The need for developing a better understanding of the behaviour of small samples by observing experiments presents a problem far beyond purely academic interest. This monograph describes the character of incomplete fuzzy information, and proposes and proves the principle of information diffusion. The focus lies in changing a traditional sample-point into a fuzzy set to partly fill the gap caused by incomplete data, so that the recognition of relationships between input and output can be improved. Part 1 examines the origins of the principle of information diffusion and describes the mathematical concepts and proofs. Topics covered include: information matrix, demonstration of information distribution, and the kernel function in terms of information diffusion. Part 2 covers applications such as earthquake engineering and risk assessment of flood, and demonstrates that the new theory is useful for studying practical cases.