Archive for inverse cdf

what does more efficient Monte Carlo mean?

Posted in Books, Kids, R, Statistics with tags , , , , , , on March 17, 2017 by xi'an

“I was just thinking that there might be a magic trick to simulate directly from this distribution without having to go for less efficient methods.”

In a simple question on X validated a few days ago [about simulating from x²φ(x)] popped up the remark that the person asking the question wanted a direct simulation method for higher efficiency. Compared with an accept-reject solution. Which shows a misunderstanding of what “efficiency” means on Monte Carlo situations. If it means anything, I would think it is reflected in the average time taken to return one simulation and possibly in the worst case. But there is no reason to call an inverse cdf method more efficient than an accept reject or a transform approach since it all depends on the time it takes to make the inversion compared with the other solutions… Since inverting the closed-form cdf in this example is much more expensive than generating a Gamma(½,½), and taking plus or minus its root, this is certainly the case here. Maybe a ziggurat method could be devised, especially since x²φ(x)<φ(x) when |x|≤1, but I am not sure it is worth the effort!

Optimization Monte Carlo: Efficient and embarrassingly parallel likelihood-free inference

Posted in Books, Statistics, Travel with tags , , , , , , , , on December 16, 2015 by xi'an

optiMC1AmstabcTed Meeds and Max Welling have not so recently written about an embarrassingly parallel approach to ABC that they call optimisation Monte Carlo. [Danke Ingmar for pointing out the reference to me.] They start from a rather innocuous rephrasing of the ABC posterior, writing the pseudo-observations as deterministic transforms of the parameter and of a vector of uniforms. Innocuous provided this does not involve an infinite number of uniforms, obviously. Then they suddenly switch to the perspective that, for a given uniform vector u, one should seek the parameter value θ that agrees with the observation y. A sort of Monte Carlo inverse regression: if

y=f(θ,u),

then invert this equation in θ. This is quite clever! Maybe closer to fiducial than true Bayesian statistics, since the prior does not occur directly [only as a weight p(θ)], but if this is manageable [and it all depends on the way f(θ,u) is constructed], this should perform better than ABC! After thinking about it a wee bit more in London, though, I realised this was close to impossible in the realistic examples I could think of. But I still like the idea and want to see if anything at all can be made of this…

“However, it is hard to detect if our optimization succeeded and we may therefore sometimes reject samples that should not have been rejected. Thus, one should be careful not to create a bias against samples u for which the optimization is difficult. This situation is similar to a sampler that will not mix to remote local optima in the posterior distribution.”

Now, the paper does not go that way but keeps the ε-ball approach as in regular ABC, to derive an approximation of the posterior density. For a while I was missing the difference between the centre of the ball and the inverse of the above equation, bottom of page 3. But then I realised the former was an approximation to the latter. When the authors discuss their approximation in terms of the error ε, I remain unconvinced by the transfer of the tolerance to the optimisation error, as those are completely different notions. This also applies to the use of a Jacobian in the weight, which seems out of place since this Jacobian appears in a term associated with (or replacing) the likelihood, f(θ,u), which is then multiplied by the prior p(θ). (Assuming a Jacobian exists, which is unclear when considering most simulation patterns use hard bounds and indicators.) When looking at the toy examples, it however makes sense to have a Jacobian since the selected θ’s are transforms of the u’s. And the p(θ)’s are simply importance weights correcting for the wrong target. Overall, the appeal of the method proposed in the paper remains unclear to me. Most likely because I did not spend enough time over it.

Tractable Fully Bayesian inference via convex optimization and optimal transport theory

Posted in Books, Statistics, University life with tags , , , , , , , , on October 6, 2015 by xi'an

IMG_0294“Recently, El Moselhy et al. proposed a method to construct a map that pushed forward the prior measure to the posterior measure, casting Bayesian inference as an optimal transport problem. Namely, the constructed map transforms a random variable distributed according to the prior into another random variable distributed according to the posterior. This approach is conceptually different from previous methods, including sampling and approximation methods.”

Yesterday, Kim et al. arXived a paper with the above title, linking transport theory with Bayesian inference. Rather strangely, they motivate the transport theory with Galton’s quincunx, when the apparatus is a discrete version of the inverse cdf transform… Of course, in higher dimensions, there is no longer a straightforward transform and the paper shows (or recalls) that there exists a unique solution with positive Jacobian for log-concave posteriors. For instance, log-concave priors and likelihoods. This solution remains however a virtual notion in practice and an approximation is constructed via a (finite) functional polynomial basis. And minimising an empirical version of the Kullback-Leibler distance.

I am somewhat uncertain as to how and why apply such a transform to simulations from the prior (which thus has to be proper). Producing simulations from the posterior certainly is a traditional way to approximate Bayesian inference and this is thus one approach to this simulation. However, the discussion of the advantage of this approach over, say, MCMC, is quite limited. There is no comparison with alternative simulation or non-simulation methods and the computing time for the transport function derivation. And on the impact of the dimension of the parameter space on the computing time. In connection with recent discussions on probabilistic numerics and super-optimal convergence rates, Given that it relies on simulations, I doubt optimal transport can do better than O(√n) rates. One side remark about deriving posterior credible regions from (HPD)  prior credible regions: there is no reason the resulting region is optimal in volume (HPD) given that the transform is non-linear.

arbitrary distributions with set correlation

Posted in Books, Kids, pictures, R, Statistics, University life with tags , , , , , , , , , , on May 11, 2015 by xi'an

