The aim of the notes is to demonstrate the potential for ideas in machine learning to impact on the fields of inverse problems and data assimilation. The perspective is one that is primarily aimed at researchers from inverse problems and/or data assimilation who wish to see a mathematical presentation of machine learning as it pertains to their fields. As a by-product of the presentation we present a succinct mathematical treatment of various topics in machine learning. The material on machine learning, along with some other related topics, is summarized in Part III, Appendix. Part I of the notes is concerned with inverse problems, employing material from Part III; Part II of the notes is concerned with data assimilation, employing material from Parts I and III.

Schneider et al. (Reports, 5 January 2018, p. 69) used an ad hoc statistical method in their calculation of the stellar initial mass function. Adopting an improved approach, we reanalyze their data and determine a power-law exponent of . Alternative assumptions regarding dataset completeness and the star formation history model can shift the inferred exponent to and, respectively. They estimate the ages and masses of individual stars with the BONNSAI Bayesian code (3), then obtain an overall mass distribution by effectively adding together the posterior probability density functions of individual stars. There is no statistical meaning to a distribution obtained in this way, which does not represent the posterior probability density function on the mass distribution. Hierarchical Bayesian inference provides the statistically justified solution to this problem (4).