In the field of statistics, the incorporation of new observational information within an existing model with some level of uncertainty is posed as a probabilistic problem, where each piece of information is described by a probability density function. When posed in this way, Bayes' theorem defines the solution. We suggest that the parameterization of microphysics should be viewed as a Bayesian problem. In other words, given the fundamental process-level uncertainty associated with microphysics combined with the considerable amount of observational data now available, we argue it is worthwhile to add elements from statistics explicitly within microphysics schemes in order to rigorously constrain them with observations. Based on this idea, we have developed a novel bulk microphysics scheme to simulate warm precipitating clouds, the Bayesian Observationally-constrained Statistical-physical Scheme (BOSS). Unlike other schemes, BOSS treats parameters and scheme structure alike as flexible, and subject to constraint by observations or prior knowledge. No assumptions are made regarding the structural or functional form of the drop size distribution, or the mathematical form of process rates, which are instead generalized as a sum of power laws. This flexibility allows for the complexity of BOSS to be tailored to any set of observational constraint and underlying theoretical knowledge. For example, BOSS can systematically vary the number and choice of prognostic DSD moments (single-, double-, triple-moment, etc.), and can also vary the number of power law terms used to model microphysical process rates. Here we investigate constraint of BOSS via synthetic polarimetric radar observations within an idealized one-dimensional rain shaft model. Special attention is given to comparing versions of BOSS with varying levels of complexity, and to what extent estimates of uncertainty in BOSS capture errors associated with inadequate microphysical models. Broader implications for using uncertain schemes to investigate microphysical processes within a rain shaft will be discussed.