Bayesianism has already gained a prominent role as a theory of probability interpretation. Its reasoning models are known as Bayesian inference. Keynes' formal induction system has made it evolved into a new era: Bayesian inference has got an extensive use in statistics, economics, psychology and artificial intelligence, and its application domain is still expanding. This is a revolution of reasoning method because Bayesian method has eliminated the problems caused by ignore-to-prior and subjective issues in classical statistic inference. Bayesian inference resolved the problem of deciding test statistic in classical statistic inference and avoided the difficulty of using stopping rule. Bayesianism replaces confidence interval by credible interval in order to obtain the prediction of the true value of a parameter. It also employs prior information through Bayes theorem on conditional probability. Though Bayesianism is very popular nowadays, several challenges have been raised, especially on its subjectivity, simplicity and on the problem of old-evidence. Therefore, there are a lot of issues need to be studied in the future. The criticism of subjectivity of Bayesanism is focused on a prior-constraint, and the question on simplicity of Bayesianism is echoed by the inconsistency between probability axiom and the simplicity postulate. Forster and Sober remarked that the simplicity postulate is ''an ad hoc method;'' Howson argued that it should not be taken as an essential guideline. Additionally, an old-evidence in the Bayesian framework could not serve as affirming the current hypothesis, which contradicts with our instinct. Howson insightfully revealed that ''evidence support'' mechanism actually contains a ternary relation between data e , hypothesis h and background knowledge k , and the problem of old-evidence only become evident when e is decided as evidence and e is included in k . These challenges have shown that the further improvement and development were needed in Bayesian inference. Recent developments in the research of Bayesian inference in the field of cognitive psychology have invoked a possible cognitive turn in the research of Bayesian inference, and they also provide various possible approaches for the development of Bayesian inference: to explore the possibility of the integration between Frequentism and Bayesianism|to bring the intentional factor in the extensional inductive logic, with an attempt to combine extensionality and non-extensionality.  Gigerenzer and Hoffrage has already offered a Bayesian model with frequency representation of information|Kahneman and Tversky proposed a non-extension logic theory, the support theory for subjective probability. This seems to be a prospective path for inductive logic that worth further exploring.