the curious incident of the inverse of the mean

Posted in R, Statistics, University life with tags , , , on July 15, 2016 by xi'an

A s I figured out while working with astronomer colleagues last week, a strange if understandable difficulty proceeds from the simplest and most studied statistical model, namely the Normal model

x~N(θ,1)

Indeed, if one reparametrises this model as x~N(υ⁻¹,1) with υ>0, a single observation x brings very little information about υ! (This is not a toy problem as it corresponds to estimating distances from observations of parallaxes.) If x gets large, υ is very likely to be small, but if x is small or negative, υ is certainly large, with no power to discriminate between highly different values. For instance, Fisher’s information for this model and parametrisation is υ⁻² and thus collapses at zero.

While one can always hope for Bayesian miracles, they do not automatically occur. For instance, working with a Gamma prior Ga(3,10³) on υ [as informed by a large astronomy dataset] leads to a posterior expectation hardly impacted by the value of the observation x:

And using an alternative estimate like the harmonic posterior mean that is associated with the relative squared error loss does not see much more impact from the observation:

There is simply not enough information contained in one datapoint (or even several datapoints for all that matters) to infer about υ.

EQUIP launch

Posted in Statistics, University life with tags , , , , , , on October 10, 2013 by xi'an

Today, as I was around (!), I attended the launch of the new Warwick research project EQUIP (which stands for Enabling quantification of uncertainty for inverse problems). This is an EPSRC funded project merging mathematics, numerical analysis, statistics and geophysics, with a primary target application [alas!] in the oil industry. It will start hiring four (4!) postdocs pretty soon. The talks were all interesting, but I particularly liked the idea that they were addressed primarily to students who were potentially interested in the positions. In addition, Mark Girolami gaves a most appreciated insight on the modelling of uncertainty in PDE models, connecting with earlier notions set by Tony O’Hagan, modelling that I hope we can discuss further when both in Warwick!

MCMC at ICMS (3)

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , on April 26, 2012 by xi'an

The intense pace of the two first days of our workshop on MCMC at ICMS had apparently taken an heavy toll on the participants as a part of the audience was missing this morning! Although not as a consequence of the haggis of the previous night at the conference dinner, nor even as a result of the above pace. In fact, the missing participants had opted ahead of time for leaving the workshop early, which is understandable given everyone’s busy schedule, esp. for those attending both Bristol and Edinburgh workshops, however slightly impacting the atmosphere of the final day. (Except for Mark Girolami who most unfortunately suffered such a teeth infection that he had to seek urgent medical assistance yesterday afternoon. Best wishes to Mark for a prompt recovery, say I with a dental appointment tomorrow…!)

$y=\sqrt{1-\beta^2}x_{t-1}+\beta\zeta\quad 0<\beta<1,\zeta\sim\varphi(|\zeta|)$