Slow feature analysis (SFA) is a time-series analysis method for extracting slowly-varying latent features from multi-dimensional data. In this paper, the probabilistic version of SFA algorithms is discussed from a theoretical point of view. First, the fundamental notions of SFA algorithms are reviewed in order to show the mechanism of extracting the slowly-varying latent features by means of the SFA. Second, recent advances in the SFA algorithms are described on the emphasis of the probabilistic version of the SFA. Third, the probabilistic SFA with rigorously derived likelihood function is derived by means of belief propagation. Using the rigorously derived likelihood function, we simultaneously extracts slow features and underlying parameters for the latent dynamics. Finally, we show using synthetic data that the probabilistic SFA with rigorously derived likelihood function can estimate the slow feature accurately even under noisy environments.