Why a *very (that means: VERY!) first conceptual introduction to Hamiltonian Monte Carlo* (HMC) on this weblog?

Effectively, in our endeavor to characteristic the assorted capabilities of TensorFlow Likelihood (TFP) / tfprobability, we began displaying examples of easy methods to match hierarchical fashions, utilizing one in all TFP’s joint distribution lessons and HMC. The technical facets being advanced sufficient in themselves, we by no means gave an introduction to the “math facet of issues.” Right here we try to make up for this.

Seeing how it’s inconceivable, in a brief weblog put up, to supply an introduction to Bayesian modeling and Markov Chain Monte Carlo generally, and the way there are such a lot of glorious texts doing this already, we are going to presuppose some prior data. Our particular focus then is on the newest and best, the magic buzzwords, the well-known incantations: Hamiltonian Monte Carlo, *leapfrog* steps, NUTS – as at all times, attempting to demystify, to make issues as comprehensible as doable.

In that spirit, welcome to a “glossary with a story.”

## So what’s it for?

Sampling, or *Monte Carlo*, methods generally are used once we need to produce samples from, or statistically describe a distribution we don’t have a closed-form formulation of. Typically, we would actually have an interest within the samples; generally we simply need them so we will compute, for instance, the imply and variance of the distribution.

What distribution? In the kind of purposes we’re speaking about, we now have a *mannequin*, a joint distribution, which is meant to explain some actuality. Ranging from essentially the most primary state of affairs, it would appear to be this:

[

x sim mathcal{Poisson}(lambda)

]

This “joint distribution” solely has a single member, a Poisson distribution, that’s presupposed to mannequin, say, the variety of feedback in a code evaluate. We even have information on precise code opinions, like this, say:

We now need to decide the *parameter*, (lambda), of the Poisson that make these information most *doubtless*. Thus far, we’re not even being Bayesian but: There isn’t any prior on this parameter. However after all, we need to be Bayesian, so we add one – think about mounted priors on *its* parameters:

[

x sim mathcal{Poisson}(lambda)

lambda sim gamma(alpha, beta)

alpha sim […]

beta sim […]

]

This being a joint distribution, we now have three parameters to find out: (lambda), (alpha) and (beta).

And what we’re fascinated about is the *posterior distribution* of the parameters given the information.

Now, relying on the distributions concerned, we normally can’t calculate the posterior distributions in closed type. As a substitute, we now have to make use of sampling methods to find out these parameters. What we’d prefer to level out as a substitute is the next: Within the upcoming discussions of sampling, HMC & co., it’s very easy to neglect *what’s it that we’re sampling*. Attempt to at all times understand that what we’re sampling isn’t the information, it’s parameters: the parameters of the posterior distributions we’re fascinated about.

## Sampling

Sampling strategies generally encompass two steps: producing a pattern (“proposal”) and deciding whether or not to maintain it or to throw it away (“acceptance”). Intuitively, in our given state of affairs – the place we now have measured one thing and are actually in search of a mechanism that explains these measurements – the latter ought to be simpler: We “simply” want to find out the probability of the information below these hypothetical mannequin parameters. However how can we provide you with options to begin with?

In concept, simple(-ish) strategies exist that could possibly be used to generate samples from an unknown (in closed type) distribution – so long as their unnormalized chances could be evaluated, and the issue is (very) low-dimensional. (For concise portraits of these strategies, resembling uniform sampling, significance sampling, and rejection sampling, see(MacKay 2002).) These are usually not utilized in MCMC software program although, for lack of effectivity and non-suitability in excessive dimensions. Earlier than HMC grew to become the dominant algorithm in such software program, the *Metropolis* and *Gibbs* strategies have been the algorithms of selection. Each are properly and understandably defined – within the case of Metropolis, usually exemplified by good tales –, and we refer the reader to the go-to references, resembling (McElreath 2016) and (Kruschke 2010). Each have been proven to be much less environment friendly than HMC, the primary matter of this put up, as a consequence of their random-walk conduct: Each proposal relies on the present place in state house, that means that samples could also be extremely correlated and state house exploration proceeds slowly.

## HMC

So HMC is in style as a result of in comparison with random-walk-based algorithms, it’s a *lot* extra environment friendly. Sadly, additionally it is much more troublesome to “get.” As mentioned in Math, code, ideas: A 3rd street to deep studying, there appear to be (at the very least) three languages to precise an algorithm: Math; code (together with pseudo-code, which can or is probably not on the verge to math notation); and one I name *conceptual* which spans the entire vary from very summary to very concrete, even visible. To me personally, HMC is totally different from most different instances in that regardless that I discover the conceptual explanations fascinating, they end in much less “perceived understanding” than both the equations or the code. For individuals with backgrounds in physics, statistical mechanics and/or differential geometry this may in all probability be totally different!

