# Statisfaction

## A healthy dose of foundational crisis

Posted in General by Rémi Bardenet on 26 March 2018

I finally took the time to read about axiomatic foundations of Bayesian statistics. I like axioms, I like Bayesians stats, so this was definitely going to be a pleasant opportunity to read some books, comfortably seated on the sofa I just added to my office. Moreover, my team in Lille includes supporters of Dempster-Shafer belief functions, another framework for uncertainty modelling and decision-making, so being precise on my own axioms was the best way to discuss more constructively with my colleagues.

Take Savage’s axioms, for instance. I’ve always heard that they were the current justification behind the saying “being Bayesian is being a coherent decision-maker”. To be precise, let $\Theta$ be the set of states of the world, that is, everything useful to make your decision. To fix ideas, in a statistical experiment, your decision might be a “credible” interval on some real parameter, so $\Theta$ should at least be the product of $\mathbb{R}$ times whatever space your data live in. Now an action $a$ is defined to be a map from $\Theta$ to some set of outcomes $\mathcal{Z}$. For the interval problem, an action $a_I$ corresponds to the choice of a particular interval $I$ and the outcomes $\mathcal{Z}$ should contain whatever you need to assess the performance of your action, say, the indicator of the parameter actually belonging to your interval $I$, and the length of $I$. Outcomes are judged by utility, that is, we consider functions $u:\mathcal{Z}\rightarrow\mathbb{R}_+$ that map outcomes to nonnegative rewards. In our example, this could be a weighted sum of the indicator and the interval length. The weights translate your preference for an interval that actually captures the value of the parameter of interest over a short interval. Now, the axioms give the equivalence between the two following bullets:

• (being Bayesian) There is a unique pair $(u,\pi)$, made of a utility function and a finitely additive probability measure $\pi$ defined on all subsets of the set $\Theta$ of states of the world, such that you choose your actions by maximizing an expected utility criterion: $\displaystyle a^\star \in \arg\max_a \mathbb{E}_\pi u(a(\theta)),$

• (being coherent) Ranking actions according to a preference relation that satisfies a few abstract properties that make intuitive sense for most applications, such as transitivity: if you prefer $a_I$ to $a_J$ and $a_J$ to $a_G$, then you prefer $a_I$ to $a_G$. Add to this a few structural axioms that impose constraints on $\Theta$ on $\mathcal{Z}$.

Furthermore, there is a natural notion of conditional preference among actions that follows from Savage’s axioms. Taken together, these axioms give an operational definition of our “beliefs” that seems to match Bayesian practice. In particular, 1) our beliefs take the form of a probability measure –which depends on our utility–, 2) we should update these beliefs by conditioning probabilities, and 3) make decisions using expected utility with respect to our belief. This is undeniably beautiful. Not only does Savage avoid shaky arguments or interpretations by using your propensity to act to define your beliefs, but he also avoids using “extraneous probabilities”. By the latter I mean any axiom that artificially brings mathematical probability structures into the picture, such as “there exists an ideal Bernoulli coin”.

But the devil is in the details. For instance, some of the less intuitive of Savage’s axioms require the set of states of the world to be uncountable and the utility bounded. Also, the measure $\pi$ is actually only required to be finitely additive, and it has to be defined on all subsets of the set of states of the world. Now-traditional notions like Lebesgue integration, $\sigma$-additivity, or $\sigma$-algebras do not appear. In particular, if you want to put a prior on the mean of a Gaussian that lives in $\Theta=\mathbb{R}$, Savage says your prior should weight all subsets of the real line, so forget about using any probability measure that has a density with respect to the Lebesgue measure! Or, to paraphrase de Finetti, $\sigma$-additive probability does not exist. Man, before reading about axioms I thought “Haha, let’s see whether someone has actually worked out the technical details to justify Bayesian nonparametrics with expected utility, this must be technically tricky”; now I don’t even know how to fit the mean of a Gaussian anymore. Thank you, Morpheus-Savage.

