Part 6: Hamiltonian Monte Carlo (HMC)

Introduction

This is the sixth part in a series of posts on MCMC-based Bayesian inference for a logistic regression model. If you are new to this series, please go back to Part 1.

In the previous post we saw how to construct an MCMC algorithm utilising gradient information by considering a Langevin equation having our target distribution of interest as its equilibrium. This equation has a physical interpretation in terms of the stochastic dynamics of a particle in a potential equal to minus the log of the target density. It turns out that thinking about the deterministic dynamics of a particle in such a potential can lead to more efficient MCMC algorithms.

Hamiltonian dynamics

Hamiltonian dynamics is often presented as an extension of a fairly general version of Lagrangian dynamics. However, for our purposes a rather simple version is quite sufficient, based on basic concepts from Newtonian dynamics, familiar from school. Inspired by our Langevin example, we will consider the dynamics of a particle in a potential function $V(q)$. We will see later why we want $V(q) = -\log \pi(q)$ for our target of interest, $\pi(\cdot)$. In the context of Hamiltonian (and Lagrangian) dynamics we typically use $q$ as our position variable, rather than $x$.

The potential function induces a (conservative) force on the particle equal to $-\nabla V(q)$ when the particle is at position $q$. Then Newton’s second law of motion, "F=ma", takes the form

$\displaystyle \nabla V(q) + m \ddot{q} = 0.$

In Newtonian mechanics, we often consider the position vector $q$ as 3-dimensional. Here it will be $n$-dimensional, where $n$ is the number of variables in our target. We can then think of our second law as governing a single $n$-dimensional particle of mass $m$, or $n$ one-dimensional particles all of mass $m$. But in this latter case, there is no need to assume that all particles have the same mass, and we could instead write our law of motion as

$\displaystyle \nabla V(q) + M \ddot{q} = 0,$

where $M$ is a diagonal matrix. But in fact, since we could change coordinates, there’s no reason to require that $M$ is diagonal. All we need is that $M$ is positive definite, so that we don’t have negative mass in any coordinate direction.

We will take the above equation as the fundamental law governing our dynamical system of interest. The motivation from Newtonian dynamics is interesting, but not required. What is important is that the dynamics of such a system are conservative, in a way that we will shortly make precise.

Our law of motion is a second-order differential equation, since it involves the second derivative of $q$ wrt time. If you’ve ever studied differential equations, you’ll know that there is an easy way to turn a second order equation into a first order equation with twice the dimension by augmenting the system with the velocities. Here, it is more convenient to augment the system with "momentum" variables, $p$, which we define as $p = M\dot{q}$. Then we can write our second order system as a pair of first order equations

$\displaystyle \dot{q} = M^{-1}p$

$\displaystyle \dot{p} = -\nabla V(q)$

These are, in fact, Hamilton’s equations for this system, though this isn’t how they are typically written.

If we define the kinetic energy as

$\displaystyle T(p) = \frac{1}{2}p^\text{T}M^{-1}p,$

then the Hamiltonian

$\displaystyle H(q,p) = V(q) + T(p),$

representing the total energy in the system, is conserved, since

$\displaystyle \dot{H} = \nabla V\cdot \dot{q} + \dot{p}^\text{T}M^{-1}p = \nabla V\cdot \dot{q} + \dot{p}^\text{T}\dot{q} = [\nabla V + \dot{p}]\cdot\dot{q} = 0.$

So, if we obey our Hamiltonian dynamics, our trajectory in $(q,p)$-space will follow contours of the Hamiltonian. It’s also clear that the system is time-reversible, so flipping the sign of the momentum $p$ and integrating will exactly reverse the direction in which the contours are traversed. Another quite important property of Hamiltonian dynamics is that they are volume preserving. This can be verified by checking that the divergence of the flow is zero.

$\displaystyle \nabla\cdot(\dot{q},\dot{p}) = \nabla_q\cdot\dot{q} + \nabla_p\cdot\dot{p} = 0,$

since $\dot{q}$ is a function of $p$ only and $\dot{p}$ is a function of $q$ only.

