In the classic Rothschild and Stiglitz (1976) model of insurance under adverse selection, it is assumed that insurance consumers are fully informed of their exact probability type, and the insurers are uninformed of the probability of accident of consumers. While it is certainly intuitive that insurance consumers should be better informed of their probability type, it is perhaps too much to assume that they are perfectly informed. In the present paper, we extend the RS model to incorporate asymmetry of information in favour of the insurance consumers over insurers, and also insurance consumers that are only partially informed of their exact type. We allow for an inter-temporal setting, in which insurance contracts maybe renewed for many periods, and in that case, information on probability types is revealed over time via Bayes' rule. Aside from other interesting results, we consider if how information gathering affects the efficiency of the model.