#### A short note on Probability for astronomers

I often feel irksome whenever I see a function being normalized over a feasible parameter space and it being used as a probability density function (pdf) for further statistical inference. In order to be a suitable pdf, normalization has to be done over a measurable space not over a feasible space. Such practice often yields biased best fits (biased estimators) and improper error bars. On the other hand, validating a measurable space under physics seems complicated. To be precise, we often lost in translation.

When I was teaching statistics, despite undergraduate courses, there were both undergraduate and graduate students of various fields except astrophysics majors. I wondered why they were not encouraged to take some basic statistics whereas they were encouraged to take some computer science courses. As there are many astronomers good at programming and designing tools, I’m sure that recommending students to take statistics courses will renovate astronomical data analysis procedures (beyond Bevington’s book) and hind theories (statistics and mathematics per se, not physics laws).

Here’s an interesting lecture for developing a curriculum for the new era in computer science and why the basic probability theory and statistics is important to raise versatile computer scientists. It could be a bit out dated now because I saw it several months ago.

About a little more than the half way through the lecture, he emphasizes that probability course partaking the computer science curriculum. I wonder any astronomy professor has similar arguments and stresses for any needs of basic probability theories to be learned among young future astrophysicists in order to prevent many statistics misuses appearing in astronomical literature. Particularly confusions between fitting (estimating) and inference (both model assessment and uncertainty quantification) are frequently observed in literature where authors claim their superior statistics and statistical data analysis. I personally sometimes attribute this confusion to the lack of distinction between what is random and what is deterministic, or strong believe in their observed and processed data absent from errors and probabilistic nature.

Many introductory books introduce very interesting problems many of which have some historical origins to introduce probability theories (many anecdotes). One can check out the very basics, probability axioms, and measurable function from wikipedia. With examples, probability is high school or lower level math that you already know but with jargon you’ll like to recite lexicons many times so that you are get used to foundations, basics, and their theories.

We often mention measurable to discuss random variables, uncertainties, and distributions without verbosity. “Assume measurable space … ” saves multiple paragraphs in an article and changes the structure of writing. This short adjective implies so many assumptions depending on statistical models and equations that you are using for best fits and error bars.

Consider a LF, that is truncated due to observational limits. The common practice I saw is drawing a histogram in a way that the adaptive binning makes the overall shape reflecting a partial bell shape curve. Thanks to its smoothed look, scientists impose a gaussian curve to partially observed data and find parameter estimates that determine the shape of this gaussian curve. There is no imputation step to fake unobserved points to comprise the full probability space. The parameter space of gaussian curves frequently does not coincide with the physically feasible space; however, such discrepancy is rarely discussed in astronomical literature and subsequent biased results look like a taboo.

Although astronomers emphasize the importance of uncertainties, factorization nor stratification of uncertainties has never been clear (model uncertainty, systematic uncertainty or bias, statistical uncertainties or variance). Hierarchical relationships or correlations among these different uncertainties are never addressed in a full measure. Basics of probability theory and the understanding of random variables would help to characterize uncertainties both in mathematical sense and astrophysical sense. This knowledge also assist appropriate quantification of these characterized uncertainties.

Statistical models are rather simple compared to models of astrophysics. However, statistics is the science of understanding uncertainties and randomness and therefore, some strategies of transcribing from complicated astrophysical models into statistical models, in order to reflect the probabilistic nature of observed (or parameters, for Bayesian modeling), are necessary. Both raw or processed data manifest the behavior of random variables. Their underlying processes determine not only physics models but also statistical models written in terms of random variables and the link functions connecting physics and uncertainties. To my best understanding, bridging and inventing statistical models for astrophysics researches seem tough due to the lack of awareness of basics of probability theory.

Once I had a chance to observe a Decadal survey meeting, which covered so diverse areas in astronomy. They discussed new projects, advancing current projects, career developments, and a little bit about educating professional astronomers apart from public reach (which often receives more importance than university curriculum. I also believe that wide spread public awareness of astronomy is very important). What I missed while I observing the meeting is that interdisciplinary knowledge transferring efforts to broaden the field of astronomy and astrophysics nor curriculum design ideas. Because of its long history, I thought astronomy is a science of everything. Marching a path for a long time made astronomy more or less the most isolated and exclusive science.

Perhaps asking astronomy majors taking multiple statistics courses is too burdensome; therefore being taught by faculty who are specialized in (statistical) data analysis organizes a data analysis course and incorporates several hours of basic probability is more realistic and what I anticipate. With a few hours of bringing fundamental notions in random variables and probability, the claims of “statistical rigorous methods and powerful results” will become more appropriate. Currently, statistics is science but in astronomy literature, it looks more or less like an adjective that modify methods and results like “powerful”, “superior”, “excellent”, “better”, “useful,” and so on. Basics of probability is easily incorporated into introduction of algorithms in designing experiments and optimization methods, which are currently used in a brute force fashion[1].

Occasionally, I see gems from arxiv written by astronomers. Their expertise in astronomy and their interest in statistics has produced intriguing accounts for statistically rigorous data analysis and inference procedures. Their papers includes explanation of fundamentals of statistics and probability more appropriate to astronomers than statistics textbooks for scientists and engineers of different fields. I wish more astronomers join this venture knowing basics and diversities of statistics to rectify many unconscious misuses of statistics while they argue that their choice of statistics is the most powerful one thanks to plausible results.

1. What I mean by a brute force fashion is that trying all methods listed in the software manual, and then later, stating that the method A gave most plausible values that matches with data in a scatter plot[]