Hamiltonian Monte Carlo (HMC)

In Hamiltonian Monte Carlo we introduce an augmented target distribution,

$\displaystyle \tilde \pi(q,p) \propto \exp[-H(q,p)]$

It is clear from this definition that moves leaving the Hamiltonian invariant will also leave the augmented target density unchanged. By following the Hamiltonian dynamics, we will be able to make big (reversible) moves in the space that will be accepted with high probability. Also, our target factorises into two independent components as

$\displaystyle \tilde \pi(q,p) \propto \exp[-V(q)]\exp[-T(p)],$

and so choosing $V(q)=-\log \pi(q)$ will ensure that the $q$-marginal is our real target of interest, $\pi(\cdot)$. It’s also clear that our $p$-marginal is $\mathcal N(0,M)$. This is also the full-conditional for $p$, so re-sampling $p$ from this distribution and leaving $q$ unchanged is a Gibbs move that will leave the augmented target invariant. Re-sampling $p$ will be necessary to properly explore our augmented target, since this will move us to a different contour of $H$.

So, an idealised version of HMC would proceed as follows: First, update $p$ by sampling from its known tractable marginal. Second, update $p$ and $q$ jointly by following the Hamiltonian dynamics. If this second move is regarded as a (deterministic) reversible M-H proposal, it will be accepted with probability one since it leaves the augmented target density unchanged. If we could exactly integrate Hamilton’s equations, this would be fine. But in practice, we will need to use some imperfect numerical method for the integration step. But just as for MALA, we can regard the numerical method as a M-H proposal and correct for the fact that it is imperfect, preserving the exact augmented target distribution.

Hamiltonian systems admit nice numerical integration schemes called symplectic integrators. In HMC a simple alternating Euler method is typically used, known as the leap-frog algorithm. The component updates are all shear transformations, and therefore volume preserving, and exact reversibility is ensured by starting and ending with a half-step update of the momentum variables. In principle, to ensure reversibility of the proposal the momentum variables should be sign-flipped (reversed) to finish, but in practice this doesn’t matter since it doesn’t affect the evaluation of the Hamiltonian and it will then get refreshed, anyway.

So, advancing our system by a time step $\epsilon$ can be done with

$\displaystyle p(t+\epsilon/2) := p(t) - \frac{\epsilon}{2}\nabla V(q(t))$

$\displaystyle q(t+\epsilon) := q(t) + \epsilon M^{-1}p(t+\epsilon/2)$

$\displaystyle p(t+\epsilon) := p(t+\epsilon/2) - \frac{\epsilon}{2}\nabla V(q(t+\epsilon))$

It is clear that if many such updates are chained together, adjacent momentum updates can be collapsed together, giving rise to the "leap-frog" nature of the algorithm, and therefore requiring roughly one gradient evaluation per $\epsilon$ update, rather than two. Since this integrator is volume preserving and exactly reversible, for reasonably small $\epsilon$ it follows the Hamiltonian dynamics reasonably well, but not exactly, and so it does not exactly preserve the Hamiltonian. However, it does make a good M-H proposal, and reasonable acceptance probabilities can often be obtained by chaining together $l$ updates to advance the time of the system by $T=l\epsilon$. The "optimal" value of $l$ and $\epsilon$ will be highly problem dependent, but values of $l=20$ or $l=50$ are not unusual. There are various more-or-less standard methods for tuning these, but we will not consider them here.

Note that since our HMC update on the augmented space consists of a Gibbs move and a M-H update, it is important that our M-H kernel does not keep or thread through the old log target density from the previous M-H update, since the Gibbs move will have changed it in the meantime.

Implementations

R

We need a M-H kernel that does not thread through the old log density.

mhKernel = function(logPost, rprop)
function(x) {
prop = rprop(x)
a = logPost(prop) - logPost(x)
if (log(runif(1)) < a)
prop
else
x
}


We can then use this to construct a M-H move as part of our HMC update.

