Hey,

I’ve just discovered this website called PhDTree, which is awesome because it shows that science-wise I am a great-great-great-great-great-great-great child of Siméon Denis Poisson.

And then I’ve looked at the genealogy of all of my French colleagues and they are all descendants of him, which is rather annoying.

Two Hessian soldiers having a ball.

Hey,

Arnaud Doucet, Sylvain Rubenthaler and I have just put a technical report on arXiv about estimating the first- and second-order derivatives of the log-likelihood (also called the score and the observed information matrix respectively) in general (intractable) statistical models, and in particular in (non-linear non-Gaussian) state-space models. We call them “derivative-free” estimates because they can be computed even if the user cannot compute any kind of derivatives related to the model (as opposed to e.g. this paper and this paper). Actually in some cases of interest we cannot even evaluate the log-likelihood point-wise (we do not have a formula for it), so forget about explicit derivatives. Would you like to know more?

Our tech report builds heavily upon the Iterated Filtering series of papers (see this first PNAS paper, then this technical Annals of Stats paper). It simply extends it to the second-order derivatives (actually we also propose an alternate estimate for the score). The main idea can be interpreted in terms of Bayesian asymptotics. Say, you have some (univariate, for clarity) parameter , and you want to evaluate the derivatives of the log-likelihood at some point ; introduce a prior distribution with mean and variance . Consider the behaviour of posterior distribution when 1) the dataset is fixed (hence the likelihood is also fixed) and 2) the prior distribution concentrates, that is . What happens to the posterior distribution?

As you can imagine it also shrinks: it looks more and more like the prior distribution when decreases. Now interestingly enough, under mild regularity assumptions and a Gaussian prior distribution we have the two following inequalities:

and

for some . It means that the shift from the prior mean to the posterior mean is proportional to the first derivative of the log-likelihood (already known from the IF papers), while the shift from the prior variance to the posterior variance is proportional to its second order derivative (new stuff). All of this up to an error term going to zero at the speed of .

In practical terms, it means that the problem of computing the first two derivatives is turned into a problem of computing posterior expectations, which can be tackled with Monte Carlo methods for a very broad class of models.

Since the classic picture of Bayes is actually not a picture of Bayes, we might as well use Ryan.

Hey,

Just a quick note about BayesComp, a new wiki about Bayesian Computational Statistics (see this outdated but well-written introduction if you really don’t know what that is), as Xian pointed out. It is organised by the ISBA Section on Bayesian Computation, notably Peter Green and Nicolas Chopin so far. If the community gets into it, it could become the nerve centre for online resources about Bayesian Computation, which so far are quite scattered and poorly advertised.

Good luck to BayesComp!

Tessellated Pavement, Eaglehawk Neck, Tasman Peninsula

Hello,

At the end of May CSIRO (Marine and Atmospheric Research, Hobart) and in particular Emlyn Jones organise a conference on this topic, subtitled:

New Pathways to Understanding and Managing Marine Ecosystems: Quantifying Uncertainty and Risk Using Biophysical-Statistical Models of the Marine Environment

Here is an example of what marine biogeochemical data assimilation is about. Suppose you want to model the population sizes of phytoplankton and zooplankton like they do in A Bayesian approach to state and parameter estimation in a Phytoplankton-Zooplankton model. Marine biologists can formulate a model based on their knowledge of the phenomenon, and in this case they propose a model as a system of different equations (they call this one the PZ model):

.

This model describes the interaction between both species’ population sizes and, although it is very simple compared to other models in the area, it already represents a non-trivial phenomenon: when there are more phytoplanktons, then the zooplanktons have more food so they grow through the term , then both population decrease through the terms and respectively, then the phytoplankton population size increases again, etc, etc. It’s a Lotka-Volterra model describing a predator/prey interaction. The randomness comes from being a Normal random variable , drawn at every integer times , corresponding to each day; it reflects that the growth rate of phytoplankton can be different from one day to the other. The observations are daily, noisy measurements of the phytoplankton population sizes, the zooplanktons never being measured.

