Dr Harry Desmond

Abstract:

Rotation curves are the key observational manifestation of the dark matter distribution around late-type galaxies. In a halo model context, the precision of constraints on halo parameters is a complex function of properties of the measurements as well as properties of the galaxy itself. Forthcoming surveys will resolve rotation curves to varying degrees of precision, or measure their integrated effect in the HI linewidth. To ascertain the relative significance of the relevant quantities for constraining halo properties, we study the information on halo mass and concentration as quantified by the Kullback–Leibler divergence of the kinematics-informed posterior from the uninformative prior. We calculate this divergence as a function of the different types of spectroscopic observation, properties of the measurement, galaxy properties, and auxiliary observational data on the baryonic components. Using the SPARC (Spitzer Photometry & Accurate Rotation Curves) sample, we find that fits to the full rotation curve exhibit a large variation in information gain between galaxies, ranging from ~1 to ~11 bits. The variation is predominantly caused by the vast differences in the number of data points and the size of velocity uncertainties between the SPARC galaxies. We also study the relative importance of the minimum HI surface density probed and the size of velocity uncertainties on the constraining power on the inner halo density slope, finding the latter to be significantly more important. We spell out the implications of these results for the optimization of galaxy surveys aiming to constrain galaxies’ dark matter distributions, highlighting the need for precise velocity measurements.

Abstract:

When choosing between competing symbolic models for a data set, a human will naturally prefer the “simpler” expression or the one which more closely resembles equations previously seen in a similar context. This suggests a non-uniform prior on functions, which is, however, rarely considered within a symbolic regression (SR) framework. In this paper we develop methods to incorporate detailed prior information on both functions and their parameters into SR. Our prior on the structure of a function is based on a ngram language model, which is sensitive to the arrangement of operators relative to one another in addition to the frequency of occurrence of each operator. We also develop a formalism based on the Fractional Bayes Factor to treat numerical parameter priors in such a way that models may be fairly compared though the Bayesian evidence, and explicitly compare Bayesian, Minimum Description Length and heuristic methods for model selection. We demonstrate the performance of our priors relative to literature standards on benchmarks and a real-world dataset from the field of cosmology.

Abstract:

Symbolic Regression (SR) algorithms attempt to learn analytic expressions which fit data accurately and in a highly interpretable manner. Conventional SR suffers from two fundamental issues which we address here. First, these methods search the space stochastically (typically using genetic programming) and hence do not necessarily find the best function. Second, the criteria used to select the equation optimally balancing accuracy with simplicity have been variable and subjective. To address these issues we introduce Exhaustive Symbolic Regression (ESR), which systematically and efficiently considers all possible equations—made with a given basis set of operators and up to a specified maximum complexity— and is therefore guaranteed to find the true optimum (if parameters are perfectly optimised) and a complete function ranking subject to these constraints. We implement the minimum description length principle as a rigorous method for combining these preferences into a single objective. To illustrate the power of ESR we apply it to a catalogue of cosmic chronometers and the Pantheon+ sample of supernovae to learn the Hubble rate as a function of redshift, finding 40 functions (out of 5.2 million trial functions) that fit the data more economically than the Friedmann equation. These low-redshift data therefore do not uniquely prefer the expansion history of the standard model of cosmology. We make our code and full equation sets publicly available.