hmcKernel = function(lpi, glpi, eps = 1e-4, l=10, dmm = 1) {
sdmm = sqrt(dmm)
leapf = function(q, p) {
p = p + 0.5*eps*glpi(q)
for (i in 1:l) {
q = q + eps*p/dmm
if (i < l)
p = p + eps*glpi(q)
else
p = p + 0.5*eps*glpi(q)
}
list(q=q, p=-p)
}
alpi = function(x)
lpi(x$q) - 0.5*sum((x$p^2)/dmm)
rprop = function(x)
leapf(x$q, x$p)
mhk = mhKernel(alpi, rprop)
function(q) {
d = length(q)
x = list(q=q, p=rnorm(d, 0, sdmm))
mhk(x)$q } }  See the full runnable script for further details. Python First a M-H kernel, def mhKernel(lpost, rprop): def kernel(x): prop = rprop(x) a = lpost(prop) - lpost(x) if (np.log(np.random.rand()) < a): x = prop return x return kernel  and then an HMC kernel. def hmcKernel(lpi, glpi, eps = 1e-4, l=10, dmm = 1): sdmm = np.sqrt(dmm) def leapf(q, p): p = p + 0.5*eps*glpi(q) for i in range(l): q = q + eps*p/dmm if (i < l-1): p = p + eps*glpi(q) else: p = p + 0.5*eps*glpi(q) return (q, -p) def alpi(x): (q, p) = x return lpi(q) - 0.5*np.sum((p**2)/dmm) def rprop(x): (q, p) = x return leapf(q, p) mhk = mhKernel(alpi, rprop) def kern(q): d = len(q) p = np.random.randn(d)*sdmm return mhk((q, p))[0] return kern  See the full runnable script for further details. JAX Again, we want an appropriate M-H kernel, def mhKernel(lpost, rprop, dprop = jit(lambda new, old: 1.)): @jit def kernel(key, x): key0, key1 = jax.random.split(key) prop = rprop(key0, x) ll = lpost(x) lp = lpost(prop) a = lp - ll + dprop(x, prop) - dprop(prop, x) accept = (jnp.log(jax.random.uniform(key1)) < a) return jnp.where(accept, prop, x) return kernel  and then an HMC kernel. def hmcKernel(lpi, glpi, eps = 1e-4, l = 10, dmm = 1): sdmm = jnp.sqrt(dmm) @jit def leapf(q, p): p = p + 0.5*eps*glpi(q) for i in range(l): q = q + eps*p/dmm if (i < l-1): p = p + eps*glpi(q) else: p = p + 0.5*eps*glpi(q) return jnp.concatenate((q, -p)) @jit def alpi(x): d = len(x) // 2 return lpi(x[jnp.array(range(d))]) - 0.5*jnp.sum((x[jnp.array(range(d,2*d))]**2)/dmm) @jit def rprop(k, x): d = len(x) // 2 return leapf(x[jnp.array(range(d))], x[jnp.array(range(d, 2*d))]) mhk = mhKernel(alpi, rprop) @jit def kern(k, q): key0, key1 = jax.random.split(k) d = len(q) x = jnp.concatenate((q, jax.random.normal(key0, [d])*sdmm)) return mhk(key1, x)[jnp.array(range(d))] return kern  There is something a little bit strange about this implementation, since the proposal for the M-H move is deterministic, the function rprop just ignores the RNG key that is passed to it. We could tidy this up by making a M-H function especially for deterministic proposals. We won’t pursue this here, but this issue will crop up again later in some of the other functional languages. See the full runnable script for further details. Scala A M-H kernel, def mhKern[S]( logPost: S => Double, rprop: S => S, dprop: (S, S) => Double = (n: S, o: S) => 1.0 ): (S) => S = val r = Uniform(0.0,1.0) x0 => val x = rprop(x0) val ll0 = logPost(x0) val ll = logPost(x) val a = ll - ll0 + dprop(x0, x) - dprop(x, x0) if (math.log(r.draw()) < a) x else x0  and a HMC kernel. def hmcKernel(lpi: DVD => Double, glpi: DVD => DVD, dmm: DVD, eps: Double = 1e-4, l: Int = 10) = val sdmm = sqrt(dmm) def leapf(q: DVD, p: DVD): (DVD, DVD) = @tailrec def go(q0: DVD, p0: DVD, l: Int): (DVD, DVD) = val q = q0 + eps*(p0/:/dmm) val p = if (l > 1) p0 + eps*glpi(q) else p0 + 0.5*eps*glpi(q) if (l == 1) (q, -p) else go(q, p, l-1) go(q, p + 0.5*eps*glpi(q), l) def alpi(x: (DVD, DVD)): Double = val (q, p) = x lpi(q) - 0.5*sum(pow(p,2) /:/ dmm) def rprop(x: (DVD, DVD)): (DVD, DVD) = val (q, p) = x leapf(q, p) val mhk = mhKern(alpi, rprop) (q: DVD) => val d = q.length val p = sdmm map (sd => Gaussian(0,sd).draw()) mhk((q, p))._1  See the full runnable script for further details. Haskell A M-H kernel: mdKernel :: (StatefulGen g