  In statistical data assimilation (SDA) and supervised machine learning (ML),
we wish to transfer information from observations to a model of the processes
underlying those observations. For SDA, the model consists of a set of
differential equations that describe the dynamics of a physical system. For ML,
the model is usually constructed using other strategies. In this paper, we
develop a systematic formulation based on Monte Carlo sampling to achieve such
information transfer. Following the derivation of an appropriate target
distribution, we present the formulation based on the standard
Metropolis-Hasting (MH) procedure and the Hamiltonian Monte Carlo (HMC) method
for performing the high dimensional integrals that appear. To the extensive
literature on MH and HMC, we add (1) an annealing method using a hyperparameter
that governs the precision of the model to identify and explore the highest
probability regions of phase space dominating those integrals, and (2) a
strategy for initializing the state space search. The efficacy of the proposed
formulation is demonstrated using a nonlinear dynamical model with chaotic
solutions widely used in geophysics.
