Risk-sensitive filtering
involves minimisation of the expectation of an exponential in quadratic cost
criteria. As opposed to $L_2$ filtering, (termed as {\em risk-neutral filtering } in \cite{SM1}), which achieves the minimisation of a quadratic error
criteria, risk-sensitive filtering robustifies the filter against plant and
noise uncertainties by penalising all the higher-order moments of the
estimation error energy. It also allows a trade-off between optimal filtering
for the nominal model case and the average noise
situation, and robustness to worst case noise and model uncertainty by
weighting the index of the exponential by a risk-sensitive parameter.
The risk-sensitive filtering problem has been addressed for linear Gauss-Markov
signal models in \cite{Speyer}. In a
companion paper \cite{SM1} to the present
one, the problem has been solved
for a general class of discrete-time nonlinear state space signal models
via the so-called reference probability method and the linear Gauss-Markov
signal model has been treated as a special case. It has been seen that
risk-sensitive filters are closely related to $H_{\infty}$ filters \cite{jrs1}.
Also, related risk-sensitive control problems are abundant in literature
\cite{EMoore} \cite{Iain}
\cite{JBE} \cite{Whittle}.
The problem of extracting
finite-state homogeneous
Markov chains hidden in white Gaussian noise
has been studied as an off-line estimation problem using the well known
Expectation Maximisation (EM) algorithm \cite{Baum}
\cite{demp} \cite{Rabiner}.
On-line
estimation schemes for
Hidden Markov Models (HMM)
have been given in \cite{Iain1} \cite{Vmoore}.
In all these estimation schemes, the so-called
``forward variable'' \cite{Rabiner}
is the true filtered estimate which is also a conditional expectation of the
state at a certain point of time given the observations up and until that
point. The smoothed estimate of the state is obtained as a maximum-likelihood
estimate based on a fixed set of observations.
These filtering schemes are essentially related to risk-neutral
filtering for HMMs.
In this paper, we address the problem of risk-sensitive filtering and
smoothing for discrete-time Hidden Markov Models with finite-discrete states.
We derive information-state filters which are linear and finite-dimensional.
The optimising state estimate is given as the minimising argument of a
finite-dimensional sum. Also, the backward filters and unnormalised smoothed
conditional probability measures are derived.
The derivation techniques are based on
a reference probability method which has been developed in \cite{MElliot} and
used in \cite{EMoore} \cite{Iain} \cite{SM1}.
In Section 2, we describe the Hidden Markov Model, formally define
the risk-sensitive filtering problem, and then deal with the change of measure
and reformulation of the problem
in the new probability measure to achieve the
filtering and smoothing results.
In Section 3, we establish the connection
between risk-sensitive and risk-neutral filtering and Section 4 presents some
concluding remarks.