Title: Retrospective Higher-Order Markov Processes for User Trails,
Abstract: Users form information trails as they browse the web, checkin with a
geolocation, rate items, or consume media. A common problem is to predict what
a user might do next for the purposes of guidance, recommendation, or
prefetching. First-order and higher-order Markov chains have been widely used
methods to study such sequences of data. First-order Markov chains are easy to
estimate, but lack accuracy when history matters. Higher-order Markov chains,
in contrast, have too many parameters and suffer from overfitting the training
data. Fitting these parameters with regularization and smoothing only offers
mild improvements. In this paper we propose the retrospective higher-order
Markov process (RHOMP) as a low-parameter model for such sequences. This model
is a special case of a higher-order Markov chain where the transitions depend
retrospectively on a single history state instead of an arbitrary combination
of history states. There are two immediate computational advantages: the number
of parameters is linear in the order of the Markov chain and the model can be
fit to large state spaces. Furthermore, by providing a specific structure to
the higher-order chain, RHOMPs improve the model accuracy by efficiently
utilizing history states without risks of overfitting the data. We demonstrate
how to estimate a RHOMP from data and we demonstrate the effectiveness of our
method on various real application datasets spanning geolocation data, review
sequences, and business locations. The RHOMP model uniformly outperforms
higher-order Markov chains, Kneser-Ney regularization, and tensor
factorizations in terms of prediction accuracy.