Abstract

Bayesian inference on sparse multinomial data using a smoothed Dirichlet distribution is a powerful technique in statistics and machine learning, particularly for analyzing categorical data. The multinomial distribution is commonly employed to model scenarios where each observation belongs to one of several distinct categories. In Bayesian inference, the multinomial likelihood function is often used to model observed data, but when the data is sparse—where some categories have few or no observations—this can lead to unreliable estimates. Smoothing techniques are essential to address this challenge by stabilizing the estimates.

In this work, we develop a Bayesian framework for estimating multinomial cell probabilities by incorporating a smoothed Dirichlet prior. The key advantage of this prior is that it induces a smoothing effect, encouraging the probabilities of neighboring cells to be closer to each other compared to the standard Dirichlet prior. This smoothing mechanism is particularly beneficial in cases of sparsity, where traditional methods may struggle.

Our Bayesian approach introduces shrinkage estimators for multinomial cell probabilities under conditions of sparsity. These estimators effectively borrow strength across other multinomial populations and categories, enhancing the reliability and accuracy of the estimated probabilities. To illustrate the effectiveness of our proposed method, we apply it to the analysis of COVID-19 data, specifically examining the distribution of cases across different age groups within Canadian health regions. This application demonstrates how the smoothed Dirichlet prior can significantly improve estimation in real-world, sparse data scenarios, offering a robust tool for statisticians and data scientists dealing with similar challenges.