We describe a technique for obtaining estimates of the short-term predictor parameters of speech under noisy conditions. We use a-priori information about speech in the form of a trained codebook of speech linear predictive coefficients. Our contribution is two-fold. First, we provide a framework where the standard vector quantization search to obtain the quantized linear predictive coefficients can be replaced by a maximum likelihood search, given the noisy observation, the speech codebook and an estimate of the noise. This results in an enhancement method that is integrated with parametric coders such as linear predictive analysis-by-synthesis coders. Second, we provide a scheme where the chosen vector is not restricted to be an element of the codebook. An interpolative search between the maximum likelihood estimate and its nearest neighbors in the codebook is used to improve the precision of the estimated parameters. Such a scheme is relevant when enhancement is considered separately from coding. Experimental results show improved performance for the proposed methods.