# Hamiltonian Monte Carlo in Scala

I'm writing a program in Scala to perform Hamiltonian Monte Carlo (HMC), coupled with Gibbs sampling of some variables. The algorithm, with the modifications such as perturbing epsilon and l and performing Gibbs updates on some variables may be found in the easily available book chapter "MCMC using Hamiltonian Dynamics" by R Neal. The program works by alternating an HMC update of a set of variables q via the hmcUpdate() method with Gibbs updates of the variables gibbsVar via some unspecified method, then putting the two sets together in a tuple. The hmcRunner() method generates these tuples and prepends them onto a list.

I'm very new to Scala (last Friday) so I would like some feedback on the idiomacity of the code, as well as advice in general on improving readability. I've tried to write this in the form of an abstract class so that myself and my other group members can use this in their own code by supplying the unspecified methods themselves - is this the most appropriate structure to use?

import breeze.linalg._
import breeze.stats.distributions.Uniform
import breeze.stats.distributions.Gaussian
import breeze.stats.distributions.Gamma
import math.log, math.sqrt
import scala.annotation.tailrec

abstract class HMC(ul: Int, ue: Double) {
val uDev = new Uniform(0.0,1.0)
val nDev = new Gaussian(0.0,1.0)
val l = ul
val ep = ue

final def metCheck(ho: Double, hn: Double): Boolean = log(uDev.sample()) < ho - hn

def calcU(q: DenseVector[Double], sigs: List[Double]): Double

def calcUgrad(q: DenseVector[Double], sigs: List[Double]): DenseVector[Double]

def gibbsUpdate(q: DenseVector[Double]): List[Double]

final def hmcUpdate(qo: DenseVector[Double], sigs: List[Double], acc: Int): (DenseVector[Double], Int) = {
val uep = ep * min(sigs) * min(sigs) * (0.9 + 0.2 * uDev.sample()) // perturb ep, l pm 10% to avoid cyclic orbits
val ul = (l.toDouble * (0.9 + 0.2 * uDev.sample())).toInt
val po = DenseVector(nDev.sample(qo.length).toArray)
val ho = calcU(qo, sigs) + (po dot po) / 2.0
@tailrec
def leapFrog(qo: DenseVector[Double], po: DenseVector[Double], ll: Int): (DenseVector[Double], DenseVector[Double]) = {
if (ll == 1) {
val qn = qo + (po :* uep)
val pn = po - (calcUgrad(qn, sigs) :* uep) / 2.0
(qn, pn)
}
else {
val qn = qo + (po :* uep)
val pn = po - (calcUgrad(qn, sigs) :* uep)
leapFrog(qn, pn, ll - 1)
}
}

val (qn, pn) = leapFrog(qo, po - (calcUgrad(qo, sigs) :* uep) / 2.0, ul)
val hn = calcU(qn, sigs) + (pn dot pn) / 2.0
if (metCheck(ho,hn)) (qn, acc + 1) else (qo, acc)
}

final def stdDevSample(sqmis: Double, n: Int) = sqrt(
sqmis / (2.0 * Gamma((n.toDouble + 3) / 2.0, 1.0).sample())
)

//Pass initial values to rl
@tailrec
final def hmcRunner(maxIt: Int, report: Int, iter: Int, acc: Int,
rl: List[(DenseVector[Double], List[Double])]):
List[(DenseVector[Double], List[Double])] = {
if (iter == maxIt) {
val (qn, accn) = hmcUpdate(rl(0)._1, rl(0)._2, acc)
val gibbsVar = gibbsUpdate(qn)
(qn, gibbsVar) :: rl
}
else {
val (qn, accn) = hmcUpdate(rl(0)._1, rl(0)._2, acc)
val gibbsVar = gibbsUpdate(qn)
val currU = calcU(qn, gibbsVar)
if (iter % report == 0) {
print("\nIteration "); print(iter); print(" of "); println(maxIt)
print("U: "); println(currU)
print("Acceptance: "); println(acc.toDouble / iter.toDouble)
}
hmcRunner(maxIt, report, iter + 1, accn, (qn, gibbsVar) :: rl)
}
}
}