m) => (s -> Double) -> (s -> s) -> g -> s -> m s mdKernel logPost prop g x0 = do let x = prop x0 let ll0 = logPost x0 let ll = logPost x let a = ll - ll0 u <- (genContVar (uniformDistr 0.0 1.0)) g let next = if ((log u) < a) then x else x0 return next  Note that here we are using a M-H kernel specifically for deterministic proposals, since there is no non-determinism signalled in the type signature of prop. We can then use this to construct our HMC kernel. hmcKernel :: (StatefulGen g m) => (Vector Double -> Double) -> (Vector Double -> Vector Double) -> Vector Double -> Double -> Int -> g -> Vector Double -> m (Vector Double) hmcKernel lpi glpi dmm eps l g = let sdmm = cmap sqrt dmm leapf q p = let go q0 p0 l = let q = q0 + (scalar eps)*p0/dmm p = if (l > 1) then p0 + (scalar eps)*(glpi q) else p0 + (scalar (eps/2))*(glpi q) in if (l == 1) then (q, -p) else go q p (l - 1) in go q (p + (scalar (eps/2))*(glpi q)) l alpi x = let (q, p) = x in (lpi q) - 0.5*(sumElements (p*p/dmm)) prop x = let (q, p) = x in leapf q p mk = mdKernel alpi prop g in (\q0 -> do let d = size q0 zl <- (replicateM d . genContVar (normalDistr 0.0 1.0)) g let z = fromList zl let p0 = sdmm * z (q, p) <- mk (q0, p0) return q)  See the full runnable script for further details. Dex Again we can use a M-H kernel specific to deterministic proposals. def mdKernel {s} (lpost: s -> Float) (prop: s -> s) (x0: s) (k: Key) : s = x = prop x0 ll0 = lpost x0 ll = lpost x a = ll - ll0 u = rand k select (log u < a) x x0  and use this to construct an HMC kernel. def hmcKernel {n} (lpi: (Fin n)=>Float -> Float) (dmm: (Fin n)=>Float) (eps: Float) (l: Nat) (q0: (Fin n)=>Float) (k: Key) : (Fin n)=>Float = sdmm = sqrt dmm idmm = map (\x. 1.0/x) dmm glpi = grad lpi def leapf (q0: (Fin n)=>Float) (p0: (Fin n)=>Float) : ((Fin n)=>Float & (Fin n)=>Float) = p1 = p0 + (eps/2) .* (glpi q0) q1 = q0 + eps .* (p1*idmm) (q, p) = apply_n l (q1, p1) \(qo, po). pn = po + eps .* (glpi qo) qn = qo + eps .* (pn*idmm) (qn, pn) pf = p + (eps/2) .* (glpi q) (q, -pf) def alpi (qp: ((Fin n)=>Float & (Fin n)=>Float)) : Float = (q, p) = qp (lpi q) - 0.5*(sum (p*p*idmm)) def prop (qp: ((Fin n)=>Float & (Fin n)=>Float)) : ((Fin n)=>Float & (Fin n)=>Float) = (q, p) = qp leapf q p mk = mdKernel alpi prop [k1, k2] = split_key k z = randn_vec k1 p0 = sdmm * z (q, p) = mk (q0, p0) k2 q  Note that the gradient is obtained via automatic differentiation. See the full runnable script for details. Next steps This was the main place that I was trying to get to when I started this series of posts. For differentiable log-posteriors (as we have in the case of Bayesian logistic regression), HMC is a pretty good algorithm for reasonably efficient posterior exploration. But there are lots of places we could go from here. We could explore the tuning of MCMC algorithms, or HMC extensions such as NUTS. We could look at MCMC algorithms that are specifically tailored to the logistic regression problem, or we could look at new MCMC algorithms for differentiable targets based on piecewise deterministic Markov processes. Alternatively, we could temporarily abandon MCMC and look at SMC or ABC approaches. Another possibility would be to abandon this multi-language approach and have a bit of a deep dive into Dex, which I think has the potential to be a great programming language for statistical computing. All of these are possibilities for the future, but I’ve a busy few weeks coming up, so the frequency of these posts is likely to substantially decrease. Remember that all of the code associated with this series of posts is available from this github repo. Advertisement Bayesian inference for a logistic regression model (Part 5) Part 5: the Metropolis-adjusted Langevin algorithm (MALA) Introduction This is the