OK as I understand it, they rate how intense the odor is, and there is an objective characteristic that is varying, namely the chemical concentration. Is the following correct? To simplify, each odor has five concentrations, odorants are presented in random order, and on each trial subjects identify which odorant concentration was presented with a rating 1-5.

Yes, of course one variable is fine. If your y variable was normal, in R you would do lm(y~x). For a normal y and one categorical x, the usual name is one-way ANOVA. BTW you could also try quasi-poisson, as you did at the start. It deals with 'over dispersion'. But I wouldn't lose sleep about it.

@charlieshades, I don't know if this helps, but let's reduce the problem by one dimension. The points around the least squares line are balanced, in that the positive errors are exactly balanced by the negative ones. The same situation holds for the 1D version: the mean. The mean is also a balance point. The sum of the distances of the points from the mean (deviations) is zero. The mean is also a sum of squares parameter: it is the point that minimises the sum of the squared distances from it.

@Neil I'm sure you are using the terminology correctly as defined in the papers you cited. It's just a different terminology from what I'm used to where SSE is energy

I am using the standard definition of energy from signal processing. Computer science / machine learning people do tend to redefine terms, I guess. I come from stats and signal processing background

@RustyStatistician The likelihood principle is a central tenet for likelihoodists. Likelihoodists are not Bayesian at all. I posted links in my answer. Your claim "if you do believe in the likelihood principle, then (I believe) you most certainly have to espouse the Bayesian paradigm" is false.

A commonly used method in ML classification is Bernoulli Naive Bayes. So I dispute the claim that ML always uses the Normal distribution.