ICML paper: Kernel Adaptive Metropolis Hastings

$\DeclareMathOperator*{\argmin}{arg\,min}$ $\def\Mz{M_{\mathbf{z},y}}$

In this year’s ICML in Bejing, Arthur Gretton, presented our paper on Kernel Adaptive Metropolis Hastings [link, poster]. His talk slides are based on Dino Sejdinovic ones [link]. Pitty that I did not have travel funding. The Kameleon is furious!

The idea of the paper is quite neat: It is basically doing Kernel-PCA to adapt a random walk Metropolis-Hastings sampler’s proposal based on the Markov chain’s history. Or, more fancy: Performing a random walk on an empirically learned non-linear manifold induced by an empirical Gaussian measure in a Reproducing Kernel Hilbert Space (RKHS). Sounds scary, but it’s not: the proposal distribution ends up being a Gaussian aligned to the history of the Markov chain.

We consider the problem of sampling from an intractable highly-nonlinear posterior distribution using MCMC. In such cases, the thing to do ™ usually is Hamiltonian Monte Carlo (HMC) (with all its super fancy extensions on Riemannian manifolds etc). The latter really is the only way to sample from complicated densities by using the geometry of the underlying space. However, in order to do so, one needs access to target likelihood and gradient.

We argue that there exists a class of posterior distributions for that HMC is not available, while the distribution itself is still highly non-linear. You get them as easy as attempting binary classification with a Gaussian Process. Machine Learning doesn’t get more applied. Well, a bit: we sample the posterior distribution over the hyper-parameters using Pseudo-Marginal MCMC. The latter makes both likelihood and gradient intractable and the former in many cases has an interesting shape. See this plot which is the posterior of parameters of an Automatic Relevance Determination Kernel on the UCI Glass dataset. This is not an exotic problem at all, but it has all the properties we are after: non-linearity and higher order information unavailable. We cannot do HMC, so we are left with a random walk, which doesn’t work nicely here due to the non-linear shape.

The idea of our algorithm is to do a random walk in an infinite dimensional space. At least almost.

  • We run some method to get a rough sketch of the distribution we are after.
  • We then embed these points into a RKHS; some infinite dimensional mean and covariance operators are going on here, but I will spare you the details here.
  • In the RKHS, the chain samples are Gaussian, which corresponds to a manifold in the input space, which in some sense aligns with what we have seen of the target density yet.
  • We draw a sample $f$ from that Gaussian – random walk in the RKHS
  • This sample is in an infinite dimensional space, we need to map it back to the original space, just like in Kernel PCA
  • It turns out that is hard (nice word for impossible). So let’s just take a gradient step along some cost function that minimises distances in the way we want: \[\argmin_{x\in\mathcal{X}}\left\Vert k\left(\cdot,x\right)-f\right\Vert _{\mathcal{H}}^{2}\]
  • Cool thing: This happens in input space and gives us a point whose embedding is in some sense close to the RKHS sample.
  • Wait, everything is Gaussian! Let’s be mathy and integrate the RKHS sample (and the gradient step) out.

Et voilĂ , we get a Gaussian proposal distribution in input space\[q_{\mathbf{z}}(\cdot|x_{t})=\mathcal{N}(x_{t},\gamma^{2}I+\nu^{2}M_{\mathbf{z},x_{t}}HM_{\mathbf{z},x_{t}}^{\top}),\]where $\Mz=2\eta\left[\nabla_{x}k(x,z_{1})|_{x=y},\ldots,\nabla_{x}k(x,z_{n})|_{x=y}\right]$ is based on kernel gradients (which are all easy to get). This proposal aligns with the target density. It is clear (to me) that using this for sampling is better than just an isotropic one. Here are some pictures of bananas and flowers.

The paper puts this idea in the form of an adaptive MCMC algorithm, which learns the manifold structure on the fly. One needs to be careful about certain details, but that’s all in the paper. Compared to existing linear adaptive samplers, which adapt to the global covariance structure of the target, our version adapts to the local covariance structure. This can all also be mathematically formalised. For example, for the Gaussian kernel, the above proposal covariance becomes \[\left[\text{cov}[q_{\mathbf{z}(\cdot|y)}]\right]_{ij} = \gamma^{2}\delta_{ij} + \frac{4\nu^{2}}{\sigma^{4}}\sum_{a=1}^{n}\left[k(y,z_{a})\right]^{2}(z_{a,i}-y_{i})(z_{a,j}-y_{j}) +\mathcal{O}(n^{-1}),\] where the previous points $z_a$ influence the covariance, weighted by their similarity $k(y,z_a)$ to current point $y$.

