  We introduce a new class of semiparametric latent variable models for long
memory discretized event data. The proposed methodology is motivated by a study
of bird vocalizations in the Amazon rain forest; the timings of vocalizations
exhibit self-similarity and long range dependence ruling out models based on
Poisson processes. The proposed class of FRActional Probit (FRAP) models is
based on thresholding of a latent process consisting of an additive expansion
of a smooth Gaussian process with a fractional Brownian motion. We develop a
Bayesian approach to inference using Markov chain Monte Carlo, and show good
performance in simulation studies. Applying the methods to the Amazon bird
vocalization data, we find substantial evidence for self-similarity and
non-Markovian/Poisson dynamics. To accommodate the bird vocalization data, in
which there are many different species of birds exhibiting their own
vocalization dynamics, a hierarchical expansion of FRAP is provided in
Supplementary Materials.
