# Difference between the methods to compute potential energy

I computed the potential energy corresponding to a model. I have used two methods to do so:

1. Using extra_fields=('potential_energy',) in mcmc.run
2. Using potential_energy under numpyro.infer.util

But, in both cases, I am getting different values. Can you please elaborate more about both method and their specific use?

Further, I also used log_density under numpyro.infer.util which gave me results same as that obtained using (1.) with a negative sign but with different digits post decimal. Isn’t log_density = -potential_energy or am I missing out on something?

Almost! Log density is the “real” measure that we care about, i.e. the log of the probability function, while potential energy is that same value after a co-ordinate transformation to the “unconstrained domain”. If your parameters have discontinuous or constrained priors, like Uniform or HalfNormal, they need to be re parameterized internally to something that’s smooth and differentiable for some algorithms, e.g. MCMC, to work properly. To transform between the two, all you have to do is multiply by the Jacobian of the transformation between constrained and unconstrained.

Regarding the differences in potential energy between utils and the MCMC chain, the problem is that potential_energy expects inputs in unconstrained form. When getting potential_energy as an extra_field in an MCMC run, this is automatically being done under the hood.

You can perform the transformation using something like:

transforms = {"x": numpyro.distributions.biject_to(x_prior.support)}

x_con = np.linspace(0, 2, 1024)
x_uncon = transform_fn(transforms, {'x': x_con}, invert=True)['x']


Where x_prior is whatever prior distribution you used to define ‘x’ in your model, e.g. numpyro.distributions.Uniform(-1,1).

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Here’s a little diagram of how you can switch between the different functions and what inputs they need.

The example graphs are for a simple model:

\pi(x) = Uniform(-1,1), \;\;\; \mathcal{L}(x) = (1+x)(1-x)

I don’t know the exact reparameterization NumPyro uses, but in this mock-up I’ve used:

z = \text{atanh}(x)

You can see how this goes from a constrained domain, x \in [-1,1], to an unconstrained domain, z \in (-\infty, \infty), which means any MCMC sampler navigating in z-space wouldn’t need to worry about sharp cliffs at the domain boundaries.

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Thank you for the fantastic explanation. Based on your writeup, won’t the potential_energy in the unconstrained space be equivalent to the log_density in the constrained space? If so, then why have you added an extra adjustment term in the form of log jacobian while comparing potential_energy and log_density in the diagram?

Note quite, because the probability density changes as you shift between coordinates. The core idea here is they are densities, and so equivalent differential elements need to have the same total power contained within them. E.g., if I have some constrained parameter x that converts to unconstrained parameter z, I need:

P_x(x) dx = P_z(z) dz

Which of course rearranges to:

P_x(x) \cdot\frac{dx}{dz} = P_z(z) \\ \ln \vert P_x(x) \vert + \ln \vert \frac{dx}{dz}\vert = \ln \vert P_z(z)\vert

I.e. to go between the two log-densities I need to add a log of the derivative.

If I have many parameters being transformed, like:

x_1 \rightarrow z_1(x_1) \\ x_2 \rightarrow z_2(x_2) \\ \vdots \\

I need to correct by the product of all of these:

\ln \vert P_z(z)\vert = \ln \vert P_x(x) \vert + \ln \vert \frac{dx_1}{dz_1} \frac{dx_2}{dz_2} \dots \vert

And in the more general case where the transformed parameters “mix”, i.e.:

x_1 \rightarrow z_1(x_1, x_2 \dots) \\ x_2 \rightarrow z_2(x_1, x_2 \dots) \\ \vdots \\

The correction factor is the determinant of the Jacobian of all of these transformations:

J = \left[ \matrix{{\frac{\partial z_1}{\partial x_1},\frac{\partial z_1}{\partial x_2}}\\{\frac{\partial z_2}{\partial x_1},\frac{\partial z_2}{\partial x_2}}} \right]
\ln \vert P_z(z)\vert = \ln \vert P_x(x) \vert + \ln \vert det(J) \vert

Throw in a negative or two to account for inverses and the difference between density and potential energy, and you have a clearer picture

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Thanks for such a comprehensive explanation

Hi everyone!