• You should also provide a reference to the algorithm you are implementing. It does not seem to be HMC. Maybe some modification of HMC? I don't think gibbsUpdate belongs in normal HMC. – toto2 Sep 1 '14 at 16:28
• You can tell its a rewrite of a Fortran program hey :p ? In seriousness, the 1 and 2 letter long variables correspond to mathematical entities (vectors of parameters etc) using the standard notation in the literature – Shikamablue Sep 1 '14 at 21:03
• It is a good point about comments though; I've been conflicted, reading different viewpoints on the amount of commenting one should do. I normally default to over commenting so I though i would try something different in this case, but now I agree with you that the lack of comments impairs readability. – Shikamablue Sep 1 '14 at 21:05

I rewrote parts of your code (see below). I did not have enough time to complete the refactoring, so it is far from I would really want to see, but at least it is not Fortran anymore.

• You can use trait instead of abstract class. It is more flexible.

• The names, pretty much all names, are very uninformative. Why uDev for a distribution?

• You should separate the printing of reports from the actual algorithm (separation of concerns). This advice applies to all languages, not just Scala.

• Avoid using recursion. It is like the assembly language of functional programming. Use reduce and such instead.

• It looks like you don't know about "string" + " " + "concantetation" or, even better, interpolation: s"Hello, ${name}". • In an if expression, when the code at the start of both if and else is the same, you should pull that code out. • You call min(sigs) twice in a row. You should set a variable with that value and use it twice. • You can shorten the imports: import math.{log, sqrt}. You should also explicitly write out all imports instead of doing a bulk import (import breeze.linalg._). I was confused by where min was coming from. • Terminating a recursive loop on l == 1 looks odd in Scala; it's usually on 0. • I'm usually a big fan of breaking everything in small functions, but metCheck is so important that I think the computation should be left as code where it is called. • Don't pass acc to hmcUpdate. That function has no business dealing with that accumulator (separation of concern). • The OO/functional design was pretty much inexistent. I did some, but it is currently very poor due to lack of time. A good design greatly improves readability. You should spend some time to properly break everything down in classes. And then think some more about it and do it, etc. After some partial refactoring:  val unifDist = new Uniform(0.0, 1.0) val normDist = new Gaussian(0.0, 1.0) // Poor names. I don't know what they are. val l = ul val ep = ue type Position = DenseVector[Double] type Momentum = DenseVector[Double] case class HamState(q: Position, p: Momentum) def leapFrogStep(potentialEnergy: PotentialEnergy, uep: Double)(hamState: HamState): HamState = { val q = hamState.q + (hamState.p :* uep) val p = hamState.p - (potentialEnergy.calcGradU(hamState.q) :* uep) HamState(q, p) } def leapFrogMomemtumHalfStep(potentialEnergy: PotentialEnergy, uep: Double)(hamState: HamState): HamState = { val p = hamState.p - (potentialEnergy.calcGradU(hamState.q) :* uep) / 2.0 HamState(hamState.q, p) } trait PotentialEnergy { def calcU(q: Position): Double def calcGradU(q: Position): DenseVector[Double] def computeTotalEnergy(hamState: HamState) = calcU(hamState.q) + (hamState.p dot hamState.p) / 2.0 // I don't have the slightest idea what uep is... // I put it in this class, but it certainly does not belong here. def uep(): Double } case class WhateverSigsIsPotentialEnergy(sigs: List[Double]) extends PotentialEnergy { override def calcU(q: Position) = calcU_(q, sigs) override def calcGradU(q: Position) = calcUgrad_(q, sigs) // uep does not belong here. It should not access "global" ep or uDev. override def uep = ep * min(sigs) * min(sigs) * (0.9 + 0.2 * unifDist.sample()) // perturb ep, l pm 10% to avoid cyclic orbits } def hmcStep(potentialEnergy: PotentialEnergy)(q: Position): Option[Position] = { val uep = potentialEnergy.uep val nSteps = (l.toDouble * (0.9 + 0.2 * unifDist.sample())).toInt val p = DenseVector(normDist.sample(q.length).toArray) val h = potentialEnergy.calcU(q) + (p dot p) / 2.0 val initState = HamState(q, p) def halfStepper = leapFrogMomemtumHalfStep(potentialEnergy, uep)(_) def fullStepper = leapFrogStep(potentialEnergy, uep)(_) def fullPathTransform = (halfStepper :: List.fill(nSteps - 1)(fullStepper)) :+ halfStepper val state = fullPathTransform.foldLeft(initState) { (state, stepper) => stepper(state) } val energyChange = potentialEnergy.computeTotalEnergy(state) - potentialEnergy.computeTotalEnergy(initState) if (unifDist.sample() < math.exp(-energyChange)) Some(state.q) else None } type GibbsSomethings = List[Double] case class PositionAndGibbs(pos: Position, gibbs: GibbsSomethings) /** * @return Boolean true if the generated position is different than the previous one. */ def hmcAndGibbsStep(potentialEnergy: PotentialEnergy)(positionAndGibbs: PositionAndGibbs): (PositionAndGibbs, Boolean) = { var posOption = hmcStep(potentialEnergy)(positionAndGibbs.pos) var pos = posOption.getOrElse(positionAndGibbs.pos) var gibbs = gibbsUpdate(pos) (PositionAndGibbs(pos, gibbs), (pos != None)) } def hmcRunner(potentialEnergy: PotentialEnergy)(maxIt: Int, prevState: PositionAndGibbs): List[PositionAndGibbs] = { val infiniteStream: Stream[(PositionAndGibbs, Boolean)] = { def loop(stateAndBool: (PositionAndGibbs, Boolean)): Stream[(PositionAndGibbs, Boolean)] = stateAndBool #:: loop(hmcAndGibbsStep(potentialEnergy)(stateAndBool._1)) loop((prevState, false)) } val finiteStream = infiniteStream.take(maxIt) val visitedStates = finiteStream.foldLeft(List[PositionAndGibbs]()) { case (visitedStates, (posAndG, isNew)) => posAndG :: visitedStates } visitedStates // To print some report every "iter" iterations: // 1) change the foldLeft above so it also counts the "isNew" // 2) instead of calling once ".take(maxIt)", write a function that // only takes "iter" at a time an then prints out some report. // Call that method until the total number of iterations reaches "maxIt". }  • Thank you very much for your answer; there is a lot to think about (I particularly liked the use of Stream)! I might point out for the sake of posterity that your hmcStep algorithm computes one too many full momentum steps (you've totalled nstep+1 momentum steps including the two half steps). – Shikamablue Sep 2 '14 at 5:27 • Fixed: I removed a full step. – toto2 Sep 2 '14 at 11:56 Overall I'd say your off to a great start. Here are some of the changes I would consider making to your code: ## Formating Output When printing strings that encapsulate values/variables you have two (idiomatic) choices in Scala. Option one is to use string interpolation:  println(s"\nIteration$iter of $maxIt") println(s"U:$currU")
// to print the result of an expression surround it with curly braces
println(s"Acceptance: \${acc.toDouble / iter.toDouble}")


