## Polarizing Bayesian approaches: why literature matters [Guest post: J. Michael Burgess]

As Bayesian buzzwords continue to pop onto arXiv daily, one often runs
across ad hoc methods for Monte Carlo integration of posteriors. Of
used for most problems, still, one occasionally finds the lack
literacy in statistics leading to methods which are, to use the words of
Ewan, without principle. Strangely, there is a concentration of
confusion concerning Bayesian methods in x-ray polarization
studies. Much of this comes from the classic misunderstanding of
credible regions and confidence regions. You’ll see astronomers
worried about not being able to detect zero polarization as it is on
a parameter boundary. This lead to some Bayesian magic here  that touches a bit on the issue, but applies only to a very fragile and
specific type of polarization data: one where the data are directly
latent polarization and everything is beautifully Gaussian. In other
words, not x-ray data! The point of the work was to derive an analytic
posterior for these types of measurements. We’ll come back to this.

Nevertheless, the idea was picked up by x-ray astronomers and non-principled methods have taken over. For example, here is an application that ignores all the fragility inherent in Poisson distributed data to derive an “analytic” posterior assuming
that low count Poisson data are Gaussian; something that is simply
not true. But, as always, throw enough fancy statistics buzzwords in
and no one will notice. But the real problem with the use of the work
of Vallincourt for x-ray data is that Vallincourt derived a
posterior for the measurements. Assuming a prior and Gaussian
likelihood, the Rice distribution was derived. But the works above,
and these here use the Rice distribution as the likehood!
They place
further priors on their parameters, do things like background subtract
Poisson data, and run some form of random number generator to obtain
trace plots. It is difficult to understand why these approaches are
wrong as they are simply made up via some form of heuristic intuition
and missing the core concepts of what a Bayesian analysis is. But the
words Bayesian are in the papers, so it must be sophisticated, right?

As a side note, we attempted to solve this issue here  by deriving
the likelihood, a conditional Poisson likelihood, and performing Bayesian
estimation of the latent parameters. We, of course, recover the Rice
distribution; because that is the posterior.

And yet, this week we had a new heuristic approach to Bayesian
estimation for x-ray polarization here. There are many
misconceptions in the work, but we can concentrate on the issues with
statistics. First, there is a claim of doing Bayesian estimation to
derive confidence intervals. The fun part is the appendix on how the
Monte Carlo was performed. It appears the authors drew random values
of the data from Gaussian distributions (yes, it is Poisson data),
then perform a least-squares fit to this randomly drawn data for the
parameter values… all stop… what the <insert favorite Bayesian
buzzword here>. Thus, we have a long way to go in communicating how
perform Bayesian analysis. Even if you can download every tested
sampler, easily write a Metropolis-Hastings algorithm, or just google
for a few interesting Bayesian blog posts. Intuition is a great tool,
principled Monte Carlo integration techniques are not proven on
intuition alone. When in doubt, consult a statistician.

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