Efficient algorithms for modifying and sampling from a categorical distribution

Probabilistic programming languages and other machine learning applications often require samples to be generated from a categorical distribution where the probability of each one of $n$ categories is specified as a parameter. If the parameters are hyper-parameters then they need to be modified, however, current implementations of categorical distributions take $\mathcal{O}(n)$ time to modify a parameter. If $n$ is large and the parameters are being frequently modified, this can become prohibitive. Here we present the insight that a Huffman tree is an efficient data structure for representing categorical distributions and present algorithms to generate samples as well as add, delete and modify categories in $\mathcal{O}(\log(n))$ time. We demonstrate that the time to sample from the distribution remains, in practice, within a few percent of the theoretical optimal value. The same algorithm may also be useful in the context of adaptive Huffman coding where computational efficiency is important.