Pretty cool! We also have a few “we beat all competing methods” plots, but I find those retarded and will spare them here – though they are needed for publication 😉

Dino and me have also written a pretty Python implementation (link) of the above, where the GP Pseudo Marginal sampling is done with Shogun’s ability to importance sample marginal likelihoods of non-conjugate GP models (link). Pretty cool!

I Like Intractable Likelihoods

Last week, I went to the i-like workshop at Oxford university. Pretty cool! All of Britain’s statisticians were there and I met many of them for the first time. Check out my two posters (Russian Roulette, Kernel Adaptive Metropolis Hastings). Talks were amazing – as in last NIPS, the main trend is on estimating likelihoods (well, that’s the name of the program), either using some other random process such as importance sampling a latent model’s marginal likelihood (aka Pseudo-Marginal MCMC), or directly sub-sampling likelihoods or gradients.

These things are important in Machine Learning too, and it is very nice to see the field growing together (even-though there was a talk by a Statistician spending lots of time on re-inventing belief propagation and Junction tree ideas – always such a pitty if this happens simply because communities do not talk to each other enough). Three talks that I really found interesting:

Remi Bardenet talked about sub-sampling approaches to speed up MCMC. This is quite related to the Austerity in MCMC land paper by Welling & Co, with the difference that his tests do not suffer from small number of points in the hypothesis test to decide accept/reject.

Chris Sherlock talked about optimal rates and scaling for Pseudo-Marginal MCMC. There finally are some nice heuristics how to scale PM estimates in a way that the number of iid samples per computation time is optimal. Interestingly, the acceptance rate and the variance of the likelihood estimate can be tweaked separately.

Jim Griffin gave a very interesting talk on adaptive MCMC on discrete, in particular binary, state-spaces – he used them for feature selection (in ML language). His algorithm automatically learns global mutations rates for each of the positions. However, it doesn’t take any correlations between the features into account. This might be a very interesting application for our fancy Kameleon sampler (arxiv, code), thinking about this!

Finally, I presented two posters, the one on Playing Russian Roulette with Intractable Likelihoods that I already presented in Reykjavik, and (with Dino) a new poster (link) on the Kernel Adaptive Metropolis Hastings Kameleon that I mentioned above. The corresponding paper is hopefully published very soon. Talking to other scientists about my own work is just great!

 

GSoC Interview with Sergey and me

Sergey and me gave an interview on Shogun and Google Summer of Code. Here it is:

The internet. More specifically #shogun on irc.freenode.net. Wasn’t IRC that thing that our big brothers used as a socialising substitute when they were teenagers back in the 90s? Anyways. We are talking to two of the hottest upcoming figures in machine learning open-source software, the Russian software entrepreneur Sergey Lisitsyn, and the big German machine Heiko Strathmann.


1.jpg2013-07-10 11.55.11.jpg

Hi guys, glad to meet you. Would you mind introducing yourself?

Sergey (S): Hey, I am Sergey. If you ask me what do I do apart from Shogun – I am currently working as a software engineer and finishing my Master’s studies at Samara State Aerospace University. I joined Shogun in 2011 as a student and now I am doing my best to help guys from the Shogun team to keep up with GSoC 2014.

Heiko (H): Hej, my name is Heiko. I do a Phd in Neuroscience & Machine Learning at the Gatsby Institute in London and joined Shogun three years ago during GSoC. I love open-source since my days in school.

 

Your project, Shogun, is about Machine Learning. That sounds scary and sexy, but what is it really?

H: My grandmother recently sent me an email asking about this ‘maschinelles Lernen’. I replied it is the art of finding structure in data in an automated way. She replied: Since when are you an artist? And what is this “data”? I showed her the movie PI by Darren Aronofsky where the main character at some point is able to predict stock prices after realising “the pattern”, and said that’s what we want to do with a computer. Since then, she is worried about me because the guy puts a drill into his head in the end
.. Another cool application is for example to model brain patterns to allow people to learn how to use a prosthesis faster.

