  We present a compartmentalized approach to finding the maximum a-posteriori
(MAP) estimate of a latent time series that obeys a dynamic stochastic model
and is observed through noisy measurements. We specifically consider modern
signal processing problems with non-Markov signal dynamics (e.g. group
sparsity) and/or non-Gaussian measurement models (e.g. point process
observation models used in neuroscience). Through the use of auxiliary
variables in the MAP estimation problem, we show that a consensus formulation
of the alternating direction method of multipliers (ADMM) enables iteratively
computing separate estimates based on the likelihood and prior and subsequently
"averaging" them in an appropriate sense using a Kalman smoother. As such, this
can be applied to a broad class of problem settings and only requires modular
adjustments when interchanging various aspects of the statistical model. Under
broad log-concavity assumptions, we show that the separate estimation problems
are convex optimization problems and that the iterative algorithm converges to
the MAP estimate. As such, this framework can capture non-Markov latent time
series models and non-Gaussian measurement models. We provide example
applications involving (i) group-sparsity priors, within the context of
electrophysiologic specrotemporal estimation, and (ii) non-Gaussian measurement
models, within the context of dynamic analyses of learning with neural spiking
and behavioral observations.
