Density Estimation via Measure Transport: Outlook for Applications in the Biological Sciences

The problem of estimating a probability distribution density from samples (e.g., observations, measurements, or simulation data) is ubiquitous in data science, uncertainty quantification, clustering and classification, and probabilistic modeling and inference tasks. Moreover, it is common among various scientific and engineering fields, including biology [38, 14, 1, 41, 5, 7, 12, 39]. Often, wellknown parametric density functions (dependent on few parameters), such as the Gaussian or Weibull density distribution functions, are adopted. While this may simplify certain tasks (e.g., computational ones), many of these known density distribution functions are not necessarily suitable for characterizing data that exhibit complex features, such as (spatial and/or temporal) correlations and non-Gaussian characteristics. For instance, as reported in [7], accounting for differences in the distribution densities of gene expressions can lead to improved interpretation of cancer transcriptomic data. Hence, a density estimation framework capable of characterizing a diverse range of properties is highly desirable. A measure transport approach [44, 37, 36] offers this possibility. Optimal measure transport, broadly defined, deals with the problem of minimizing the cost of transporting one (probability) measure to another.