Archive for University of Amsterdam

are there a frequentist and a Bayesian likelihoods?

Posted in Statistics with tags , , , , , , , , , , on June 7, 2018 by xi'an

A question that came up on X validated and led me to spot rather poor entries in Wikipedia about both the likelihood function and Bayes’ Theorem. Where unnecessary and confusing distinctions are made between the frequentist and Bayesian versions of these notions. I have already discussed the later (Bayes’ theorem) a fair amount here. The discussion about the likelihood is quite bemusing, in that the likelihood function is the … function of the parameter equal to the density indexed by this parameter at the observed value.

“What we can find from a sample is the likelihood of any particular value of r, if we define the likelihood as a quantity proportional to the probability that, from a population having the particular value of r, a sample having the observed value of r, should be obtained.” R.A. Fisher, On the “probable error’’ of a coefficient of correlation deduced from a small sample. Metron 1, 1921, p.24

By mentioning an informal side to likelihood (rather than to likelihood function), and then stating that the likelihood is not a probability in the frequentist version but a probability in the Bayesian version, the W page makes a complete and unnecessary mess. Whoever is ready to rewrite this introduction is more than welcome! (Which reminded me of an earlier question also on X validated asking why a common reference measure was needed to define a likelihood function.)

This also led me to read a recent paper by Alexander Etz, whom I met at E.J. Wagenmakers‘ lab in Amsterdam a few years ago. Following Fisher, as Jeffreys complained about

“..likelihood, a convenient term introduced by Professor R.A. Fisher, though in his usage it is sometimes multiplied by a constant factor. This is the probability of the observations given the original information and the hypothesis under discussion.” H. Jeffreys, Theory of Probability, 1939, p.28

Alexander defines the likelihood up to a constant, which causes extra-confusion, for free!, as there is no foundational reason to introduce this degree of freedom rather than imposing an exact equality with the density of the data (albeit with an arbitrary choice of dominating measure, never neglect the dominating measure!). The paper also repeats the message that the likelihood is not a probability (density, missing in the paper). And provides intuitions about maximum likelihood, likelihood ratio and Wald tests. But does not venture into a separate definition of the likelihood, being satisfied with the fundamental notion to be plugged into the magical formula

posteriorprior×likelihood

JASP, a really really fresh way to do stats

Posted in Statistics with tags , , , , , , on February 1, 2018 by xi'an

Bayesian workers, unite!

Posted in Books, Kids, pictures, Statistics, University life with tags , , , , , , , on January 19, 2018 by xi'an

This afternoon, Alexander Ly is defending his PhD thesis at the University of Amsterdam. While I cannot attend the event, I want to celebrate the event and a remarkable thesis around the Bayes factor [even though we disagree on its role!] and the Jeffreys’s Amazing Statistics Program (!), otherwise known as JASP. Plus commend the coolest thesis cover I ever saw, made by the JASP graphical designer Viktor Beekman and representing Harold Jeffreys leading empirical science workers in the best tradition of socialist realism! Alexander wrote a post on the JASP blog to describe the thesis, the cover, and his plans for the future. Congratulations!

absolutely no Bayesians inside!

Posted in Statistics with tags , , , , , , , , on December 11, 2017 by xi'an

bridgesampling [R package]

Posted in pictures, R, Statistics, University life with tags , , , , , , , , , on November 9, 2017 by xi'an

Quentin F. Gronau, Henrik Singmann and Eric-Jan Wagenmakers have arXived a detailed documentation about their bridgesampling R package. (No wonder that researchers from Amsterdam favour bridge sampling!) [The package relates to a [52 pages] tutorial on bridge sampling by Gronau et al. that I will hopefully comment soon.] The bridge sampling methodology for marginal likelihood approximation requires two Monte Carlo samples for a ratio of two integrals. A nice twist in this approach is to use a dummy integral that is already available, with respect to a probability density that is an approximation to the exact posterior. This means avoiding the difficulties with bridge sampling of bridging two different parameter spaces, in possibly different dimensions, with potentially very little overlap between the posterior distributions. The substitute probability density is chosen as Normal or warped Normal, rather than a t which would provide more stability in my opinion. The bridgesampling package also provides an error evaluation for the approximation, although based on spectral estimates derived from the coda package. The remainder of the document exhibits how the package can be used in conjunction with either JAGS or Stan. And concludes with the following words of caution:

