Typo in "MCMC Using Hamiltonian Dynamics" (Neal, 2012)?

Questions
1

Reading through Radford Neal’s 2012 review of HMC, “MCMC Using Hamiltonian Dynamics,” I got stumped by the following:

In Bayesian statistics, the posterior distribution for the model parameters is the usual focus of interest, and hence these parameters will take the role of the position, q. We can express the posterior distribution as a canonical distribution (with T = 1) using a potential energy function defined as follows:

U(q) = -\log [ \pi(q) L(q|D)]

where \pi(q) is the prior density, and L(q|D) is the likelihood function given data D.

Am I wrong in thinking at the equation in \log[\cdot] should be the posterior distribution P(q|D) and thus \pi(q) L(D|q). Note that I am using the pre-print available on arXiv, so it may be different in the actual book Handbook of Markov Chain Monte Carlo.

Thank you for any clarification!

2

Yeah this is definitely an unusual way to write the posterior density function - maybe Neal wants to emphasis we are interested in the parameter q here. I guess from the programming point of view it makes little differences as the likelihood function is usually written as likelihood_fn(*data, *parameters) (btw it is written the same way in the book)

3

Thank you @junpenglao. Do you think it could have something to do with his notation of using L for the likelihood function instead of P for probability? In other words, perhaps L(q|D) = P(D|q)?

4

Yes, I think that’s the idea. I find it very confusing, but it’s just another of the 38 different ways of writing down Bayes’ rule.

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