There are axiomatic ways around these shortcomings. From what I’ve read they all either include extraneous probabilities or rather artificial mathematical constructions. Extraneous probabilities lead to philosophically beautiful axioms and interpretations, see e.g. Chapter 2 of Bernardo and Smith (2000), and they can get you finite and countably finite sets of states of the world, for instance, whereas Savage’s axioms do not. Stronger versions also give you $\sigma$-additivity, see below. Loosely speaking, I understand extraneous probabilities as measuring uncertainty with respect to an ideal coin, similarly to measuring heat in degrees Celsius by comparing a physical system to freezing or boiling water. However, I find extraneous probability axioms harder to swallow than (most of) Savage’s axioms, and they involve accepting a more general notion of probability than personal propensity to act.

If you want to bypass extraneous probability and still recover $\sigma$-additivity, you could follow Villegas (1964), and try to complete the state space $\Theta$ so that well-behaved measures $\pi$ extend uniquely to $\sigma$-additive measures on a $\sigma$-algebra on this bigger set of states $\hat\Theta$. Defining the extended $\hat\Theta$ involves sophisticated functional analysis, and requires to add potentially hard-to-intepret states of the world, so losing some of the interpretability of Savage’s construction. Authors of reference books seem reluctant to go in that direction: De Groot (1970), for instance, solves the issue by using a strong extraneous probability axiom that allows working in the original set $\Theta$ with $\sigma$-additive beliefs. Bernardo & Smith use extraneous probabilities, but keep their measures finitely additive until the end of Chapter 2. Then they admit departing from axioms for practical purposes and define “generalized beliefs” in Chapter 3, defined on a $\sigma$-algebra of the original $\Theta$. Others seem to more readily accept the gap between axioms and practice, and look for a more pragmatic justification of the combined use of expected utility and countably additive probabilities. For instance, Robert (2007) introduces posterior expected utility, and then argues that it has desirable properties among decision-making frameworks, such as respecting the likelihood principle. This is unlike Savage’s approach, for whom the (or rather, a finitely additive version of the) likelihood principle is a consequence of the axioms. I think this is an interesting subtlelty.

To conclude, I just wanted to share my excitement for having read some fascinating works on decision-theoretic axioms for Bayesian statistics. There still is some unresolved tension between having both an applicable and axiomatized Bayesian theory of belief. I would love this post to generate discussions, and help me understand the different thought processes behind each Bayesian being Bayesian (and each non-Bayesian being non-Bayesian). For instance, I had not realised how conceptually different the points of view in the reference books of Robert and Bernardo & Smith were. This definitely helped me understand (Xi’an) Robert’s short three answers to this post.

If this has raised your interest, I will mention here a few complementary sources that I have found useful, ping me if you want more. Chapters 2 and 3 of Bernardo and Smith (2000) contain a detailed description of their set of axioms with extraneous probability, and they give great lists of pointers on thorny issues at the end of each chapter. A lighter read is Parmigiani and Inoue (2009), which I think is a great starting point, with emphasis on the main ideas of de Finetti, Ramsey, Savage, and Anscombe and Aumann, how they apply, and how they relate to each other, rather than the technical details. Technical details and exhaustive reviews of sets of axioms for subjective probability can be found in their references to Fishburn’s work, which I have found to be beautifully clear, rigorous and complete, although like many papers involving low-level set manipulations, the proofs sometimes feel like they are written for robots. But after all, a normative theory of rationality is maybe only meant for robots.

## AI in Grenoble, 2nd to 6th July 2018

Posted in General, Seminar/Conference by Julyan Arbel on 22 March 2018 This is an advertisement for on conference on AI organised at Inria Grenoble by Thoth team and Naver labs : https://project.inria.fr/paiss/. This AI summer school comprises lectures and practical sessions conducted by renowned experts in different areas of artificial intelligence.

This event is the revival of a past series of very successful summer schools which took place in Grenoble and Paris. The latest edition of this series was held in 2013. While originally focusing on computer vision, the summer school now targets a broader AI audience, and will also include presentations about machine learning, natural language processing, robotics, and cognitive science.

Note that NAVER LABS is funding a number of students to attend PAISS. Apply before 4th April. (more…)

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## Do you really need to work 60+ hours per week to be a good researcher ?