Thanks also from my side @Hourglass, for this super helpful explanation! I am currently at a bit of a loss on how to implement the transformation, though:

Let’s say I have a model with two parameters sampled from a HalfNormal Distribution:

x = np.linspace(0, 1, 100)
y_true = 0.5 + 0.3 * x

def model_1(y):
beta_0 = numpyro.sample('beta_0', dist.HalfNormal(1))
beta_1 = numpyro.sample('beta_1', dist.HalfNormal(1))
proj = beta_0 + beta_1 * x
with numpyro.plate('data', len(y)):
numpyro.sample('yl', dist.Normal(proj), obs=y)

nuts_kernel = NUTS(model_1)
mcmc = MCMC(nuts_kernel, num_samples=1000, num_warmup=1000)
mcmc.run(rng_key,y_true, extra_fields=('potential_energy',))
mcmc_samples = mcmc.get_samples()
np.mean(mcmc.get_extra_fields()['potential_energy'] * -1)


While the average negative potential energy equals -95.41983, what would be the correct value for the log marginal likelihood here, and how would I compute it?

Thank you!

if by marginal likelihood you mean the quantity also called the evidence that quantity cannot be computed in a straightforward way from a bag of samples and requires a different approach (i.e. additional computation)

Thanks for the comment!

I initially replicated the tutorial by PyMC here in Numpyro and noticed that the values for the marginal likelihood PyMC provides are very similar to the potential energy numbers (but not identical) which in turn lead me to this post.

I’m interested in precisely what you are referring to: the evidence. Are there any examples of how to do this in Numpyro?

Thank you!

Best
Niklas

i think you misunderstand me. mcmc samples on their own do not give you marginal likelihood estimates. you can try to post-process mcmc samples, but in any case this is an active area of research (see e.g. random paper).

for a simple estimate of the log marginal likelihood you can use the elbo lower bound in variational inference. the only trouble is that it is biased and may not work very well depending on details.

i believe you can use the nested sampler in numpyro or in jaxns if you want evidence estimates: statistics — jaxns 2.0.0 documentation (not sure if numpyro wraps the necessary jaxns utilities)

note that for a simple model like yours with a 2-dimensional latent space it is sufficient to draw a bunch of samples from the prior and thus get a MC estimate of the evidence. the problem is that fails horrendously in higher dimensions

Thanks again, I will have a look into the ELBO calculation, but your second comment matches my intuition:

Let’s say I samle from my prior to get some predictions for the target variable, which I then use to calculate a loss (e.g. MSE) or something like the accuracy (in case of binary targets). Intuitively, this should be related to the model evidence. But how can I calculate the actual evidence? This is a minimal example of what I have in mind:

from numpyro.infer import Predictive

x = np.linspace(0, 1, 100)
y_true = 0.5 + 0.3 * x

def model_1():
beta_0 = numpyro.sample('beta_0', dist.HalfNormal(1))
beta_1 = numpyro.sample('beta_1', dist.HalfNormal(1))
proj = beta_0 + beta_1 * x

numpyro.sample('yl', dist.Normal(proj))

prior_predictive = Predictive(model_1, num_samples=1000)
prior_samples = prior_predictive(rng_key=jax.random.PRNGKey(0))

prior_predictions = prior_samples['yl']
squared_error = (y_true - prior_predictions)**2
np.mean(squared_error)


Calculation of evidence from MCMC samples is non-trivial; there are some ways to do it but you need to be sure it can work for your problem. This is one example method: [1301.3156] A remarkably simple and accurate method for computing the Bayes Factor from a Markov chain Monte Carlo Simulation of the Posterior Distribution in high dimension
And here is a pretty comprehensive discussion: machine learning - Computation of the marginal likelihood from MCMC samples - Cross Validated

The main issue is that marginal likelihoods require one to integrate across the parameter space. Theoretically, MCMC samples can be used for this, but a naive harmonic mean of posterior samples often has infinite variance (this is discussed in the links above).