Dr Harry Desmond

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.

Abstract:

We set constraints on the dark matter halo mass and concentration of ∼22 000 individual galaxies visible both in H I (from the ALFALFA survey) and optical light (from the Sloan Digital Sky Survey). This is achieved by combining two Bayesian models, one for the H I line width as a function of the stellar and neutral hydrogen mass distributions in a galaxy using kinematic modelling, and the other for the galaxy’s total baryonic mass using the technique of inverse subhalo abundance matching. We hence quantify the constraining power on halo properties of spectroscopic and photometric observations, and assess their consistency. We find good agreement between the two sets of posteriors, although there is a sizeable population of low-line width galaxies that favour significantly smaller dynamical masses than expected from abundance matching (especially for cuspy halo profiles). Abundance matching provides significantly more stringent bounds on halo properties than the H I line width, even with a mass–concentration prior included, although combining the two provides a mean gain of 40 per cent for the sample when fitting an NFW profile. We also use our kinematic posteriors to construct a baryonic mass–halo mass relation, which we find to be near power law, and with a somewhat shallower slope than expected from abundance matching. Our method demonstrates the potential of combining photometric and spectroscopic observations to precisely map out the dark matter distribution at the galaxy scale using upcoming H I surveys such as the SKA.