S: Or have you seen your iPhone detects faces? That’s just a Support Vector Machine (SVM). It employs kernels which are inner products of non-linear mappings of Haar features into a reproducing kernel Hilbert Space so that it minimizes 
.

 

Yeah, okok… What is the history of Shogun in the GSoC?

S: The project got started by Sören in his student days around 15 years ago. It was a research only tool for a couple of years before being made public. Over the years, more and more people joined, but the biggest boost came from GSoC…

H: We just got accepted into our 4th year in that program. We had 5+8+8 students so far who all successfully did the program with us. Wow I guess that’s a few million dollars. (EDITOR: actually 105,000$.) GSoC students forced Shogun to grow up in many ways: github, a farm of buildbots, proper unit-testing, a cloud-service, web-demos, etc were all set up by students. Also, the diversity of algorithms from latest research increased a lot. From the GSoC money, we were able to fund our first Shogun workshop in Berlin last summer.

 

How did you two got into Shogun and GSoC? Did the money play a role?

H: I was doing my undergraduate project back in 2010, which actually involved kernel SVMs, and used Shogun. I thought it would be a nice idea of putting my ideas into it — also I was lonely coding just on my own. 2010, they were rejected from GSoC, but I eventually implemented my ideas in 2011. The money to me was very useful as I was planning to move to London soon. Being totally broke in that city one year later, I actually paid my rent from my second participation’s stipend – which I got for implementing ideas from my Master’s project at uni. Since 2013, I mentor other students and help organising the project. I think I would have stayed around without the money, but it would have been a bit tougher.

S: We were having a really hard winter in Russia. While I was walking my bear and clearing the roof of the snow, I realised I forgot to turn off my nuclear missile system
..

H: Tales!

S: Okay, so on another cold night I noticed a message on GSoC somewhere and then I just glanced over the list of accepted organizations and Shogun’s description was quite interesting so I joined a chat and started talking to people – the whole thing was breathtaking for me. As for the money – well, I was a student and was about to start my first part-time job as a developer – it was like a present for me but it didn’t play the main role!

H: To make it short: Sergey suddenly appeared and rocked the house coding in lightspeed, drinking Vodka.

 

But now you are not paid anymore, while still spending a lot of time on the project. What motivates you to do this?

S: This just involves you and you feels like you participate in something useful. Such kind of appreciation is important!

H: Mentoring students is very rewarding indeed! Some of those guys are insanely motivated and talented. It is very nice to interact with the community with people from all over the world sharing the same interest. Trying to be a scientist, GSoC is also very useful in producing tools that myself or my colleagues need, but that nobody has the time to build properly. You see, there are all sorts of synergic effects in GSoC and my day-job at university, such as meeting new people or getting a job since you know how to code in a team.

 

How does this work? Did you ever publish papers based on GSoC work?

S: Yeah, I actually published a paper based on my GSoC 2011 work. It is called ‘Tapkee: An Efficient Dimension Reduction Library’ and was recently published in the Journal of Machine Learning Research. We started writing it up with my mentor Christian (Widmer) and later Fernando (Iglesias) joined our efforts. It took enormous amount of time but we did it! Tapkee by the way is a Russian word for slippers.

H: I worked on a project on statistical simulation of global ozone data last year. The code is mainly based on one of my last year’s student’s project – a very clever and productive guy from Mumbai who I would never have met without the program, see http://www.ucl.ac.uk/roulette/ozoneexample

 

So you came all the way from being a student with GSoC up to being an organisation admin. How does the perspective change during this path?

H: I first had too much time so I coded open-source, then too little money so I coded open-source, then too much work so I mentor people coding it open-source. At some point I realised I like this stuff so much that I would like to help organising Shogun and bring together the students and scientists involved. It is great to give back to the community which played a major role for me in my studies. It is also sometimes quite amusing to get those emails by students applying, being worried about the same unimportant things that I worried about back then.

S: It seems to be quite natural actually. You could even miss the point when things change and you became a mentor. Once you are into the game things are going pretty fast. Especially if you have full-time job and studies!

 

Are there any (forbidden) substances that you exploit to keep up with the workload?

S: It would sound strange but I am not addicted to vodka. Although I bet Heiko is addicted to beer and sausages.

H: Coffeecoffeecoffeee

 Well, to be honest GSoC definitely reduces your sleep no matter whether you are either student, mentor, or admin. By the way, our 3.0 release was labelled: Powered by Vodka, Mate, and beer.