Posted in General, publishing by JB Salomond on 6 February 2018 There is a controversy these days on social media about academics claiming that “if you do not feel like working 60+ hours per week including weekends and evenings you should probably find another job”. This is utterly frustrating. As an academic myself, I clearly do not feel like working days and nights, even if I am neck deep in a project. Does it makes me a poor assistant prof ? It is no secret that academic research is not, in general, a 9 to 5 job (not saying that it cannot be). I myself usually work on weekends, when commuting or during holidays. I always carry a few papers that I did not had time to read or relentlessly write equations in my notebook when an idea strikes me during the morning commute. I do think about how I could improve my next day lecture and make some changes in my slides late in the evening. That is partly because I am disorganised, also because the job sort of requires it.  We lack time to do all the things we want to do while at the lab. Conferences, seminars and meetings wreck your schedule if you had any. So you might end up seeing any time off as a lost opportunity to do more research.

This situation is clearly not good, and many academics, including me, have or had a hard time dealing with this. In particular when you are still a PhD/postdoc/tenure track (you name it) and need to stick out of the pack to get a position in an extremely competitive environment. And when senior full professors are telling you that you need to work even harder if you want to be considered as worthy, that is clearly not helping.

My view is that even if we all want to shoot for the moon (and hit the damn thing), it is totally fine to simply do some good research. Each of your paper does not have to be ground breaking, as long as it contribute to the field, it should be good enough and acknowledge as such. If your goal is to have that-many JRSS/Biometrika/Annals of Stats papers and a sky-rocketing h-index no matter what, you are probably doing it wrong. Competition between academics can be a good boost from time to time, but it should not be the end of it. More importantly, it should not be what drives an academic careerer. The negative externalities of this system are depressed junior researchers and limited scientific research squeezed out of our brains to extend our publication record.

So what should we do about this ? Well first, stop bragging about how many hours to work per week, if you feel like working 24/7, good for you, but it does not have to be that way for everyone. Secondly, stop judging academics (especially junior ones) on some dumb metrics such as h-index or so, if you need to evaluate a candidate, read their research. In short, cut the competition and be supportive. Make academia fun again !

## Sub-Gaussian property for the Beta distribution (part 3, final)

Posted in General, R by Julyan Arbel on 26 December 2017 When a Beta random variable wants to act like a Bernoulli: convergence of optimal proxy variance.

In this third and last post about the Sub-Gaussian property for the Beta distribution  (post 1 and post 2), I would like to show the interplay with the Bernoulli distribution as well as some connexions with optimal transport (OT is a hot topic in general, and also on this blog with Pierre’s posts on Wasserstein ABC). (more…)

## Sub-Gaussian property for the Beta distribution (part 2)

Posted in R by Julyan Arbel on 20 December 2017 Left: What makes the Beta optimal proxy variance (red) so special? Right: The difference function has a double zero (black dot).

As a follow-up on my previous post on the sub-Gaussian property for the Beta distribution , I’ll give here a visual illustration of the proof.

A random variable $X$ with finite mean $\mu=\mathbb{E}[X]$ is sub-Gaussian if there is a positive number $\sigma$ such that: $\mathbb{E}[\exp(\lambda (X-\mu))]\le\exp\left(\frac{\lambda^2\sigma^2}{2}\right)\,\,\text{for all } \lambda\in\mathbb{R}.$

We focus on X being a Beta $(\alpha,\beta)$ random variable. Its moment generating function $\mathbb{E}[\exp(\lambda X)]$ is known as the Kummer function, or confluent hypergeometric function $_1F_1(\alpha,\alpha+\beta,\lambda)$. So is $\sigma^2$-sub-Gaussian as soon as the difference function $u_\sigma(\lambda)=\exp\left(\frac{\alpha}{\alpha+\beta}\lambda+\frac{\sigma^2}{2}\lambda^2\right)-_1F_1(\alpha,\alpha+\beta,\lambda)$

remains positive on $\mathbb{R}$. This difference function $u_\sigma(\cdot)$ is plotted on the right panel above for parameters $(\alpha,\beta)=(1,1.3)$. In the plot, $\sigma^2$ is varying from green for the variance $\text{Var}[X]=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$ (which is a lower bound to the optimal proxy variance) to blue for the value $\frac{1}{4(\alpha+\beta+1)}$, a simple upper bound given by Elder (2016), . The idea of the proof is simple: the optimal proxy-variance corresponds to the value of $\sigma^2$ for which $u_\sigma(\cdot)$ admits a double zero, as illustrated with the red curve (black dot). The left panel shows the curves with $\mu = \frac{\alpha}{\alpha+\beta}$ varying, interpolating from green for $\text{Var}[X]=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$ to blue for $\frac{1}{4(\alpha+\beta+1)}$, with only one curve qualifying as the optimal proxy variance in red.

