Conditional Density Estimation with Deep Neural Networks

Abstract

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Estimating conditional density functions is a fundamental problem in statistics.This task is crucial for understanding the underlying relationships between variables and for making informed predictions in various applications.In this paper, we introduce a novel deep nonparametric approach for estimating conditional density functions from data.Our method leverages the flexibility and expressiveness of deep neural networks to model the conditional density without imposing restrictive parametric assumptions.We formulate the problem of conditional density estimation as a nonparametric least squares problem, which allows us to harness the strengths of deep learning in a principled manner.By framing the problem this way, we can effectively utilize deep neural networks to approximate the conditional density function.We demonstrate that our proposed approach achieves the minimax optimal convergence rate for conditional density estimation.Additionally, we show that the convergence rate can be further improved for high-dimensional data satisfying a low-dimensional manifold assumption.To validate the performance of our approach, we conduct extensive numerical evaluations on both simulated and real-world datasets.These experiments reveal that our method consistently outperforms several established techniques, highlighting its superior accuracy and robustness in diverse scenarios.