In any case, bodily analogies make for the perfect begin.

## Bodily analogies

The basic bodily analogy is given within the reference article, Radford Neal’s “MCMC utilizing Hamiltonian dynamics” (Neal 2012), and properly defined in a video by Ben Lambert.

So there’s this “factor” we need to maximize, the loglikelihood of the information below the mannequin parameters. Alternatively we will say, we need to decrease the detrimental loglikelihood (like loss in a neural community). This “factor” to be optimized can then be visualized as an object sliding over a panorama with hills and valleys, and like with gradient descent in deep studying, we would like it to finish up deep down in some valley.

In Neal’s personal phrases

In two dimensions, we will visualize the dynamics as that of a frictionless puck that slides over a floor of various peak. The state of this technique consists of the place of the puck, given by a 2D vector q, and the momentum of the puck (its mass instances its velocity), given by a 2D vector p.

Now while you hear “momentum” (and on condition that I’ve primed you to consider deep studying) you might really feel that sounds acquainted, however regardless that the respective analogies are associated the affiliation doesn’t assist that a lot. In deep studying, momentum is usually praised for its avoidance of ineffective oscillations in imbalanced optimization landscapes.

With HMC nevertheless, the main target is on the idea of *power*.

In statistical mechanics, the chance of being in some state (i) is inverse-exponentially associated to its power. (Right here (T) is the *temperature*; we gained’t concentrate on this so simply think about it being set to 1 on this and subsequent equations.)

[P(E_i) sim e^{frac{-E_i}{T}} ]

As you would possibly or won’t keep in mind from faculty physics, power is available in two varieties: potential power and kinetic power. Within the sliding-object state of affairs, the thing’s potential power corresponds to its peak (place), whereas its kinetic power is expounded to its momentum, (m), by the method

[K(m) = frac{m^2}{2 * mass} ]

Now with out kinetic power, the thing would slide downhill at all times, and as quickly because the panorama slopes up once more, would come to a halt. By way of its momentum although, it is ready to proceed uphill for some time, simply as if, going downhill in your bike, you decide up pace you might make it over the subsequent (quick) hill with out pedaling.

In order that’s kinetic power. The opposite half, potential power, corresponds to the factor we actually need to know – the *detrimental log posterior* of the parameters we’re actually after:

[U(theta) sim – log (P(x | theta) P(theta))]

So the “trick” of HMC is augmenting the state house of curiosity – the vector of posterior parameters – by a momentum vector, to enhance optimization effectivity. Once we’re completed, the momentum half is simply thrown away. (This side is very properly defined in Ben Lambert’s video.)

Following his exposition and notation, right here we now have the power of a state of parameter and momentum vectors, equaling a sum of potential and kinetic energies:

[E(theta, m) = U(theta) + K(m)]

The corresponding chance, as per the connection given above, then is

[P(E) sim e^{frac{-E}{T}} = e^{frac{- U(theta)}{T}} e^{frac{- K(m)}{T}}]

We now substitute into this equation, assuming a temperature (T) of 1 and a mass of 1:

[P(E) sim P(x | theta) P(theta) e^{frac{- m^2}{2}}]

Now on this formulation, the distribution of momentum is simply a regular regular ((e^{frac{- m^2}{2}}))! Thus, we will simply combine out the momentum and take (P(theta)) as samples from the posterior distribution:

[

begin{aligned}

& P(theta) =

int ! P(theta, m) mathrm{d}m = frac{1}{Z} int ! P(x | theta) P(theta) mathcal{N}(m|0,1) mathrm{d}m

& P(theta) = frac{1}{Z} int ! P(x | theta) P(theta)

end{aligned}

]

How does this work in observe? At each step, we

- pattern a brand new momentum worth from its marginal distribution (which is similar because the conditional distribution given (U), as they’re impartial), and
- clear up for the trail of the particle. That is the place
*Hamilton’s equations*come into play.

## Hamilton’s equations (equations of movement)

For the sake of much less confusion, must you resolve to learn the paper, right here we swap to Radford Neal’s notation.

Hamiltonian dynamics operates on a d-dimensional place vector, (q), and a d-dimensional momentum vector, (p). The state house is described by the *Hamiltonian*, a operate of (p) and (q):

[H(q, p) =U(q) +K(p)]

Right here (U(q)) is the potential power (known as (U(theta)) above), and (Okay(p)) is the kinetic power as a operate of momentum (known as (Okay(m)) above).

