I am an astronomer studying galaxies. I have observations of about 500 galaxies. For each galaxy, I can measure two quantities (call them $X$ and $Y$).

I want to compare my observations to some theoretical (physical, not statistical) models. I have about 10,000 numerical models of all kinds of galaxies (describing their size, structure, and observable properties), some of which are quite different than the ones I am studying.

I want to "thin down" the sample of theoretical models until the distributions of $X$ and $Y$ across all the models look like the observed distributions of $X$ and $Y$. Is there a good algorithm to do this?

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    $\begingroup$ Why would you want to do this? What would you learn? $\endgroup$
    – jbowman
    Jan 8 '19 at 14:55
  • $\begingroup$ @jbowman My telescope is not sensitive to certain things, and so the sample of galaxies I have observed are not representative of the true sample of galaxies in the Universe. In astronomy we refer to this as the selection function. Galactic studies actually usually perform this step, but manually and often in some ad hoc way. So I was just wondering if the statisticians have a better way of doing it... $\endgroup$ Jan 8 '19 at 14:57
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    $\begingroup$ Maybe you should edit the question to clarify exactly what a "model" is. In Statistics the term "model" usually refers to a probability distribution that generates data. $\endgroup$ Jan 8 '19 at 16:39
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    $\begingroup$ Couldn't you view your set of physical models as describing (more or less) a space of statistical models in which observational error is incorporated? That would make your situation amenable to standard approaches, of which the most appealing might be Maximum Likelihood: simply compute the likelihood of your data for each of your models, identify the one that produces the largest likelihood, and select those those likelihoods are reasonably close to the largest one (using the usual chi-squared theory). $\endgroup$
    – whuber
    Jan 9 '19 at 14:46
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    $\begingroup$ I'm having a very hard time making sense of that request, because I understand all models to differ: no more than one can "belong to the same population." Regardless, my suggestion was not to select the best fit, but to select a subset of models that all fit reasonably well compared to the best fit. $\endgroup$
    – whuber
    Jan 9 '19 at 16:24

There are many statistical approaches to gauging the match between observed and theoretical distributions. Now, with thousands of distributions to evaluate, you will need to devise some automated procedures, each to be run as many times as there are models you want to test. If you use R or are willing to learn, this should be workable, perhaps with some advice from an expert.

To get you started in this effort, you could try the descdist() function in the R package fitdistrplus. It helps characterize a distribution. For a good, detailed example of a way it can be used, see this answer by COOLSerdash. It draws on the Cullen and Frey graphing technique as well as the more commonly used Kolmogorov-Smirnov test [available in base R via the function ks.test()].

Another, briefer source on descdist(), perhaps another good place to start, is bill_080's answer.

  • $\begingroup$ Hi @rolando2, thanks for your answer. Yes I am happy to program in R. I am familiar with the Kolmogorov-Smirnov test and others, but (as far as I know) what they will tell me is whether the distributions match or not (sorry if I am "handwaving" this, but you know what I mean). What I am wondering is how to select a subset such that the KS test (or another test) says that they match? $\endgroup$ Jan 8 '19 at 15:40
  • $\begingroup$ One algorithm I can think of is: while (True) select a random subset of models of random size, compute the KS test, and terminate if they match below a tolerance, otherwise repeat. However, this algorithm may never terminate. $\endgroup$ Jan 8 '19 at 15:42
  • $\begingroup$ Even if this algorithm terminates, the resulting subset is not necessarily distributed from the distribution of interest. $\endgroup$
    – Xi'an
    Apr 16 '19 at 6:58

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