s take on the matter.) As illustrated by both quotes, above and below:

“The two key issues of modern Bayesian statistics are: (i) establishing principled approach for distilling statistical prior that is consistent with the given data from an initial believable scientific prior; and (ii) development of a Bayes-frequentist consolidated data analysis work ow that is more effective than either of the two separately.”

(I wonder who else in this Universe would characterise “modern Bayesian statistics” in such a non-Bayesian way! And love the notion of distillation applied to priors!) The setup is actually one of empirical Bayes inference where repeated values of the parameter θ drawn from the prior are behind independent observations. Which is not the usual framework for a statistical analysis, where a single value of the parameter is supposed to hide behind the data, but most convenient for frequency based arguments behind empirical Bayes methods (which is the case here). The paper adopts a far-from-modern discourse on the “truth” of “the” prior… (Which is always conjugate in that Universe!) Instead of recognising the relativity of a statistical analysis based on a given prior.

When I tried to read the paper any further, I hit a wall as I could not understand the principle described therein. And how it “consolidates Bayes and frequentist, parametric and nonparametric, subjective and objective, quantile and information-theoretic philosophies.”. Presumably the lack of oxygen at the altitude of Chamonix…. Given an “initial guess” at the prior, g, a conjugate prior (in dimension one with an invertible cdf), a family of priors is created in what first looks like a form of non-parametric exponential tilting of g. But a closer look [at (2.1)] exposes the “family” as the tautological π(θ)=g(θ)x π(θ)/g(θ). The ratio is expanded into a Legendre polynomial series. Which use in Bayesian statistics dates a wee bit further back than indicated in the paper (see, e.g., Friedman, 1985; Diaconis, 1986). With the side issue that the resulting approximation does not integrate to one. Another side issue is that the coefficients of the Legendre truncated series are approximated by simulations from the prior [Step 3 of the Type II algorithm], rarely an efficient approach to the posterior.

double-yolkers

Last night I was cooking buckwheat pancakes (galettes de sarrasin) from Brittany with an egg-and-ham filling. The first egg I used contained a double yolk, a fairly rare occurrence, at least in my kitchen! Then came the second pancake and, unbelievably!, a second egg with a double yolk! This sounded too unbelievable to be…unbelievable! The experiment stopped there as no one else wanted another galette, but tonight, when making chocolate mousse, I checked whether or not the four remaining eggs also were double-yolkers…and indeed they were. Which does not help when separating yolks from white, by the way. Esp. with IX fingers. At some stage, during the day, I remembered a talk by Prof of Risk David Spiegelhalter mentioning the issue, even including a picture of an egg-box with the double-yolker guarantee, as in the attached picture. But all I could find first was this explanation on BBC News. Which made sense for my eggs, as those were from a large calibre egg-box (which I usually do not buy)… (And then I typed David Spiegelhalter plus ‘double-yolker” on Google and all those references came out!)