We consider the problem of jointly estimating the states and sparse inputs of a linear dynamical system using noisy low-dimensional observations. We exploit the underlying sparsity in the inputs using fictitious sparsity-promoting Gaussian priors with unknown variances (as hyperparameters). We develop two Bayesian learning-based techniques to estimate states and inputs: sparse Bayesian learning and variational Bayesian inference. Through numerical simulations, we illustrate that our algorithms outperform the conventional Kalman filtering based algorithm and other state-of-the-art sparsity-driven algorithms, especially in the low-dimensional measurement regime.