A question recently posted on X Validated by Antoni Parrelada: given two arbitrary cdfs F and G, how can we simulate a pair (X,Y) with marginals  F and G, and with set correlation ρ? The answer posted by Antoni Parrelada was to reproduce the Gaussian copula solution: produce (X’,Y’) as a Gaussian bivariate vector with correlation ρ and then turn it into (X,Y)=(F⁻¹(Φ(X’)),G⁻¹(Φ(Y’))). Unfortunately, this does not work, because the correlation does not keep under the double transform. The graph above is part of my answer for a χ² and a log-Normal cdf for F amd G: while corr(X’,Y’)=ρ, corr(X,Y) drifts quite a  lot from the diagonal! Actually, by playing long enough with my function

tacor=function(rho=0,nsim=1e4,fx=qnorm,fy=qnorm)
{
  x1=rnorm(nsim);x2=rnorm(nsim)
  coeur=rho
  rho2=sqrt(1-rho^2)
  for (t in 1:length(rho)){
     y=pnorm(cbind(x1,rho[t]*x1+rho2[t]*x2))
     coeur[t]=cor(fx(y[,1]),fy(y[,2]))}
  return(coeur)
}

Playing further, I managed to get an almost flat correlation graph for the admittedly convoluted call

tacor(seq(-1,1,.01),
      fx=function(x) qchisq(x^59,df=.01),
      fy=function(x) qlogis(x^59))

zerocorNow, the most interesting question is how to produce correlated simulations. A pedestrian way is to start with a copula, e.g. the above Gaussian copula, and to twist the correlation coefficient ρ of the copula until the desired correlation is attained for the transformed pair. That is, to draw the above curve and invert it. (Note that, as clearly exhibited by the graph just above, all desired correlations cannot be achieved for arbitrary cdfs F and G.) This is however very pedestrian and I wonder whether or not there is a generic and somewhat automated solution…

simulating correlated Binomials [another Bernoulli factory]

Posted in Books, Kids, pictures, R, Running, Statistics, University life with tags , , , , , , , on April 21, 2015 by xi'an

This early morning, just before going out for my daily run around The Parc, I checked X validated for new questions and came upon that one. Namely, how to simulate X a Bin(8,2/3) variate and Y a Bin(18,2/3) such that corr(X,Y)=0.5. (No reason or motivation provided for this constraint.) And I thought the following (presumably well-known) resolution, namely to break the two binomials as sums of 8 and 18 Bernoulli variates, respectively, and to use some of those Bernoulli variates as being common to both sums. For this specific set of values (8,18,0.5), since 8×18=12², the solution is 0.5×12=6 common variates. (The probability of success does not matter.) While running, I first thought this was a very artificial problem because of this occurrence of 8×18 being a perfect square, 12², and cor(X,Y)x12 an integer. A wee bit later I realised that all positive values of cor(X,Y) could be achieved by randomisation, i.e., by deciding the identity of a Bernoulli variate in X with a Bernoulli variate in Y with a certain probability ϖ. For negative correlations, one can use the (U,1-U) trick, namely to write both Bernoulli variates as

X_1=\mathbb{I}(U\le p)\quad Y_1=\mathbb{I}(U\ge 1-p)

in order to minimise the probability they coincide.

I also checked this result with an R simulation

> z=rbinom(10^8,6,.66)
> y=z+rbinom(10^8,12,.66)
> x=z+rbinom(10^8,2,.66)
cor(x,y)
> cor(x,y)
[1] 0.5000539

Searching on Google gave me immediately a link to Stack Overflow with an earlier solution with the same idea. And a smarter R code.

simulation by inverse cdf

Posted in Books, Kids, R, Statistics, University life with tags , , , , , on January 14, 2015 by xi'an

Another Cross Validated forum question that led me to an interesting (?) reconsideration of certitudes! When simulating from a normal distribution, is Box-Muller algorithm better or worse than using the inverse cdf transform? My first reaction was to state that Box-Muller was exact while the inverse cdf relied on the coding of the inverse cdf, like qnorm() in R. Upon reflection and commenting by other members of the forum, like William Huber, I came to moderate this perspective since Box-Muller also relies on transcendental functions like sin and log, hence writing

X=\sqrt{-2\log(U_1)}\,\sin(2\pi U_2)

also involves approximating in the coding of those functions. While it is feasible to avoid the call to trigonometric functions (see, e.g., Algorithm A.8 in our book), the call to the logarithm seems inescapable. So it ends up with the issue of which of the two functions is better coded, both in terms of speed and precision. Surprisingly, when coding in R, the inverse cdf may be the winner: here is the comparison I ran at the time I wrote my comments

> system.time(qnorm(runif(10^8)))
sutilisateur     système      écoulé
 10.137           0.120      10.251
> system.time(rnorm(10^8))
utilisateur     système      écoulé
 13.417           0.060      13.472`

However re-rerunning it today, I get opposite results (pardon my French, I failed to turn the messages to English):

> system.time(qnorm(runif(10^8)))
utilisateur     système      écoulé
     10.137       0.144      10.274
> system.time(rnorm(10^8))
utilisateur     système      écoulé
      7.894       0.060       7.948

(There is coherence in the system time, which shows rnorm as twice as fast as the call to qnorm.) In terms, of precision, I could not spot a divergence from normality, either through a ks.test over 10⁸ simulations or in checking the tails:

“Only the inversion method is inadmissible because it is slower and less space efficient than all of the other methods, the table methods excepted”. Luc Devroye, Non-uniform random variate generation, 1985

Update: As pointed out by Radford Neal in his comment, the above comparison is meaningless because the function rnorm() is by default based on the inversion of qnorm()! As indicated by Alexander Blocker in another comment, to use an other generator requires calling RNG as in

RNGkind(normal.kind = “Box-Muller”)

(And thanks to Jean-Louis Foulley for salvaging this quote from Luc Devroye, which does not appear to apply to the current coding of the Gaussian inverse cdf.)