# NIPS Proceedingsβ

## Fast Bidirectional Probability Estimation in Markov Models

A note about reviews: "heavy" review comments were provided by reviewers in the program committee as part of the evaluation process for NIPS 2015, along with posted responses during the author feedback period. Numerical scores from both "heavy" and "light" reviewers are not provided in the review link below.

[PDF] [BibTeX] [Supplemental] [Reviews]

### Abstract

We develop a new bidirectional algorithm for estimating Markov chain multi-step transition probabilities: given a Markov chain, we want to estimate the probability of hitting a given target state in $\ell$ steps after starting from a given source distribution. Given the target state $t$, we use a (reverse) local power iteration to construct an expanded target distribution', which has the same mean as the quantity we want to estimate, but a smaller variance -- this can then be sampled efficiently by a Monte Carlo algorithm. Our method extends to any Markov chain on a discrete (finite or countable) state-space, and can be extended to compute functions of multi-step transition probabilities such as PageRank, graph diffusions, hitting/return times, etc. Our main result is that in sparse' Markov Chains -- wherein the number of transitions between states is comparable to the number of states -- the running time of our algorithm for a uniform-random target node is order-wise smaller than Monte Carlo and power iteration based algorithms; in particular, our method can estimate a probability $p$ using only $O(1/\sqrt{p})$ running time.