The questions are numerous: can we estimate the zooplankton population size from the phytoplankton population size (a problem called filtering) under parameter uncertainty? Can we predict both time-series under parameter uncertainty? Can we estimate the parameters, which all have biological interpretation (grazing rates, growth rates, mortality rates etc)? If we have competing models, can we use the data to decide which one is the most accurate under parameter uncertainty (a problem called model choice)? If we can for this simple model, can we also do all of that for more complex models, where both the processes and the parameters are high-dimensional?

The fact that we want to do these things “under parameter uncertainty” obviously makes everything way more challenging. To be clear it is meant that parameters are not fixed (or estimated in a first stage), they are simply given a prior distribution. At the moment scientists can perform most of the tasks mentioned above for reasonably simple models such as the PZ model, not for the craziest ones with million-dimensional hidden processes (e.g. spatial state space models). At least not without additional approximations. The conference will be an opportunity to discuss future improvements to make the methods scalable and computational issues currently experienced by practitioners.

Just a quick post to advertise the following list of statistics-related blogs and websites. Click on the badge to access it:

We will be back soon with more content!

Cheers,

Pierre

I presented today at INSEE R user group (FLR) how to use knitr (Sweave evolution) for writing documents which are self contained with respect to the source code: your data changed? No big deal, just compile your .Rnw file again and you are done with an updated version of your paper![Ctrl+Shift+I] is easy. Some benefits with respect to having two separate .R and .tex files: it is integrated in a single software (RStudio), you can call variables in your text with the \Sexpr{} command. The slow speed at compilation is no more a real matter as one can put “cache=TRUE” in code chunk options not to reevaluate unchanged chunks, which fastens things.

I share the (brief) slides below. They won’t help much those who already use knitr, but they give the first steps for those who would like to give it a try.

Hi there !

Like Pierre a while ago, I got fed up with printing articles, annotating them, losing them, re-printing them, and so on. Moreover, I also wanted to be able to carry more than one or two books in my bag without ruining my back. E-Ink readers seemed good but at some point I changed my mind…

After the ISBA conference in Kyoto, where I saw bazillions of IPads, I thought that tablets really worth the shot. I am cool with reading on a LCD screen, I probably won’t read scientific articles/books outside in the sun, and I like the idea of a light device that can replace my laptop in conferences. Furthermore, there is now a large choice of apps to annotate pdf which is crucial for me.

The device I chose run on Android (mainly because there is no memory extension on Apple devices), combined with a good capacitive pen, an annotation app such as eZreader that get your pdf directly from Dropbox (which is simply awesome). You can even use LaTeX (without fancy packages…) which may become handy.

I hope that I will not experience the same disappointment as Pierre did with his reader, but for the moment a tablet seems just what I needed !

Hello everyone,

and of course Happy New Year (2013 is the international year of statistics!).

Last week the ISBA Regional Meeting was held in Banaras / Varanasi, in the North of India. The conference was well attended, with leading figures such as Jayanta K. Ghosh, José Bernardo, James Berger, Peter Green, Christian Robert who blogged about it, and an overall ~350 participants.

As always I really liked the conference, as much for the opportunity to spend time with colleagues out of the usual environment, as for the content of the talks. Since the conference was held in the University, it was also an opportunity to discover a very different academic environment and to talk with the local students. The highlight of the conference happened when a stray dog, out of nowhere, walked nimbly behind the speaker during one of the talks.

Somewhere in Banaras Hindu University.

I gave a talk about Score and Observed Information Matrix estimation in general state space models (those with intractable transition densities). This is on-going work with colleagues and I will surely explain all about it on this blog once it is finished.

Namastey!

Gaussian density biased such that 75% of the mass is in the negative values and 25% in the positive values

Hey,

With Robin Ryder we wrote a paper titled The Wang-Landau Algorithm Reaches the Flat Histogram in Finite Time and it has been accepted in Annals of Applied Probability (arXiv preprint  here). I’m especially happy about it since it was the last remaining unpublished chapter of my PhD thesis. In this post I’ll try to explain what we proved here on a simple example.

