\section{Limiting Results}
In this section, we consider the case when the risk-sensitive parameter
$\theta$ approaches $0$. It is known that for linear and nonlinear state-space
signal models with continuous-range states, the risk-neutral (or
$L_2$) filtering
problem is recovered as a special case of risk-sensitive filtering when
$\theta \rightarrow 0$ \cite{SM1}. Since, in the case of $L_2$ filtering,
we know that the conditional mean estimate is the minimum variance estimate,
we can define the unnormalised information state $\alpha_k(e_j)$ as
$$
\alpha_k(e_j) = \bar E [\bar \Lambda_{k} \theta \exp (\theta \hat \Psi_{0,k-1}
)
<{\cal X}_k, e_j> | {\cal Y}_k] $$ instead of using Definition \ref{def:alphahmm}. It is not difficult to show that with this definition, $\alpha_k(e_j)$
will obey the following recursion
$$
\alpha_{k+1}(e_j) = \frac{\phi_{k+1}(y_{k+1} - C(e_j))}{\phi_{k+1}(y_{k+1})}
\exp \left( \frac{\theta}{2} (e_j - \hat {\cal X}_{k})^{\p} Q_{k+1} (e_j - \hat {\cal X}_{k}) \right) \sum_{i=1}^N a_{ij} \alpha_k(e_i) $$
It follows immediately that as $\theta \rightarrow 0$, we have
$$
\alpha_{k+1}(e_j) = \frac{\phi_{k+1}(y_{k+1} - C(e_j))}{\phi_{k+1}(y_{k+1})}
\sum_{i=1}^N a_{ij} \alpha_k(e_i) $$
which is the well-known risk-neutral recursive filter for HMMs.
A variation of this
recursion in the so-called forward variable (analogous to $\alpha_k(e_j)$ here) appears in \cite{Rabiner}.
The MAP estimate is defined as $\hat X_k = e_{m^*},\; m^* = \argmax_{m}
\alpha_k(e_m)$.
This implies that the standard filtering equations for HMMs can be obtained
as a special case of the risk-sensitive filtering equations when
$\theta \rightarrow 0$.
\section{Conclusion}
The problem of discrete-time filtering and smoothing for Hidden Markov Models
with finite-discrete states with an exponential of quadratic cost criteria,
termed risk-sensitive filtering in \cite
{SM1} is addressed in this paper using the
reference probability method. A new probability measure is defined where observations are {\em i.i.d} and the reformulated cost-criteria is
minimised to give filtering and smoothing results for HMMs.
Finite-dimensional linear recursions are obtained in the
information state.
Closed form
results for the optimising state estimate
and unnormalised smoothed conditional probability measure are given
and connection between risk-sensitive filtering and risk-neutral filtering
for HMM's has been obtained as a limit result when $\theta$ approaches $0$.
Simulation studies (not reported here due to lack of space)
have confirmed that risk-sensitive filters are more robust to uncertain noise
environments, specially to coloured and occasionally high noise.
They show that there is a certain range of $\theta$ where the risk-sensitive filter outperforms the $L_2$ or risk-neutral filter
in terms of desirable robustness properties, but increasing $\theta$ beyond this range actually degrades the
performance of the risk-sensitive filter.