Indeed the second explanation is what I was intending: X and Y are samples from two nodes. Isn’t it the case though that if we wanted to do the maximum likelihood estimation of the parameter w, we would take the product of the probabilities of each observation, each of which would have the multiplicative factor e^(w*(2*X-1)*(2*Y-1)), and thus following through the logarithms, the sum without normalization is correct to give the likelihood of the series of observations? Then, if you want to incorporate in the prior, that would be an additional term?
I’m sure I’m missing an obvious statement here right now.