The partial derivatives of the Hamiltonian decide how (p) and (q) change over time, (t), in response to Hamilton’s equations:

[

begin{aligned}

& frac{dq}{dt} = frac{partial H}{partial p}

& frac{dp}{dt} = – frac{partial H}{partial q}

end{aligned}

]

How can we clear up this technique of partial differential equations? The fundamental workhorse in numerical integration is *Euler’s methodology*, the place time (or the impartial variable, generally) is superior by a step of measurement (epsilon), and a brand new worth of the dependent variable is computed by taking the (partial) by-product and including it to its present worth. For the Hamiltonian system, doing this one equation after the opposite seems like this:

[

begin{aligned}

& p(t+epsilon) = p(t) + epsilon frac{dp}{dt}(t) = p(t) − epsilon frac{partial U}{partial q}(q(t))

& q(t+epsilon) = q(t) + epsilon frac{dq}{dt}(t) = q(t) + epsilon frac{p(t)}{m})

end{aligned}

]

Right here first a brand new place is computed for time (t + 1), making use of the present momentum at time (t); then a brand new momentum is computed, additionally for time (t + 1), making use of the present place at time (t).

This course of could be improved if in step 2, we make use of the *new* place we simply freshly computed in step 1; however let’s straight go to what’s really utilized in modern software program, the *leapfrog* methodology.

## Leapfrog algorithm

So after *Hamiltonian*, we’ve hit the second magic phrase: *leapfrog*. Not like *Hamiltonian* nevertheless, there may be much less thriller right here. The leapfrog methodology is “simply” a extra environment friendly solution to carry out the numerical integration.

It consists of three steps, principally splitting up the Euler step 1 into two components, earlier than and after the momentum replace:

[

begin{aligned}

& p(t+frac{epsilon}{2}) = p(t) − frac{epsilon}{2} frac{partial U}{partial q}(q(t))

& q(t+epsilon) = q(t) + epsilon frac{p(t + frac{epsilon}{2})}{m}

& p(t+ epsilon) = p(t+frac{epsilon}{2}) − frac{epsilon}{2} frac{partial U}{partial q}(q(t + epsilon))

end{aligned}

]

As you’ll be able to see, every step makes use of the corresponding variable-to-differentiate’s worth computed within the previous step. In observe, a number of leapfrog steps are executed earlier than a proposal is made; so steps 3 and 1 (of the following iteration) are mixed.

*Proposal* – this key phrase brings us again to the higher-level “plan.” All this – Hamiltonian equations, leapfrog integration – served to generate a proposal for a brand new worth of the parameters, which could be accepted or not. The way in which that call is taken isn’t explicit to HMC and defined intimately within the above-mentioned expositions on the Metropolis algorithm, so we simply cowl it briefly.

## Acceptance: Metropolis algorithm

Beneath the Metropolis algorithm, proposed new vectors (q*) and (p*) are accepted with chance

[

min(1, exp(−H(q∗, p∗) +H(q, p)))

]

That’s, if the proposed parameters yield a better probability, they’re accepted; if not, they’re accepted solely with a sure chance that is determined by the ratio between outdated and new likelihoods.

In concept, power staying fixed in a Hamiltonian system, proposals ought to at all times be accepted; in observe, lack of precision as a consequence of numerical integration could yield an acceptance charge lower than 1.

## HMC in a number of traces of code

We’ve talked about ideas, and we’ve seen the mathematics, however between analogies and equations, it’s straightforward to lose observe of the general algorithm. Properly, Radford Neal’s paper (Neal 2012) has some code, too! Right here it’s reproduced, with just some further feedback added (many feedback have been preexisting):

```
# U is a operate that returns the potential power given q
# grad_U returns the respective partial derivatives
# epsilon stepsize
# L variety of leapfrog steps
# current_q present place
# kinetic power is assumed to be sum(p^2/2) (mass == 1)
HMC <- operate (U, grad_U, epsilon, L, current_q) {
q <- current_q
# impartial customary regular variates
p <- rnorm(size(q), 0, 1)
# Make a half step for momentum initially
current_p <- p
# Alternate full steps for place and momentum
p <- p - epsilon * grad_U(q) / 2
for (i in 1:L) {
# Make a full step for the place
q <- q + epsilon * p
# Make a full step for the momentum, besides at finish of trajectory
if (i != L) p <- p - epsilon * grad_U(q)
}
# Make a half step for momentum on the finish
p <- p - epsilon * grad_U(q) / 2
# Negate momentum at finish of trajectory to make the proposal symmetric
p <- -p
# Consider potential and kinetic energies at begin and finish of trajectory
current_U <- U(current_q)
current_K <- sum(current_p^2) / 2
proposed_U <- U(q)
proposed_K <- sum(p^2) / 2
# Settle for or reject the state at finish of trajectory, returning both
# the place on the finish of the trajectory or the preliminary place
if (runif(1) < exp(current_U-proposed_U+current_K-proposed_K)) {
return (q) # settle for
} else {
return (current_q) # reject
}
}
```

Hopefully, you discover this piece of code as useful as I do. Are we by way of but? Effectively, up to now we haven’t encountered the final magic phrase: NUTS. What, or who, is NUTS?

