  In forecasting problems it is important to know whether or not recent events
represent a regime change (low long-term predictive potential), or rather a
local manifestation of longer term effects (potentially higher predictive
potential). Mathematically, a key question is about whether the underlying
stochastic process exhibits "memory", and if so whether the memory is "long" in
a precise sense. Being able to detect or rule out such effects can have a
profound impact on speculative investment (e.g., in financial markets) and
inform public policy (e.g., characterising the size and timescales of the earth
system's response to the anthropogenic CO2 perturbation). Most previous work on
inference of long memory effects is frequentist in nature. Here we provide a
systematic treatment of Bayesian inference for long memory processes via the
Autoregressive Fractional Integrated Moving Average (ARFIMA) model. In
particular, we provide a new approximate likelihood for efficient parameter
inference, and show how nuisance parameters (e.g., short memory effects) can be
integrated over in order to focus on long memory parameters and hypothesis
testing more directly than ever before. We illustrate our new methodology on
both synthetic and observational data, with favorable comparison to the
standard estimators.
