Rigorous book on probability theory and statistics

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I would like some examples of very rigorous books on probability and statistics. I know there are lots of rigorous books on probability by itself, but I am also looking for a book that makes rigorous the theory behind statistics. From my understanding, there is a difference between probability theory and statistics.

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E. T. Jaynes is very thoughtful and addresses such matters and properly highlights the role of Bayesian probability and statistics.