Is Jeffreys’ prior unique?

Posted in Books, Statistics, University life with tags , , , , , on March 3, 2015 by xi'an

“A striking characterisation showing the central importance of Fisher’s information in a differential framework is due to Cencov (1972), who shows that it is the only invariant Riemannian metric under symmetry conditions.” N. Polson, PhD Thesis, University of Nottingham, 1988

Following a discussion on Cross Validated, I wonder whether or not the affirmation that Jeffreys’ prior was the only prior construction rule that remains invariant under arbitrary (if smooth enough) reparameterisation. In the discussion, Paulo Marques mentioned Nikolaj Nikolaevič Čencov’s book, Statistical Decision Rules and Optimal Inference, Russian book from 1972, of which I had not heard previously and which seems too theoretical [from Paulo’s comments] to explain why this rule would be the sole one. As I kept looking for Čencov’s references on the Web, I found Nick Polson’s thesis and the above quote. So maybe Nick could tell us more!

However, my uncertainty about the uniqueness of Jeffreys’ rule stems from the fact that, f I decide on a favourite or reference parametrisation—as Jeffreys indirectly does when selecting the parametrisation associated with a constant Fisher information—and on a prior derivation from the sampling distribution for this parametrisation, I have derived a parametrisation invariant principle. Possibly silly and uninteresting from a Bayesian viewpoint but nonetheless invariant.

market static

Posted in Kids, Travel with tags , on March 2, 2015 by xi'an

[Heard in the local market, while queuing for cheese:]

– You took too much!

– Maybe, but remember your sister is staying for two days.

– My sister…, as usual, she will take a big serving and leave half of it!

– Yes, but she will make sure to finish the bottle of wine!

trans-dimensional nested sampling and a few planets

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , on March 2, 2015 by xi'an

This morning, in the train to Dauphine (train that was even more delayed than usual!), I read a recent arXival of Brendon Brewer and Courtney Donovan. Entitled Fast Bayesian inference for exoplanet discovery in radial velocity data, the paper suggests to associate Matthew Stephens’ (2000)  birth-and-death MCMC approach with nested sampling to infer about the number N of exoplanets in an exoplanetary system. The paper is somewhat sparse in its description of the suggested approach, but states that the birth-date moves involves adding a planet with parameters simulated from the prior and removing a planet at random, both being accepted under a likelihood constraint associated with nested sampling. I actually wonder if this actually is the birth-date version of Peter Green’s (1995) RJMCMC rather than the continuous time birth-and-death process version of Matthew…

“The traditional approach to inferring N also contradicts fundamental ideas in Bayesian computation. Imagine we are trying to compute the posterior distribution for a parameter a in the presence of a nuisance parameter b. This is usually solved by exploring the joint posterior for a and b, and then only looking at the generated values of a. Nobody would suggest the wasteful alternative of using a discrete grid of possible a values and doing an entire Nested Sampling run for each, to get the marginal likelihood as a function of a.”

This criticism is receivable when there is a huge number of possible values of N, even though I see no fundamental contradiction with my ideas about Bayesian computation. However, it is more debatable when there are a few possible values for N, given that the exploration of the augmented space by a RJMCMC algorithm is often very inefficient, in particular when the proposed parameters are generated from the prior. The more when nested sampling is involved and simulations are run under the likelihood constraint! In the astronomy examples given in the paper, N never exceeds 15… Furthermore, by merging all N’s together, it is unclear how the evidences associated with the various values of N can be computed. At least, those are not reported in the paper.

The paper also omits to provide the likelihood function so I do not completely understand where “label switching” occurs therein. My first impression is that this is not a mixture model. However if the observed signal (from an exoplanetary system) is the sum of N signals corresponding to N planets, this makes more sense.

ice-climbing Niagara Falls

Posted in Mountains, pictures with tags , , , , , , on March 1, 2015 by xi'an

I had missed these news that a frozen portion of the Niagara Falls had been ice-climbed. By Will Gadd on Jan. 27. This is obviously quite impressive given the weird and dangerous nature of the ice there, which is mostly frozen foam from the nearby waterfall. (I once climbed an easy route on such ice at the Chutes Montmorency, near Québec City, and it felt quite strange…) He even had a special ice hook designed for that climb as he did not trust the usual ice screws. Will Gadd has however climbed much more difficult routes like Helmcken Falls in British Columbia, which may be the hardest mixed route in the World!

Ubuntu issues

Posted in Kids, Linux with tags , , , on February 28, 2015 by xi'an

screen shot with ubuntu 10.10It may be that weekends are the wrong time to tamper with computer OS… Last Sunday, I noticed my Bluetooth icon had a “turn off” option and since I only use Bluetooth for my remote keyboard and mouse when in Warwick, I turned it off, thinking I would turn it on again next week. This alas led to a series of problems, maybe as a coincidence since I also updated the Kubuntu 14.04 system over the weekend.

