On the significance of the reference ratio method in inferential structure determination of biomolecules

The inference of biomolecular structure from biophysical data is an important task in molecular biology. Rigorous Bayesian inference requires the formulation of a joint posterior distribution regarding structure and nuisance parameters given the observed data. The relationship between biomolecule, typically a vector of atomic coordinates, and data is typically many-to-one. Consequently, forward models are needed to relate a given biomolecular structure to its associated data. Thus, the calculation of the likelihood involves projecting a high dimensional manifold to a lower dimensional one, respectively concerning structure and data. This projection is a reduced or coarse grained representation of the structure. Given the nature of the data obtained from the biophysical experiments, the use of prior distributions concerning biomolecular structure is indispensable. A prior on biomolecular structure necessarily also induces a prior on its reduced or coarse-grained representation. We call this induced prior the reference distribution. The reference distribution induced by a fine-grained prior is typically assumed to be suitable for the coarse-grained variable. Often, this assumption is invalid. Here, we quantify the impact of the induced reference distribution on the posterior distribution and discuss its possible implications. 1 Background Bio-molecular function is closely connected to structure. Consequently, the inference of structure from biophysical experiments is an important problem in molecular biology. However, the procedure is complicated by the nature of the experimental data which, are incomplete, averaged and subject to experimental noise. This fact makes it difficult, if not impossible, to use experimental observations to determine structures without the use of strong prior information. The determination of bio-molecular structure usually concerns an atomic-level, or finegrained representation, x ∈ N . The experimental observations, d, provide information of some projection f of x. The relationship between f and x is given by a forward model as f = F(x,θ) ∈ M where F : N ,O → M, dim(N ) >> dim(M), with nuisance parameters θ ∈ O. That is, there is a deterministic relationship between the fine-grained x and coarsegrained representation f , through F . In a practical structure inference setting, prior information on the fine-grained space N is typically introduced to compensate for the noisy and incomplete nature of experimental data. This inadvertently results in a prior distribution on the coarse-grained space M, as a consequence of the variables’ deterministic relationship. While the prior information about N may seem entirely appropriate, this is not guaranteed for the resulting induced prior distribution on M. Moreover, if the induced prior is inappropriate, other free parameters, such as θ ∈ O, may compensate for this. This will in turn introduce a bias which may hamper the direct interpretation of θ. Here, the aim is to demonstrate that the prior information used needs to be appropriate with respect to both N and M.