 

Do you crazy Nerds actually ever go away from your computers?

H: No.

S: Once we all met at our workshop in Berlin – but we weren’t really away from our computers. Why on earth to do that?

 

Any tips for upcoming members of the open-source community? For students? Mentors? Admins?

H: Students: Do GSoC! You will learn a lot. Mentors: Do GSoC! You will get a lot. Admins/Mentors: Don’t do GSoC, it ruins your health. Rather collect stamps!

S: He is kidding. (whispers: “we need this 
 come on 
 just be nice to them”)

H: Okay to be honest: just have fun of what you are doing!

 

Due to the missing interest in the community, Sergey and Heiko interviewed themselves on their own.

 

Shogun: http://www.shogun-toolbox.org

GSoC 2013 blog: http://herrstrathmann.de/shogun-blog/110-shogun-3-0.html

GSoC 2014 ideas: http://www.shogun-toolbox.org/page/Events/gsoc2014_ideas

Heiko: http://herrstrathmann.de/

Sergey: http://cv.lisitsyn.me/

 

Google Summer of Code 2014


Yeah! Shogun this week got accepted to be an organisation participating in the 10th Google Summer of Code. This year, besides mentoring a few projects, I am one of the three project administrators. I am curious how this will be. One first thing to do was to write the application for Shogun – I’m glad it worked! I also will spend a little more time organising things. Apart from trying to find mentors (which requires a lot of talking people into it), I also want to make Shogun (and the students) having more from the program. Last year, I pushed the team to ask all students

  • to write a project report in the form of IPython notebooks (link). These are absolutely great for talking about the GSoC work, impressing people, and having a final piece of work to show for the students.
  • To fully unit-test every module of their algorithm/framework. This is absolutely essential in order to not loose the student’s work a few years later when a re-factoring change breaks their code and nobody knows how to fix it. Those tests already saved lots of life since last year.
  • To peer-review each other in pairs of students. This improved documentation here and there and solved some bugs. I want to emphasise this more this year as I think it is a great way of enabling synergistic effects between students.

In addition, we will again screen all the applicants via a set of entrance tasks on our github page (link). I just wrote a large number of such smaller or larger tasks that get students started on a particular project, fix bugs in Shogun, or prepare some larger change. In order to get the students started a bit more easily (contributing to Shogun these days is a non-trivial task), I wrote a little how-to (link) that is supposed to point out our expectations, and what are the first steps towards participating in GSoC. 

Finally, I wrote descriptions for quite a few possible projects, some of them with a number of interesting co-mentors. The full list is here (link). If you are a talented student interested in any of those topics, consider working with us during the summer. It’s usually very fun!

  • Variational Learning for Recommendation with Big Data. With Emtiyaz Khan, who I met at last year’s workshop for latent Gaussian models. Matrix factorisation and Gaussian Processes, ultra-cool project.
  • Generic Framework for Markov Chain Monte Carlo Algorithms and Stan Interface. With Theo Papamarkou, who I know from my time at UCL Statistics. It’s about a modular representation of MCMC within Shogun and a possible interface to STAN for the actual sampling. This would be a major step of Shogun towards probabilistic models.
  • Testing and Measuring Variable Interactions With Kernels. With Dino, who is post-doc at Gatsby and co-author of our optimal kernel for MMD paper. This project is to implement all kernel based interaction measures in Shogun in a unified way. We’ll probably use this for research later.
  • A Meta-Language for Shogun examples. With Sören. Write example once, press button to generate in any modular language binding. This would be so useful to have in Shogun!
  • Lobbying Shogun in MLPACK’s automatic benchmarking system. Joint project with Ryan from MLPACK. He already can compare speed of different toolboxes. Now let’s compare results.
  • Shogun Missionary & Shogun in Education. With Sören. Write high quality notebooks and eye-candy examples. Very different project as this is about creative technical writing and illustrating methods on cool data rather than hacking new algorithms. I would be very excited if this happened!

Some of the other projects involve cool buzzwords such as Deep Learning, Structured Output, Kernel, Dual solvers, Cluster backends, etc. Join us! 🙂

MLOSS workshop at NIPS 2013

Last week, I went to the Advances in Neural Information Processing Systems (NIPS) for the first time. That was a very nice experience due to the incredibly density of people whose names I know from research papers. In fact, it was too much to take so I had to pick things that sounded interesting – still loads.