  1. I cannot turn Bluetooth on again! My keyboard and mouse are no longer recognised or detected. No Bluetooth adapter is found by the system setting. Similarly, sudo modprobe bluetooth shows nothing. I have installed a new interface called Blueman but to no avail. The fix suggested on forums to run rfkill unblock bluetooth does not work either… Actually rfkill list all only returns the wireless device. Which is working fine.
  2. My webcam vanished as well. It was working fine before the weekend.
  3. Accessing some webpages, including all New York Times articles, now takes forever on Firefox! If less on Chrome.

Is this a curse of sorts?!

As an aside, I also found this week that I cannot update Adobe reader from version 9 to version 11, as Adobe does not support Linux versions any more… Another bummer. If one wants to stick to acrobat.

Update [03/02]

Thanks to Ingmar and Thomas, I got  both my problems solved! The Bluetooth restarted after I shut down my unplugged computer, in connection with an USB over-current protection. And Thomas figured out my keyboard had a key to turn the webcam off and on, key that I had pressed when trying to restart the Bluetooth device. Et voilà!

je suis Avijit Roy

Posted in Uncategorized with tags , , , , , on February 27, 2015 by xi'an

আমরা শোকাহত
কিন্তু আমরা অপরাজিত

[“We mourn but we are not defeated”]

Unbiased Bayes for Big Data: Path of partial posteriors [a reply from the authors]

Posted in Statistics, University life with tags , , , , , , , , , on February 27, 2015 by xi'an

[Here is a reply by Heiko Strathmann to my post of yesterday. Along with the slides of a talk in Oxford mentioned in the discussion.]

Thanks for putting this up, and thanks for the discussion. Christian, as already exchanged via email, here are some answers to the points you make.

First of all, we don’t claim a free lunch — and are honest with the limitations of the method (see negative examples). Rather, we make the point that we can achieve computational savings in certain situations — essentially exploiting redundancy (what Michael called “tall” data in his note on subsampling & HMC) leading to fast convergence of posterior statistics.

Dan is of course correct noticing that if the posterior statistic does not converge nicely (i.e. all data counts), then truncation time is “mammoth”. It is also correct that it might be questionable to aim for an unbiased Bayesian method in the presence of such redundancies. However, these are the two extreme perspectives on the topic. The message that we want to get along is that there is a trade-off in between these extremes. In particular the GP examples illustrate this nicely as we are able to reduce MSE in a regime where posterior statistics have *not* yet stabilised, see e.g. figure 6.

“And the following paragraph is further confusing me as it seems to imply that convergence is not that important thanks to the de-biasing equation.”

To clarify, the paragraph refers to the additional convergence issues induced by alternative Markov transition kernels of mini-batch-based full posterior sampling methods by Welling, Bardenet, Dougal & co. For example, Firefly MC’s mixing time is increased by a factor of 1/q where q*N is the mini-batch size. Mixing of stochastic gradient Langevin gets worse over time. This is not true for our scheme as we can use standard transition kernels. It is still essential for the partial posterior Markov chains to converge (if MCMC is used). However, as this is a well studied problem, we omit the topic in our paper and refer to standard tools for diagnosis. All this is independent of the debiasing device.

About MCMC convergence.
Yesterday in Oxford, Pierre Jacob pointed out that if MCMC is used for estimating partial posterior statistics, the overall result is not unbiased. We had a nice discussion how this bias could be addressed via a two-stage debiasing procedure: debiasing the MC estimates as described in the “Unbiased Monte Carlo” paper by Agapiou et al, and then plugging those into the path estimators — though it is (yet) not so clear how (and whether) this would work in our case.
In the current version of the paper, we do not address the bias present due to MCMC. We have a paragraph on this in section 3.2. Rather, we start from a premise that full posterior MCMC samples are a gold standard. Furthermore, the framework we study is not necessarily linked to MCMC – it could be that the posterior expectation is available in closed form, but simply costly in N. In this case, we can still unbiasedly estimate this posterior expectation – see GP regression.

“The choice of the tail rate is thus quite delicate to validate against the variance constraints (2) and (3).”

It is true that the choice is crucial in order to control the variance. However, provided that partial posterior expectations converge at a rate n with n the size of a minibatch, computational complexity can be reduced to N1-α (α<β) without variance exploding. There is a trade-off: the faster the posterior expectations converge, more computation can be saved; β is in general unknown, but can be roughly estimated with the “direct approach” as we describe in appendix.

About the “direct approach”
It is true that for certain classes of models and φ functionals, the direct averaging of expectations for increasing data sizes yields good results (see log-normal example), and we state this. However, the GP regression experiments show that the direct averaging gives a larger MSE as with debiasing applied. This is exactly the trade-off mentioned earlier.

I also wonder what people think about the comparison to stochastic variational inference (GP for Big Data), as this hasn’t appeared in discussions yet. It is the comparison to “non-unbiased” schemes that Christian and Dan asked for.


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