  Conditional particle filters (CPFs) are powerful smoothing algorithms for
general nonlinear/non-Gaussian hidden Markov models. However, CPFs can be
inefficient or difficult to apply with diffuse initial distributions, which are
common in statistical applications. We propose a simple but generally
applicable auxiliary variable method, which can be used together with the CPF
in order to perform efficient inference with diffuse initial distributions. The
method only requires simulatable Markov transitions that are reversible with
respect to the initial distribution, which can be improper. We focus in
particular on random-walk type transitions which are reversible with respect to
a uniform initial distribution (on some domain), and autoregressive kernels for
Gaussian initial distributions. We propose to use on-line adaptations within
the methods. In the case of random-walk transition, our adaptations use the
estimated covariance and acceptance rate adaptation, and we detail their
theoretical validity. We tested our methods with a linear-Gaussian random-walk
model, a stochastic volatility model, and a stochastic epidemic compartment
model with time-varying transmission rate. The experimental findings
demonstrate that our method works reliably with little user specification, and
can be substantially better mixing than a direct particle Gibbs algorithm that
treats initial states as parameters.
