  The hidden Markov model (HMM) is a classic modeling tool with a wide swath of
applications. Its inception considered observations restricted to a finite
alphabet, but it was quickly extended to multivariate continuous distributions.
In this article, we further extend the HMM from mixtures of normal
distributions in $d$-dimensional Euclidean space to general Gaussian measure
mixtures in locally convex topological spaces. The main innovation is the use
of the Onsager-Machlup functional as a proxy for the probability density
function in infinite dimensional spaces. This allows for choice of a
Cameron-Martin space suitable for a given application. We demonstrate the
versatility of this methodology by applying it to simulated diffusion processes
such as Brownian and fractional Brownian sample paths as well as the
Ornstein-Uhlenbeck process. Our methodology is applied to the identification of
sleep states from overnight polysomnography time series data with the aim of
diagnosing Obstructive Sleep Apnea in pediatric patients. It is also applied to
a series of annual cumulative snowfall curves from 1940 to 1990 in the city of
Edmonton, Alberta.
