This thesis considers the problem of density estimation when the variables of interest are subject to measurement error. The measurement error is assumed to be additive and homoscedastic. We specify the density of interest by a Dirichlet Process Mixture Model and establish variational approximation approaches to the density deconvolution problem. Gaussian and Laplacian error distributions are considered, which are representatives of supersmooth and ordinary smooth distributions, respectively. We develop two variational approximation algorithms for Gaussian error deconvolution and one variational approximation algorithm for Laplacian error deconvolution. Their performances are compared to deconvoluting kernels and Monte Carlo Markov Chain method by simulation experiments. A conjecture based on hidden variables categorization is proposed to explain why two variational approximation algorithms for Gaussian error deconvolution perform differently. We establish a stochastic variational approximation algorithm for Gaussian error deconvolution, which improves the performance of variational approximation algorithm and performs as well as MCMC method at faster speed. The stochastic variational approximation algorithm is applied to simulation experiments and an example of physical activity measurements.