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Nonparametric Density Estimation via Diffusion Mixing

Suppose we are given empirical data and a prior density about the distribution of the data. We wish to construct a nonparametric density estimator that incorporates the prior information. We propose an estimator that allows for the incorporation of prior information in the density estimation procedure within a non-Bayesian framework. The prior density is mixed with the available empirical data via a Langevin diffusion process. The diffusion process is constructed so that the prior density is the limiting and stationary distribution of the process. We analyze the asymptotic bias and variance properties of the estimator and compare them with the properties of the standard density estimators. We present simulation examples in which the proposed estimator outperforms the standard estimation procedures in terms of accuracy.