A Determination-Based Correlation Coefficient for Multivalued Dependencies via Bayesian Mixture Models

Measuring statistical dependence between two random variables is a fundamental problem of data analysis, yet no single measure captures every type of relationship while remaining interpretable. Classical coefficients detect only narrow classes of dependence: linear for Pearson, monotonic for Spearman. More general measures are either computationally expensive, prone to overestimation, or uninformative about the structure of the relationship. This paper makes two contributions. First, we propose a unified framework for comparing dependence measures of different mathematical natures, based on the functional class of a coefficient and a set of linking properties anchored to the Pearson correlation as a common reference. Second, building on the correlation ratio, we develop a determination-based correlation coefficient (DBCC) that captures multivalued dependencies, in which a single value of the predictor may correspond to several states of the response. The coefficient decomposes the conditional distribution within local bins through a soft-assignment Bayesian Gaussian mixture model, replacing the discontinuous mode-seeking of an earlier kernel-based realisation. We prove the consistency of the estimator via an oracle decomposition that separates the algorithmic and statistical components of the error, and we show that the coefficient is computed in linear time. The construction retains the transparent variance-explained interpretation of the classical coefficient of determination $R^{2}$.