I think the first three chapters of Jaynes are really excellent, and the chapters on updating (4 and 6 I think?), symmetry properties, and maximum entropy methods are also highly useful.

You will need some calculus (On the advanced end, you might need to understand how to use Lagrange multipliers, but it's not strictly necessary), but I think that's it. If you are unfamiliar with other concepts like Shannon entropy, what it might require is more practice of the material.

If you want to actually use this for data analysis, another modern textbook would be helpful (don't be afraid to read multiple books on the same thing).