The main three buzzwords of the conference for me were: Deep Learning (even Mark Zuckerberg is interested in that these days), Mini-batch, and stochastic gradient descent (aka on-line whatever).

One very interesting workshop I attended on Tuesday was on Machine Learning Open-Source Software (MLOSS), organised by Cheng Soon Ong (who could not be there unfortunately) and Antti Honkela. I presented a short spotlight for Shogun (slide) and had a one hour demo, showing off with our cool IPython notebooks (link) and the cloud Shogun server (link). I got some very encouraging feedback for this, including from Fernando Perez.
I also met a few nice fellow open-source ML coders from scikit-learn.

During the workshop, there was a quite lively discussion about licensing issues, in particular whether to choose GPL or BSD. The python universe for example seems to gain a lot from being BSD-style licensed.

Finally, NIPS is was held close to Lake Tahoe, which is surrounded by incredibly beautiful mountains to hike in. One evening, I met the guy who left those traces … very exciting, slightly scary…

GSoC 2013 brings Shogun 3.0

Shogun’s third Google Summer of Code just ended with our participation in the mentor summit at Google’s headquarter in Mountain View and the release of Shogun 3.0 (link) What a great summer! But let’s start at the beginning


Shogun is a toolbox that offers a unified framework for data-analysis, or in buzz words: machine learning, for a broad range of data types and analysis problems. Those not only include standard tools such as regression, classification, clustering, etc, but also cutting edge techniques from recent developments in research. One of Shogun’s most unique features is its interfaces to a wide range of mainstream computing languages.

In our third GSoC, we continued most of the directions taken in previous years such as asking students to contribute code in the application process for them to be considered. For that, we created a list of smaller introductory tasks for each of the GSoC projects that would become useful later in the project. While allowing students to get used to our development process, and increasing the quality of the applications, this also pushed the projects forward a bit before GSoC even started. The number of applications did not suffer through that (57 proposals from 52 students) but even increased compared to the previous year (48 proposals from 38 students) — this seems to be a trend.

This summer, we also had former GSoC students mentoring for the first time: Sergey Lisitsyn and me (mentoring two projects). Both of us joined in 2011. In addition, the former student Fernando Iglesias participated again and former student Viktor Gal stayed around to work on Shogun during GSoC (and did some massive infrastructure improvements). These are very nice long term effects of continuous GSoC participation. Thanks to GSoC, Shogun is growing constantly both in terms of code and developers.

As in 2012, we eventually could give away 8 slots to some very talented students. All of them did an awesome job on some highly involved projects covering a large number of topics. Two projects were extensions of previous ones:

 

Roman Votjakov extended last year’s project on the popular Gaussian Processes for handling classification problems and Shell Hu implemented a collection of algorithms within last year’s structured output framework (for example for OCR)


Fernando Iglesias implemented a new algorithm called metric learning, which plays well together with existing methods in Shogun.


Another new algorithm came from Soumyajit De, who has implemented an estimation method for log-determinants of large sparse matrices (needed for example for large-scale Gaussian distributions), and implemented a framework for linear operators and solvers, and fundamentals of an upcoming framework for distributed computing (which is used by his algorithm) on the fly. 


Evangelos Anagnostopoulos worked on feature hashing and random kitchen sinks, two very cool tricks to speed up linear and kernel-based learning methods in Shogun. Kevin Hughes implemented methods for independent component analysis, which can be used to separate mixtures of signals (for example audio, heart-beats, or images) and are well known in the community.


Last but not least, Liu Zhengyang created a pretty web-framework for running Shogun demos from the web browser and did add support for directly loading data from the mldata website. Evgeniy Andreev improved Shogun’s usability via integrating native support for various popular file formats such as CSV and protobuf.

 

 

You might have noticed the links in the above text (and images). Most of them are the final reports of the students in the form of IPython notebooks, an awesome new open-source tool that we started using for documentation. We are very proud of these.  See http://shogun-toolbox.org/page/documentation/notebook/ for a list of all notebooks. Also check out the web-demo framework at http://www.shogun-toolbox.org/page/documentation/demo/ if you haven’t yet.

