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Yes, that’s just what I did. Instead of minimizing badness I switched to maximizing goodness. I’m just that kind of guy.

And again

Probabilities are never greater than one, so log probabilities are always less than or equal to zero.  So a negative log likelihood is always a positive quantity, and smaller negative log likelihood values are associated with more likely outcomes.  So one way to understand the minus sign in the Gibbs-Boltzmann distribution is that it makes H(x) correspond to a negative log likelihood.But I think one can give a more detailed explanation.

In a Gibbs-Boltzmann distribution p(x) = k*exp(–H(x)/T), H(x) is the energy of a configuration x.

Because energies H(x) are non-negative (which follows from the definition of energy?), and given a couple of other assumptions (e.g., that there are an infinite number of configurations and energies are unbounded — maybe other assumptions will do?), it follows that probability must decrease with energy, otherwise the inverse partition function k would not exist (i.e., the probability distribution p(x) would not sum to 1).

So if the minus sign were not there, the temperature T (which relates energy and probability) would need to be negative.  There’s no mathematical reason why we couldn’t allow negative temperatures, but the minus sign makes the factor T in the formula correspond much closer with our conventional understanding of temperature.

In fact, I think it is amazing that the constant T in the Gibbs-Boltzmann formula denotes exactly the pre-statistical mechanics concept of temperature (well, absolute temperature in Kelvin).  In many other domains there’s a complex relationship between a physical quantity and our perception of it; what is the chance of a simple linear relationship like this for temperature?

But perhaps it’s not a huge coincidence.  Often our perceptual quantities are logarithmically related to physical quantities, so perhaps its no accident that T is inside the exp() rather than outside (where it would show up as an “exponential temperature” term).  And the concept of temperature we had before Gibbs and Boltzmann wasn’t just a naive perception of warmth; there had been several centuries of careful empirical work on properties of gases, heat engines, etc., which presumably lead scientists to the right notion of temperature well before the Gibbs-Boltzmann relationship was discovered.