Bayesian inference has shown powerful impacts on understanding and explaining data and their generating mechanisms, but misspecification of the model is a major threat to the validity of the inference. Although methods that deal with misspecification have been developed and their properties have been studied, these methods are mainly established based on the premise that the whole model is misspecified. Since the real mechanism of the data generating process is often complex and many factors can affect the observation and collection of data, the reliability of the model may widely vary across its components and lead to partial misspecification. Dealing with such partial misspecification for a robust inference remains challenging and requires comprehensive studies of its methodology, algorithm, theory and potential application.

Modularized Bayesian inference has been developed as a robust alternative to standard Bayesian inference for partial misspecification. As a particular form, cut inference completely removes the influence from misspecified components and involves a cut distribution which differs from the standard posterior distribution. Existing algorithms which sample from this cut distribution suffer from unclear convergence properties or slow computations. A novel algorithm named the stochastic approximation cut algorithm (SACut) is proposed in this thesis. The theoretical and computational properties of the SACut algorithm are studied.

A general framework of cut inference beyond a generic two-module case, where one component is assumed to be misspecified, is not clear. In particular, the definition of what a ``module'' is remains vague in the literature. Furthermore, implementing cut inference for an arbitrary multiple-module case remains an open question. Solving these basic questions is appealing and necessary. This thesis formulates rules including the definition of modules; determination of relationships between modules and building the cut distribution that one should follow to implement cut inference within an arbitrary model structure.

Semi-Modular inference bridges the gap between standard Bayesian inference and cut inference through the use of a likelihood with a power term. Interestingly, this feature corresponds to a geographically weighted regression (GWR) model that has been developed to handle the spatial non-stationarity but hitherto not been extended to Bayesian inference except for the Gaussian regression. This thesis proposes the Bayesian GWR model as a certain multiple-module case of Semi-Modular inference. The theory of Semi-Modular inference is extended to the multiple-module case to justify the Bayesian GWR model.

Modularized Bayesian inference remains a young and emerging topic. Being one of the many pioneering works that promote the modularized Bayesian inference to a broader range of statistical models, it is hoped that this thesis will enlighten future developments of methodology and algorithm, and stimulate applications of modularized Bayesian inference.