IPython also features Shogun in the cloud: Former student Viktor Gal did setup http://cloud.shogun-toolbox.org which is an IPython notebook server ran by us. It allows you to play with Shogun-python from any web-browser without having to install it. You can try the existing notebooks or write your own. Give it a shot and let us know what you think!

This year’s GSoC also was the most productive one for us ever. We got  more than 2000 commits changing almost 400000 lines in more than 7000 files since our last release before GSoC.

Students! You all did a great job and we are more than amazed what you all have achieved. Thank you very much and we hope some of you will stick around.

Besides all the above individual projects, we encouraged students to work together a bit more to enable synergistic effects. One way we tried to implement this was through a peer review where we paired students to check each others interface documentation and final notebooks. We held the usual meetings with both mentors and students every few weeks to monitor progress and happiness, as well as asking students to write weekly reports. Keeping our IRC channel active every day also helped a lot in keeping things going.

My personal experience with mentoring was very positive. It is very nice to give back to the community. I tried to give them the same useful guidance that I received back then, and probably learned as much as my students did on the way. Having participated in GSoC 2011 and 2012, the change of perspective as a mentor was interesting, in particular regarding the selection process. Time wise, I think Google’s official statement of 5 hours per student per week is underestimating things quite a bit (if you want to get things done), and of course there is no upper bound on time you can spend.

Our plan of pairing external mentors with internal developers worked smoothly. As most of our mentors are scientists who tend to be very busy, it is sometimes hard for them to review all code on their own. Combining  big-picture guidance with the in-depth framework knowledge of the paired core developers allowed for more flexibility when allocating mentors for projects. Keep in mind that Shogun is still being organised by only five people (4 former students) plus a hand full of occasional developers, which makes it challenging to supervise 8 projects.

Another change this year was that writing unit-tests were mandatory to get code merged, which made the number of unit tests grew from 50 to more than 600. In the past years, we had seen how difficult it is to write tests at the end of projects, or maintain untested code. Making students do this on-the-fly drastically increased the stability of their code. A challenging side-effect of this was that many bugs within Shogun were discovered (and eventually fixed) which kept students and developers busy.

 As for Shogun itself, GSoC also boosts our community of users, which became so active this year that decided to organise a the first Shogun workshop in Berlin this summer. We had something over 30 participants from all over the world. The Shogun core team also met for the first time in real life, which was nice! We had a collection of talks, discussions, and hands-on sessions. Click here and here for videos and slides.

October brought the mentor summit, which I attended for the first time. This was such a cool event! There was a hotel with hot-tub, lots of goodies on the google campus as for example an on-site barista (!), a GSoC mentor with a robot-dog, and loads of loads of interesting people from interesting open-source projects. Some of these were new to me, some of them are projects that I have been checking out for more than 10 years now.I attended a few fruitful sessions, for example on open-source software for science. Sören hang out with the people he knew from previous years and the cool Debian guys (for which he is a developer too).

After the summit, the Shogun mentor team went hiking in the south Californian desert – I even climbed a rock.

What a great summer!

 

 

 

 

 

 

 

 

 

 

Shogun Workshop 2013

Last weekend, our Shogun workshop finally took place in Berlin. It was really cool to meet all those guys in person. We have been working together for quite some time now. The core-team an Shogun’s supporters are absolutely awesome. It is great to be part of that.

We had a nice afternoon at c-base (who were so friendly to host us) with some talks by all of our developers, followed by two days of hands-on workshop at the TU-Berlin.

I gave a little talk on two random things you can do with kernels (that are completely unrelated): Gaussian Processes and the kernel MMD. Slides are (download). I also wrote some IPython notebooks for GP-regression (link), GP-probit-classification (link), and two-sample testing with the kernel MMD (link).
One of the results of our discussions was that we will start using those notebook for Shogun’s documentation as they allow to combined code, plots, and maths in a web-based viewer.

Finally, here are some picture of us, (pretty nerdy)

 

Russian Roulette for intractable Likelihoods

Updates:
December 2015
: The journal version of the paper finally got published — after just three years.

While I was working at UCL’s Statistics Department in winter, I got involved into a very exciting project in the group of Mark Girolami. It is based around the Pseudo-Marginal Metropolis-Hastings algorithm. In 2003, a Genetics paper [1] described an approach to sample a distribution using the standard Metropolis-Hastings algorithm when the density function is not available by simply replacing it with an unbiased estimate.

