Robots operating in an open world will encounter novel objects with unknown physical properties, such as mass, friction, or size. These robots will need to sense these properties through interaction prior to performing downstream tasks with the objects. We propose a method that autonomously learns tactile exploration policies by developing a generative world model that is leveraged to 1) estimate the object's physical parameters using a differentiable Bayesian filtering algorithm and 2) develop an exploration policy using an information-gathering model predictive controller. We evaluate our method on three simulated tasks where the goal is to estimate a desired object property (mass, height or toppling height) through physical interaction. We find that our method is able to discover policies that efficiently gather information about the desired property in an intuitive manner. Finally, we validate our method on a real robot system for the height estimation task, where our method is able to successfully learn and execute an information-gathering policy from scratch.