## NUTS

NUTS, added to Stan in 2011 and a couple of month in the past, to TensorFlow Likelihood’s grasp department, is an algorithm that goals to bypass one of many sensible difficulties in utilizing HMC: The selection of variety of leapfrog steps to carry out earlier than making a proposal. The acronym stands for No-U-Flip Sampler, alluding to the avoidance of U-turn-shaped curves within the optimization panorama when the variety of leapfrog steps is chosen too excessive.

The reference paper by Hoffman & Gelman (Hoffman and Gelman 2011) additionally describes an answer to a associated issue: selecting the step measurement (epsilon). The respective algorithm, *twin averaging*, was additionally not too long ago added to TFP.

NUTS being extra of algorithm within the laptop science utilization of the phrase than a factor to clarify conceptually, we’ll go away it at that, and ask the reader to learn the paper – and even, seek the advice of the TFP documentation to see how NUTS is applied there. As a substitute, we’ll spherical up with one other conceptual analogy, Michael Bétancourts crashing (or not!) satellite tv for pc (Betancourt 2017).

## How you can keep away from crashes

Bétancourt’s article is an superior learn, and a paragraph specializing in a single level made within the paper could be nothing than a “teaser” (which is why we’ll have an image, too!).

To introduce the upcoming analogy, the issue begins with excessive dimensionality, which is a given in most real-world issues. In excessive dimensions, as typical, the density operate has a *mode* (the place the place it’s maximal), however essentially, there can’t be a lot *quantity* round it – identical to with k-nearest neighbors, the extra dimensions you add, the farther your nearest neighbor will probably be.

A product of quantity and density, the one important chance mass resides within the so-called typical set, which turns into an increasing number of slim in excessive dimensions.

So, the standard set is what we need to discover, however it will get an increasing number of troublesome to search out it (and keep there). Now as we noticed above, HMC makes use of gradient data to get close to the mode, but when it simply adopted the gradient of the log chance (the *place*) it might go away the standard set and cease on the mode.

That is the place momentum is available in – it counteracts the gradient, and each collectively be certain that the Markov chain stays on the standard set. Now right here’s the satellite tv for pc analogy, in Bétancourt’s personal phrases:

For instance, as a substitute of attempting to motive a couple of mode, a gradient, and a typical set, we will equivalently motive a couple of planet, a gravitational area, and an orbit (Determine 14). The probabilistic endeavor of exploring the standard set then turns into a bodily endeavor of putting a satellite tv for pc in a secure orbit across the hypothetical planet. As a result of these are simply two totally different views of the identical mathematical system, they’ll endure from the identical pathologies. Certainly, if we place a satellite tv for pc at relaxation out in house it’ll fall within the gravitational area and crash into the floor of the planet, simply as naive gradient-driven trajectories crash into the mode (Determine 15). From both the probabilistic or bodily perspective we’re left with a catastrophic consequence.

The bodily image, nevertheless, gives a right away resolution: though objects at relaxation will crash into the planet, we will keep a secure orbit by endowing our satellite tv for pc with sufficient momentum to counteract the gravitational attraction. We’ve to watch out, nevertheless, in how precisely we add momentum to our satellite tv for pc. If we add too little momentum transverse to the gravitational area, for instance, then the gravitational attraction will probably be too sturdy and the satellite tv for pc will nonetheless crash into the planet (Determine 16a). However, if we add an excessive amount of momentum then the gravitational attraction will probably be too weak to seize the satellite tv for pc in any respect and it’ll as a substitute fly out into the depths of house (Determine 16b).

And right here’s the image I promised (Determine 16 from the paper):

And with this, we conclude. Hopefully, you’ll have discovered this useful – except you knew all of it (or extra) beforehand, by which case you in all probability wouldn’t have learn this put up 🙂

Thanks for studying!

*arXiv e-Prints*, January, arXiv:1701.02434. https://arxiv.org/abs/1701.02434.

*Journal of the American Statistical Affiliation*112 (518): 859–77. https://doi.org/10.1080/01621459.2017.1285773.

Kruschke, John Okay. 2010. *Doing Bayesian Information Evaluation: A Tutorial with r and BUGS*. 1st ed. Orlando, FL, USA: Educational Press, Inc.

MacKay, David J. C. 2002. *Data Concept, Inference & Studying Algorithms*. New York, NY, USA: Cambridge College Press.

*Statistical Rethinking: A Bayesian Course with Examples in r and Stan*. CRC Press. http://xcelab.internet/rm/statistical-rethinking/.

*arXiv e-Prints*, June, arXiv:1206.1901. https://arxiv.org/abs/1206.1901.