Bayesian Curve Estimation with Overcomplete Wavelet Dictionary

Jen-hwa Chu

We describe a Bayesian approach for curve estimation based on
overcomplete wavelet dictionary proposed by Abramovich, Sapatinas 
and Silverman (1999), where the function is modeled by a sum of the 
wavelet components at arbitrary
location and scale and some prior distributions are imposed on the
location and scale of the wavelet components and the coefficients. It
has been shown that with an overcomplete basis we can often
achieve greater sparsity and robustness against noises. (Lewicki
and Sejnowski, 1998; Donoho and Elad, 2002; Donoho, Elad and
Temlyakov, 2004; Wolfe at al, 2004) We believe that by avoiding the
dyadic constraints for orthornormal wavelet bases, the overcomplete
wavelet dictionary will have greater flexibility to match the
structure of the data, and give sparser representations.
The main challenge here is to efficiently search over an infinite number
of basis elements. We will discuss a reversible jump Markov Chain Monte Carlo
algorithm for estimating the posterior distribution of the wavelet
coefficients, locations and scales, and various strategies to give
better convergence. We will also present some simulated examples to
illustrate our method, which has promising results.