Rising above chaotic likelihoods

Berliner [J. Amer. Statist. Assoc., 86 (1991), pp. 983--952] identified a number of difficulties in using the likelihood function within the Bayesian paradigm which arise both for state estimation and for parameter estimation of chaotic systems. Even when the equations of the system are given, he demonstrated “chaotic likelihood functions” both of initial conditions and of parameter values in the logistic map. Chaotic likelihood functions, while ultimately smooth, have such complicated small scale structure as to cast doubt on the possibility of identifying high likelihood states in practice. In this paper, the challenge of chaotic likelihoods is overcome by embedding the observations in a higher dimensional sequence space; this allows good state estimation with finite computational power. An importance sampling approach is introduced, where pseudo-orbit data assimilation is employed in the sequence space, first to identify relevant pseudo-orbits and then relevant trajectories. Estimates are identified with likelihoods orders of magnitude higher than those previously identified in the examples given by Berliner. The pseudo-orbit data assimilation importance sampler exploits the information both from the model dynamics and from the observations. While sampling from the relevant prior (here, the natural measure) will, of course, eventually yield an accountable sample, given the realistic computational resource this traditional approach would provide no high likelihood points at all. While one of the challenges Berliner posed is overcome, his central conclusion is supported. Chaotic likelihood functions for parameter estimation still pose a challenge; this fact helps clarify why physical scientists maintain a strong distinction between the initial condition uncertainty and parameter uncertainty.