As Bayesian buzzwords continue to pop onto arXiv daily, one often runs

across ad hoc methods for Monte Carlo integration of posteriors. Of

course, there are ready-made tools that can be downloaded and easily

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.