“It should also be kept in mind that there may be cases in which the bridge sampling procedure may not be the ideal choice for conducting Bayesian model comparisons. For instance, when the models are nested it might be faster and easier to use the Savage-Dickey density ratio (Dickey and Lientz 1970; Wagenmakers et al. 2010). Another example is when the comparison of interest concerns a very large model space, and a separate bridge sampling based computation of marginal likelihoods may take too much time. In this scenario, Reversible Jump MCMC (Green 1995) may be more appropriate.”

Bayesian spectacles

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , on October 4, 2017 by xi'an

E.J. Wagenmakers and his enthusiastic team of collaborators at University of Amsterdam and in the JASP software designing team have started a blog called Bayesian spectacles which I find a fantastic title. And not only because I wear glasses. Plus, they got their own illustrator, Viktor Beekman, which sounds like the epitome of sophistication! (Compared with resorting to vacation or cat pictures…)

In a most recent post they addressed the criticisms we made of the 72 author paper on p-values, one of the co-authors being E.J.! Andrew already re-addressed some of the address, but here is a disagreement he let me to chew on my own [and where the Abandoners are us!]:

Disagreement 2. The Abandoners’ critique the UMPBTs –the uniformly most powerful Bayesian tests– that features in the original paper. This is their right (see also the discussion of the 2013 Valen Johnson PNAS paper), but they ignore the fact that the original paper presented a series of other procedures that all point to the same conclusion: p-just-below-.05 results are evidentially weak. For instance, a cartoon on the JASP blog explains the Vovk-Sellke bound. A similar result is obtained using the upper bounds discussed in Berger & Sellke (1987) and Edwards, Lindman, & Savage (1963). We suspect that the Abandoners’ dislike of Bayes factors (and perhaps their upper bounds) is driven by a disdain for the point-null hypothesis. That is understandable, but the two critiques should not be mixed up. The first question is Given that we wish to test a point-null hypothesis, do the Bayes factor upper bounds demonstrate that the evidence is weak for p-just-below-.05 results? We believe they do, and in this series of blog posts we have provided concrete demonstrations.

Obviously, this reply calls for an examination of the entire BS blog series, but being short in time at the moment, let me point out that the upper lower bounds on the Bayes factors showing much more support for H⁰ than a p-value at 0.05 only occur in special circumstances. Even though I spend some time in my book discussing those bounds. Indeed, the [interesting] fact that the lower bounds are larger than the p-values does not hold in full generality. Moving to a two-dimensional normal with potentially zero mean is enough to see the order between lower bound and p-value reverse, as I found [quite] a while ago when trying to expand Berger and Sellker (1987, the same year as I was visiting Purdue where both had a position). I am not sure this feature has been much explored in the literature, I did not pursue it when I realised the gap was missing in larger dimensions… I must also point out I do not have the same repulsion for point nulls as Andrew! While considering whether a parameter, say a mean, is exactly zero [or three or whatever] sounds rather absurd when faced with the strata of uncertainty about models, data, procedures, &tc.—even in theoretical physics!—, comparing several [and all wrong!] models with or without some parameters for later use still makes sense. And my reluctance in using Bayes factors does not stem from an opposition to comparing models or from the procedure itself, which is quite appealing within a Bayesian framework [thus appealing per se!], but rather from the unfortunate impact of the prior [and its tail behaviour] on the quantity and on the delicate calibration of the thing. And on a lack of reference solution [to avoid the O and the N words!]. As exposed in the demise papers. (Which main version remains in a publishing limbo, the onslaught from the referees proving just too much for me!)