Deep Neural Networks (DNNs) deployed to the real world are regularly subject to out-of-distribution (OoD) data, various types of noise, and shifting conceptual objectives. This paper proposes a framework for adapting to data distribution drift modeled by benchmark Continual Learning datasets. We develop and evaluate a method of Continual Learning that leverages uncertainty quantification from Bayesian Inference to mitigate catastrophic forgetting. We expand on previous approaches by removing the need for Monte Carlo sampling of the model weights to sample the predictive distribution. We optimize a closed-form Evidence Lower Bound (ELBO) objective approximating the predictive distribution by propagating the first two moments of a distribution, i.e. mean and covariance, through all network layers. Catastrophic forgetting is mitigated by using the closed-form ELBO to approximate the Minimum Description Length (MDL) Principle, inherently penalizing changes in the model likelihood by minimizing the KL Divergence between the variational posterior for the current task and the previous task's variational posterior acting as the prior. Leveraging the approximation of the MDL principle, we aim to initially learn a sparse variational posterior and then minimize additional model complexity learned for subsequent tasks. Our approach is evaluated for the task incremental learning scenario using density propagated versions of fully-connected and convolutional neural networks across multiple sequential benchmark datasets with varying task sequence lengths. Ultimately, this procedure produces a minimally complex network over a series of tasks mitigating catastrophic forgetting.