Suppose you are given a target density , and you wish to sample from it BUT with the additional constraint that you want some parts of the state space to be favoured and some other to be penalized. For instance, you are given an univariate Gaussian density. It is symmetric so if you sample directly from it, in average you will get half of your samples in the negative values, and half in the positive values. Suppose that you would like to have 75% of your samples in the negative values instead. This kind of constraint can be interesting if you want to explore multimodal distributions, in which case you might want to penalize already-visited regions of the state space, in order to explore yet unvisited regions (see this old post for instance, which was about a related paper now accepted in JCGS).

In the case of a Gaussian distribution, it is easy because you can compute exactly the required quantities For instance if you partition the state space in two regions and , and you want to have a proportion of samples on each bin (say and ). You can compute for each bin

and then if you define the “biased” density as follows (up to a multiplicative constant):

then you can check that the biased density has a mass of on bin and on bin , ie . Hence drawing from this biased density should yield the desired samples. However in general it is not possible to compute , so you need an algorithm to estimate these quantities. The Wang-Landau algorithm does this in an online manner, which means that it both gets the desired draws from and estimates the penalties simultaneously. It’s an iterative algorithm that you can run for as long as you want. Now we say that the Flat Histogram criterion is reached when we obtain (approximately) the desired frequencies in each bin, ie when the empirical proportions of samples in each bin are close to the desired ones. What we proved with Robin is that the Flat Histogram criterion is reached in a random time that has a finite expectation, i.e. you’re not going to wait forever in front of your computer waiting for the Flat Histogram criterion.

The following plot shows the empirical proportions converging to the desired ones.

The empirical proportions of samples in each bin (red lines, one for each bin) converge to the desired ones (dotted lines), as the iterations go along.

To produce these figures you can use the PAWL package that is on CRAN. Here’s a bit of code producing both figures of this post.

library(PAWL)
theme_set(theme_bw())
rinit <- function(size) rep(0, size)
parameters <- list(mean = 0, sd = 1)
logdensity <- function(x, parameters){
  return(dnorm(x, parameters$mean, parameters$sd, log = TRUE) * (x > -10 & x < 10))
}
gaussiantarget <- target(name = "gaussian", dimension = 1,
                         rinit = rinit, logdensity = logdensity,
                         parameters = parameters)
Nchains <- 1; Niterations <- 10000
proposal <- createAdaptiveRandomWalkProposal(Nchains, gaussiantarget@dimension,
                                             adaptiveproposal = FALSE)
pawlparameters <- tuningparameters(nchains = Nchains, niterations = Niterations, storeall = TRUE)
getPos <- function(points, logdensity) points
desfreq <- c(0.75, 0.25)
positionbinning <- binning(position = getPos,
                           name = "position",
                           bins = c(-Inf, 0),
                           desiredfreq = desfreq,
                           useLearningRate = FALSE,
                           autobinning = FALSE)
pawlresults <- pawl(gaussiantarget, binning = positionbinning, AP = pawlparameters,
                    proposal = proposal)
chains <- ConvertResults(pawlresults)
alllocations <- positionbinning@getLocations(positionbinning@bins, pawlresults$allreaction)
locationsdf <- data.frame(cbind(cumsum(alllocations == 1) / 1:(Niterations+1),
                                cumsum(alllocations == 2) / 1:(Niterations+1),
                                1:(Niterations+1)))
names(locationsdf) <- c("proportion1", "proportion2",
                        "iterations")
meltedlocations <- melt(data=locationsdf, id = c("iterations"))

g <- ggplot(data = meltedlocations, aes(x = iterations, y = value, colour = variable))
g <- g + geom_line() + theme(legend.position = "none")
g <- g + scale_colour_manual(values= c("red", "red"))
g <- g + geom_hline(yintercept = positionbinning@desiredfreq, col = "black", linetype = 3)
g <- g + xlab("iterations") + ylab("Proportions of visits to each bin")
print(g)

ghist <- qplot(chains[,1], geom = "histogram", binwidth = 0.3) + geom_vline(xintercept = 0, colour = "red", size = 3)
ghist <- ghist + xlab("X")
print(ghist)

Hi,

additionally to the referral program (you refer a new user, you win an extra .5 Go), the Dropbox Space Race will give you 3 Go extra space (for 2 years) if you register with your email from a competing university. The best schools will get more space. Here are the 100 top schools. Com’ on, there is no french school in the 100 top !

Thanks Nicolas for the info.