A Variational Approach to Parameter Estimation for Characterizing 2-D Cluster Variation Method Topographies

One of the biggest challenges in characterizing 2-D topographies is succinctly communicating the dominant nature of local configurations. In a 2-D grid composed of bistate units, this could be expressed as finding the characteristic configuration variables such as nearest-neighbor pairs and triplet combinations. The 2-D cluster variation method (CVM) provides a theoretical framework for associating a set of configuration variables with only two parameters, for a system that is at free energy equilibrium. This work presents a method for determining which of many possible two-parameter sets provides the ``most suitable'' match for a given 2-D topography, drawing from methods used for variational inference. This particular work focuses exclusively on topographies for which the activation enthalpy parameter (epsilon_0) is zero, so that the distribution between two states is equiprobable. This condition is used since, when the two states are equiprobable, there is an analytic solution giving the configuration variable values as functions of the h-value, where we define h in terms of the interaction enthalpy parameter (epsilon_1) as h = exp(2*epsilon_1). This allows the computationally-achieved configuration variable values to be compared with the analytically-predicted values for a given h-value. The method is illustrated using four patterns derived from three different naturally-occurring black-and-white topographies, where each pattern meets the equiprobability criterion. We achieve expected results, that is, as the patterns progress from having relatively low numbers of like-near-like nodes to increasing like-near-like masses, the h-values for each corresponding free energy-minimized model also increase. Further, the corresponding configuration variable values for the (free energy-minimized) model patterns are in approximate alignment with the analytically-predicted values.