We develop Bayesian learning algorithms for estimation of time-varying linear prediction (TVLP) coefficients of speech. Estimation of TVLP coefficients is a naturally underdetermined problem. We consider sparsity and subspace based approaches for dealing with the corresponding underdetermined system. Bayesian learning algorithms are developed to achieve better estimation performance. Expectationmaximization (EM) framework is employed to develop the Bayesian learning algorithms where we use a combined prior to model a driving noise (glottal signal) that has both sparse and dense statistical properties. The efficiency of the Bayesian learning algorithms is shown for synthetic signals using spectral distortion measure and formant tracking of real speech signals.