simulating from the joint cdf

An X validated question (what else?!) brought back (to me) the question of handling a bivariate cdf for simulation purposes. In the specific case of a copula when thus marginals were (well-)known…. And led me to an erroneous chain of thought, fortunately rescued by Robin Ryder! When the marginal distributions are set, the simulation setup is indeed equivalent to a joint Uniform simulation from a copula

In specific cases, as for instance the obvious example of Gaussian copulas, there exist customised simulation algorithms. Looking for more generic solutions, I turn to the Bible, where Chapter XI[an], has two entire sections XI.3.2. and XI.3.3 on the topic (even though Luc Devroye does not use the term copula there despite them being introduced in 1959 by A, Sklar, in response to a query of M. Fréchet). In addition to a study of copulas, both sections contain many specific solutions (as for instance in the [unnumbered] Table on page 585) but I found no generic simulation method. My [non-selected] answer to the question was thus to propose standard solutions such as finding one conditional since the marginals are Uniform. Which depends on the tractability of the derivatives of C(·,·).

However, being dissatisfied with this bland answer, I thought further about the problem and came up with a fallacious scheme, namely to first simulate the value p of C(U,V) by drawing a Uniform, and second simulate (U,V) conditional on C(U,V)=p. Going as far as running an R code on a simple copula, as shown above. Fallacious reasoning since (as I knew already!!!), C(U,V) is not uniformly distributed! But has instead a case-dependent distribution… As a (connected) aside, I wonder if the generator attached with Archimedean copulas has any magical feature that help with the generation of the associated copula.

analogy between mean and median [X validated]

A recent question on X validated was looking rather naïvely for the median being equal to the mean, as earlier questions there, but this made me realise a certain symmetry in the two notions, namely that the median ς is solution of the balance condition

namely the areas under the density are equal, while the mean μ is solution of a parallel balance condition

which is an equality of the same kind, i.e., of areas under the cdf or its complement…. Courtesy of the general representation

The general problem of having the mean and median equal is not particularly interesting as it is a matter of parameterisation. For instance, using the cdf transform turns the random variate X into a Uniform F(X), with mean and median equal to ½.

on approximations of Φ and Φ⁻¹

As I was working on a research project with graduate students, I became interested in fast and not necessarily very accurate approximations to the normal cdf Φ and its inverse. Reading through this 2010 paper of Richards et al., using for instance Polya’s

(with another version replacing 2/π with the squared root of π/8) and

not to mention a rational faction. All of which are more efficient (in R), if barely, than the resident pnorm() function.

      test replications elapsed relative user.self 
3 logistic       100000   0.410    1.000     0.410 
2    polya       100000   0.411    1.002     0.411 
1 resident       100000   0.455    1.110     0.455 

For the inverse cdf, the approximations there are involving numerical inversion except for

which proves slightly faster than qnorm()

       test replications elapsed relative user.self 
2 inv-polya       100000   0.401    1.000     0.401
1  resident       100000   0.450    1.000     0.450

warped Cauchys

A somewhat surprising request on X validated about the inverse cdf representation of a wrapped Cauchy distribution. I had not come across this distribution, but its density being

means that it is the superposition of shifted Cauchys on the unit circle (with nice complex representations). As such, it is easily simulated by re-shifting a Cauchy back to (-π,π), i.e. using the inverse transform

nested sampling: any prior anytime?!

A recent arXival by Justin Alsing and Will Handley on “nested sampling with any prior you like” caught my attention. If only because I was under the impression that some priors would not agree with nested sampling. Especially those putting positive weight on some fixed levels of the likelihood function, as well as improper priors.

“…nested sampling has largely only been practical for a somewhat restrictive class of priors, which have a readily available representation as a transform from the unit hyper-cube.”

Reading from the paper, it seems that the whole point is to demonstrate that “any proper prior may be transformed onto the unit hypercube via a bijective transformation.” Which seems rather straightforward if the transform is not otherwise constrained: use a logit transform in every direction. The paper gets instead into the rather fashionable direction of normalising flows as density representations. (Which suddenly reminded me of the PhD dissertation of Rob Cornish at Oxford, which I examined last year. Even though nested was not used there in the same understanding.) The purpose appearing later (in the paper) or in fine to express a random variable simulated from the prior as the (generative) transform of a Uniform variate, f(U). Resuscitating the simulation from an arbitrary distribution from first principles.

“One particularly common scenario where this arises is when one wants to use the (sampled) posterior from one experiment as the prior for another”

But I remained uncertain at the requirement for this representation in implementing nested sampling as I do not see how it helps in bypassing the hurdles of simulating from the prior constrained by increasing levels of the likelihood function. It would be helpful to construct normalising flows adapted to the truncated priors but I did not see anything related to this version in the paper.

The cosmological application therein deals with the incorporation of recent measurements in the study of the ΛCDM cosmological model, that is, more recent that the CMB Planck dataset we played with 15 years ago. (Time flies, even if an expanding Universe!) Namely, the Baryon Oscillation Spectroscopic Survey and the SH₀ES collaboration.