Option two:

   println("\nInteration "+ iter +" of "+ maxIt)
println("U: "+ currU)
println("Acceptance: "+ acc.toDouble / iter.toDouble)


I haven't tested it, but option one or two (or some slight variation of them) will mirror the functionality you have in your code while also being easier to write and read.

## Pattern Matching

Within the helper function leapFrog each layer of the recursion checks the value of ll against some constant value using an if statement. The more idiomatic way to do this in Scala is to use pattern matching. For example:

    @tailrec
def leapFrog(qo: T, po: T, ll: Int): (T, T) = ll match {
case c =>
val qn = ...
val pn = ...
(qn, pn)

case _ =>
val qn = ...
val pn = ...
leapFrog(qn, pn, ll - 1)
}


Note that in the first case clause you can replace c with any arbitrary constant that you want to match ll against. In the second case clause _ simply means match ll against the wildcard, that is, ll can have any value or type and the block will get evaluated.
As a general Scala tip I'd recommend learning about the many other ways in which pattern matching can be used to create concise code. There are a couple of other places in your code where you could apply pattern matching, but I'll leave those to you as an exercise :)

## Code Legibility

Many of the values and functions in your program have non-descriptive names. If you plan on only sharing this code with your colleagues who have familiarity with the domain, similar algorithms, and the mathematics behind it all then there may be no need to be descriptive. However, it is also easy to cite whatever source you got this from at the top of the code so that future users can have a frame of reference.

• Thank you for your answer; the case matching especially has helped to pare things down. – Shikamablue Sep 2 '14 at 5:27