In this talk, I will present two concrete problems related to sequentially testing and estimating an unknown parameter within the exponential family in discrete time, incorporating observation costs within the Bayesian setting. Specifically, we will examine the entire one-parameter exponential family with an arbitrary prior distribution, and, therefore, we will not rely on conjugate priors. These problems can be embedded in Markovian frameworks. In the absence of explicit solutions, we will discuss the properties of the value functions and their implications for the structure of continuation regions. Beyond the obvious statistical applications, I will briefly discuss their relevance in stochastic control problems with learning features. Part of this talk is based on joint work with Erik Ekström.