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The more important aim of this conversion is that now the minima of the term in the exponent, , are equal to 0. If we manage to find a way to express  as a polynomial, this lets us to pull in the powerful machinery of algebraic geometry, which studies the zeros of polynomials. We've turned our problem of probability theory and statistics into a problem of algebra and geometry.

Wait... but  just isn't a polynomial most of the time. Right? From its definition above, ) differs by a constant from the log-likelihood .  So the log-likelihood has to be a polynomial too? If the network has, say, a ReLU layer, then I wouldn't even expect  to be smooth.  And I can't see any reason to think that  or swishes or whatever else we use would make  happen to be a polynomial either.