In this paper, a joint feature selection and parameter estimation algorithm is presented for hidden Markov models (HMMs) and hidden semi-Markov models (HSMMs). New parameters, feature saliencies, are introduced to the model and used to select features that distinguish between states. The feature saliencies represent the probability that a feature is relevant by distinguishing between state-dependent and state-independent distributions. An expectation maximization algorithm is used to calculate maximum a posteriori estimates for model parameters. An exponential prior on the feature saliencies is compared with a beta prior. These priors can be used to include cost in the model estimation and feature selection process. This algorithm is tested against maximum likelihood estimates and a variational Bayesian method. For the HMM, four formulations are compared on a synthetic data set generated by models with known parameters, a tool wear data set, and data collected during a painting process. For the HSMM, two formulations, maximum likelihood and maximum a posteriori, are tested on the latter two data sets, demonstrating that the feature saliency method of feature selection can be extended to semi-Markov processes. The literature on feature selection specifically for HMMs is sparse, and non-existent for HSMMs. This paper fills a gap in the literature concerning simultaneous feature selection and parameter estimation for HMMs using the EM algorithm, and introduces the notion of selecting features with respect to cost for HMMs.