Traditionally viewed as a motor control center, the cerebellum is increasingly recognized as a crucial component of a neural network that enables adaptive behavior across various domains, including cognition, affect, emotion, and social interactions. Recent clinical studies have linked cerebellar dysfunction to impairments in mentalizing and narrative coherence in autism spectrum disorder (ASD). Given that narratives involve the temporal sequencing of causally related events and actions, these findings imply the potential role of the cerebellum in predictive sequence. The aim of this review is to dissect the neural circuitry and computational mechanisms underlying cerebellar predictive cognitive control. We propose that the Kalman filter model, which has been applied successfully to the motor cerebrocerebellum, can be extended to the non-motor (cognitive/affective/social) regions. In sharp contrast, the cerebral cortex employs a recurrent network architecture, as evidenced by intracortical connections, which underlies hierarchical processing in areas such as the visual and motor cortices. Surprisingly, the computational principles of the cerebrocerebellar loop have received relatively little attention in computational and theoretical neuroscience. We stress the need for a comprehensive theory on cerebrocerebellar connectivity that integrates the distinct neural mechanisms of the cerebrum and cerebellum, to help understand the role of this network in cognitive, affective, and social functions. Our theory provides a theoretical framework that explains how the cerebellum deals with motor and non-motor operations. The Kalman filter theory fits with two major requirements: sequencing and predictions. We propose a core operational mechanism for motor, cognitive, affective and social operations handled by the cerebellar circuitry.