The connection isn't immediately obvious, but it's simply because solving for the maximum entry distribution that achieves a given expectation value produces the Botlzmann distribution. In stat mech, our "classifier" over (micro-)states is energy; in A.I. the classifier is labels.
For details, the keyword is Lagrange multiplier [0]. The specific application here is maximizing f as the entropy with the constraint g the expectation value.
If you're like me at all, the above will be a nice short rabbit hole to go down!
[0]:https://tutorial.math.lamar.edu/classes/calciii/lagrangemult...