#### References

 Marchal and Arbel (2017), On the sub-Gaussianity of the Beta and Dirichlet distributions. Electronic Communications in Probability, 22:1–14, 2017. Code on GitHub.
 Elder (2016), Bayesian Adaptive Data Analysis Guarantees from Subgaussianity, https://arxiv.org/abs/1611.00065

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## A big problem in our community

Posted in General, Seminar/Conference, Statistics by Pierre Jacob on 14 December 2017 “Tout va très bien”, meaning “all is well”, by Franquin.

Hi all,

Kristian Lum, who was already one of my Statistics superheroes for her many interesting papers and great talks, bravely wrote the following text about her experience as a young statistician going to conferences:

https://medium.com/@kristianlum/statistics-we-have-a-problem-304638dc5de5

I can’t thank Kristian enough for speaking out. Her experience is both shocking and hardly surprising. Many, many academics report similar stories. This simply can’t go on like that.

I happen to have gone to the conferences mentioned by Kristian, and my experience as a young man was completely different. It was all about meeting interesting people, discussing ideas, being challenged, and having good times. Nobody harassed, touched or assaulted me. There was some flirting, as I guess is natural when hundreds of people are put in sunny places far away from home, but I was never the victim of any misconduct or abuse of power. So instead of driving me out of the field, conferences became important, enriching and rewarding moments of my professional life.

Looking back at those conferences I feel sick, and heartbroken, at the thought that some of my peers were having such a difficult time, because of predators who don’t ever  face the consequences of their actions. Meanwhile I was part of the silent majority.

The recent series of revelations about sexual harassment and assaults in other professional environments indicate that this is not specific to our field, nor to academia. But this does not make it any more acceptable. I know for a fact that many leaders of our field take this issue extremely seriously (as Kristian mentions too),  but clearly much much more needs to be done. The current situation is just shameful; strong and  coordinated actions will be needed to fix it. Thanks again to Kristian for the wake-up call.

## nrow, references and copies

Posted in R by Pierre Jacob on 10 December 2017 Claude Monet’s paintings have nothing to do with the topic of this post.

Hi all,

This post deals with a strange phenomenon in R that I have noticed while working on unbiased MCMC. Reducing the problem to a simple form, consider the following code, which iteratively samples a vector ‘x’ and stores it in a row of a large matrix called ‘chain’ (I’ve kept the MCMC terminology).

dimstate = 100
nmcmc = 1e4
chain = matrix(0, nrow = nmcmc, ncol = dimstate)
for (imcmc in 1:nmcmc){
if (imcmc == nrow(chain)){ #call to nrow
}
x = rnorm(dimstate, mean = 0, sd = 1)
chain[imcmc,] = x #copying of x in chain
}


If you execute this code, you will see that it is surprisingly slow: it takes close to a minute on my computer. Now, consider the next block, which does exactly the same except that the vector ‘x’ is not copied into the matrix ‘chain’.

dimstate = 100
nmcmc = 1e4
chain = matrix(0, nrow = nmcmc, ncol = dimstate)
for (imcmc in 1:nmcmc){
if (imcmc == nrow(chain)){ #call to nrow
}
x = rnorm(dimstate, mean = 0, sd = 1)
# chain[imcmc,] = x #no more copying
}


This code runs nearly instantaneously. Could it be that just copying a vector in a matrix takes a lot of time? Sounds unlikely. Now consider this third block.

dimstate = 100
nmcmc = 1e4
chain = matrix(0, nrow = nmcmc, ncol = dimstate)
for (imcmc in 1:nmcmc){
if (imcmc == nmcmc){ #no call to nrow
}
x = rnorm(dimstate, mean = 0, sd = 1)
chain[imcmc,] = x #copying of x in chain
}


This code runs nearly instantaneously as well; this time ‘x’ is copied into ‘chain’, but the call to the nrow function is removed….?! What is nrow doing? It is meant to simply return dim(chain), the first dimension of chain. So consider this fourth block.

dimstate = 100
nmcmc = 1e4
chain = matrix(0, nrow = nmcmc, ncol = dimstate)
for (imcmc in 1:nmcmc){
if (imcmc == dim(chain)){ #call to dim instead of nrow
}
x = rnorm(dimstate, mean = 0, sd = 1)
chain[imcmc,] = x #copying of x in chain
}


This one also runs instantaneously! So replacing nrow(chain) by dim(chain) solves the problem. Why?