For a standard Bayesian inference problem with likelihood $\pi(y|\theta) $, prior $\pi(\theta)$, and a proposal $Q$, rather than using the standard M-H ratio $$\frac{\pi(y|\theta^{\text{new}})}{pi(y|\theta)}\times\frac{\pi(\theta^{\text{new}})}{\pi(\theta)}\times \frac{Q(\theta|\theta^{\text{new}})}{Q(\theta^{\text{new}}|\theta)},$$ the likelihood is replaced by an unbiased estimator as

$$\frac{\hat{\pi}(y|\theta^{\text{new}})}{\hat{\pi}(y|\theta)}\times\frac{\pi(\theta^{\text{new}})}{\pi(\theta)}\times \frac{Q(\theta|\theta^{\text{new}})}{Q(\theta^{\text{new}}|\theta)}.$$ Remarkably  the resulting Markov chain converges to the same posterior distribution as the exact algorithm. The approach was later formalised and popularised in [2].

In our project, we exploited this idea to perform inference over models whose likelihood functions are intractable. Example of such intractable likelihoods are for example Ising models or, even simpler, very large Gaussian models. Both of those models’ normalising constants are very hard to compute. We came up with a way of producing unbiased estimators for the likelihoods, which are based on writing likelihoods as an infinite sum, and then truncating it stochastically.

Producing unbiased estimators for the Pseudo-Marginal approach is a very challenging task. Estimates have to be strictly positive. This can be achieved via pulling out the sign of the estimates in the final Monte-Carlo integral estimate and add a correction term (which increases the variance of the estimator). This problem is studied under the term Sign problem. The next step is to write the likelihood function as an infinite sum. In our paper, we do this for a geometrically titled correction of a biased estimator obtained by an approximation such as importance sampling estates, upper bounds, or deterministic approximations, and for likelihoods based on the exponential function.

I in particular worked on the exponential function estimate. We took a very nice example from spatial statistics: a worldwide grid of ozone measurements from a satellite that consists of a about 173,405 measurements. We fitted a simple Gaussian model whose covariance matrices are massive (and sparse). In such models of the form $$ \log \mathcal{N}_x(\mu,\Sigma))=-\log(\det(\Sigma)) – (\mu-x)^T \Sigma^{-1}(\mu-x) + C, $$ the normalising constant involves a log-determinant of such a large matrix. This is impossible using classical methods such as Cholesky factorisation $$\Sigma=LL^T \Rightarrow \log(\det(\Sigma))=2\sum_i\log(L_{ii}),$$ due to memory shortcomings: It is not possible to store the Cholesky factor $L$ since it is not in general sparse. We therefore constructed an unbiased estimator using a very neat method based on graph colourings and Krylov methods from [3].

This unbiased estimator of the log-likelihood is then turned into a (positive) unbiased estimator of the likelihood itself via writing the exponential function as an infinite series $$\exp(\log(\det(\Sigma)))=1+\sum_{i=1}^\infty \frac{\log(\det(\Sigma))^i}{i!}. $$

We then construct an unbiased estimator of this series by playing Russian Roulette: We evaluate the terms in the series and plug in a different estimator for $\log(\det(\Sigma))$ for every $i$; once those values are small, we start flipping a coin every whether we continue the series or not. If we do continue, we add some weights that ensure unbiasedness. We also ensure that it is less likely to continue in every iteration so that the procedure eventually stops. This basic idea (borrowed from Physics papers from some 20 years ago) and some technical details and computational tricks then give an unbiased estimator of the likelihood of the log-determinant of our Gaussian model and can therefore be plugged into Pseudo-Marginal M-H. This allows to perform Bayesian inference over models of sizes where it has been impossible before.

More details can be found on our project page (link, see ozone link), and in our paper draft on arXiv (link). One of my this year’s Google summer of Code projects for the Shogun Machine-Learning toolbox is about producing a sophisticated implementation of log-determinant estimators (link). Pretty exciting!

[1]: Beaumont, M. A. (2003). Estimation of population growth or decline in genetically monitored populations. Genetics 164 1139–1160.
[2]: Andrieu, C., & Roberts, G. O. (2009). The pseudo-marginal approach for efficient Monte Carlo computations. The Annals of Statistics, 37(2), 697–725.
[3]: Aune, E., Simpson, D., & Eidsvik, J. (2012). Parameter Estimation in High Dimensional Gaussian Distributions.