  In this work, we will investigate a Bayesian approach to estimating the
parameters of long memory models. Long memory, characterized by the phenomenon
of hyperbolic autocorrelation decay in time series, has garnered significant
attention. This is because, in many situations, the assumption of short memory,
such as the Markovianity assumption, can be deemed too restrictive.
Applications for long memory models can be readily found in fields such as
astronomy, finance, and environmental sciences. However, current parametric and
semiparametric approaches to modeling long memory present challenges,
particularly in the estimation process.
  In this study, we will introduce various methods applied to this problem from
a Bayesian perspective, along with a novel semiparametric approach for deriving
the posterior distribution of the long memory parameter. Additionally, we will
establish the asymptotic properties of the model. An advantage of this approach
is that it allows to implement state-of-the-art efficient algorithms for
nonparametric Bayesian models.