The answer comes from R guru and terrific statistician Louis Aslett. I directly quote from an exchange of emails, since he brilliantly explains the phenomenon.

You probably know R stores everything by reference, so if I do:

x <- matrix(0, nrow=1e5, ncol=100)
y <- x

I actually only have one copy of the matrix in memory with two references to it. If I then do:

x[1,1] <- 1

R will first make a copy of the whole matrix, update x to point to that and then change the first element to one. This idea is used when you pass a variable to a standard (i.e. non-core, non-primitive) R function, which nrow is: it creates a reference to the variable you pass so that it doesn’t have to copy and the function call is very fast …. as long as you don’t write to it inside the function, no copy need ever happen. But the “bad design” bit is that R makes a decision whether to copy on write based only on a reference count and crucially that reference count stays increased even after a function returns, irrespective of whether or not the function has touched the variable.

So:

x <- matrix(0, nrow=1e5, ncol=100) # matrix has ref count 1
x[1,1] <- 1 # ref count is 1, so write with no copy
nrow(x) # ref count is 2 even though nothing was touched
x[1,1] <- 1 # ref count still 2, so R copies before writing first element. Now the ref count drops to 1 again
x[2,2] <- 1 # this writes without a copy as ref count got reset on last line
nrow(x) # ref count jumps
x[3,3] <- 1 # copy invoked again! Aaaargh!

So by calling nrow in the loop for the first example, the chain matrix is being copied in full on every iteration. In the second example, chain is never written to so there is no negative side effect to the ref count having gone up. In the third example, chain only ever has ref count 1 so there are no copies and each row is written in-place. I did a quick bit of profiling and indeed in the slow example, the R garbage collector allocates and tidies up nearly 9GB of RAM when executing the loop!

The crazy thing is that dim(chain) works full speed even though that is all that nrow is doing under the hood, but the reason is that dim is a so-called “primitive” core R function which is special because it doesn’t affect the reference counter of its arguments. If you want to dig into this yourself, there’s a function refs() in the pryr package which tells you the current reference count to any variable.

Thanks Louis!

## Bayesian model comparison with vague or improper priors

Posted in Statistics by Pierre Jacob on 6 November 2017 Synthetic data from a Lévy-driven stochastic volatility model (top), log-Bayes factor between two such models (middle) and “Hyvärinen factor” (proposed approach, bottom). Each line represents a different Monte Carlo estimate, obtained sequentially over time.

Hi,

With Stephane Shao, Jie Ding and Vahid Tarokh we have just arXived a tech report entitled “Bayesian model comparison with the Hyvärinen score: computation and consistency“. Here I’ll explain the context, that is, scoring rules and Hyvärinen scores (originating in Hyvärinen’s score matching approach to inference), and then what we actually do in the paper.

## Dynamic publication list for research webpage using arXiv, HAL, or bibtex2html

Posted in General by Julyan Arbel on 23 October 2017

Well of course, dynamic is conditional upon some manual feeding. If you put your papers on arXiv or HAL, then those two propose dynamic widgets. If you maintain a .bib file of your papers, you can use tools like bibtex2html. This is not dynamic at all, but it allows for finer tuning of url links you might want to add than with arXiv or HAL options. I review below those three options. (more…)

## ISBA elections, let’s go voting

Posted in General by Julyan Arbel on 16 October 2017 So it seems even Thomas B. went voting.

The International Society for Bayesian Analysis (ISBA), is running elections until November, 15. This year, two contributors on this blog, Nicolas Chopin and myself, are running for an ISBA Section office. The sections of the society, nine in number as of today, gather researchers with common research interests: Computation, Objective Bayes, Nonparametrics, etc.

Here are our candidate statements:

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