fifth part in a series of posts on MCMC-based Bayesian inference for a logistic regression model. If you are new to this series, please go back to Part 1. In the previous post we saw how to use Langevin dynamics to construct an approximate MCMC scheme using the gradient of the log target distribution. Each step of the algorithm involved simulating from the Euler-Maruyama approximation to the transition kernel of the process, based on some pre-specified step size, $\Delta t$. We can improve the accuracy of this approximation by making the step size smaller, but this will come at the expense of a more slowly mixing MCMC chain. Fortunately, there is an easy way to make the algorithm "exact" (in the sense that the equilibrium distribution of the Markov chain will be the exact target distribution), for any finite step size, $\Delta t$, simply by using the Euler-Maruyama approximation as the proposal distribution in a Metropolis-Hastings algorithm. This is the Metropolis-adjusted Langevin algorithm (MALA). There are various ways this could be coded up, but here, for clarity, a HoF for generating a MALA kernel will be used, and this function will in turn call on a HoF for generating a Metropolis-Hastings kernel. Implementations R First we need a function to generate a M-H kernel. mhKernel = function(logPost, rprop, dprop = function(new, old, ...) { 1 }) function(x, ll) { prop = rprop(x) llprop = logPost(prop) a = llprop - ll + dprop(x, prop) - dprop(prop, x) if (log(runif(1)) < a) list(x=prop, ll=llprop) else list(x=x, ll=ll) }  Then we can easily write a function for returning a MALA kernel that makes use of this M-H function. malaKernel = function(lpi, glpi, dt = 1e-4, pre = 1) { sdt = sqrt(dt) spre = sqrt(pre) advance = function(x) x + 0.5*pre*glpi(x)*dt mhKernel(lpi, function(x) rnorm(p, advance(x), spre*sdt), function(new, old) sum(dnorm(new, advance(old), spre*sdt, log=TRUE))) }  Notice that our MALA function requires as input both the gradient of the log posterior (for the proposal) and the log posterior itself (for the M-H correction). Other details are as we have already seen – see the full runnable script. Python Again, we need a M-H kernel def mhKernel(lpost, rprop, dprop = lambda new, old: 1.): def kernel(x, ll): prop = rprop(x) lp = lpost(prop) a = lp - ll + dprop(x, prop) - dprop(prop, x) if (np.log(np.random.rand()) < a): x = prop ll = lp return x, ll return kernel  and then a MALA kernel def malaKernel(lpi, glpi, dt = 1e-4, pre = 1): p = len(init) sdt = np.sqrt(dt) spre = np.sqrt(pre) advance = lambda x: x + 0.5*pre*glpi(x)*dt return mhKernel(lpi, lambda x: advance(x) + np.random.randn(p)*spre*sdt, lambda new, old: np.sum(sp.stats.norm.logpdf(new, loc=advance(old), scale=spre*sdt)))  See the full runnable script for further details. JAX If we want our algorithm to run fast, and if we want to exploit automatic differentiation to avoid the need to manually compute gradients, then we can easily convert the above code to use JAX. def mhKernel(lpost, rprop, dprop = jit(lambda new, old: 1.)): @jit def kernel(key, x, ll): key0, key1 = jax.random.split(key) prop = rprop(key0, x) lp = lpost(prop) a = lp - ll + dprop(x, prop) - dprop(prop, x) accept = (jnp.log(jax.random.uniform(key1)) < a) return jnp.where(accept, prop, x), jnp.where(accept, lp, ll) return kernel def malaKernel(lpi, dt = 1e-4, pre = 1): p = len(init) glpi = jit(grad(lpost)) sdt = jnp.sqrt(dt) spre = jnp.sqrt(pre) advance = jit(lambda x: x + 0.5*pre*glpi(x)*dt) return mhKernel(lpi, jit(lambda k, x: advance(x) + jax.random.normal(k, [p])*spre*sdt), jit(lambda new, old: jnp.sum(jsp.stats.norm.logpdf(new, loc=advance(old), scale=spre*sdt))))  See the full runnable script for further details. Scala def mhKernel[S]( logPost: S => Double, rprop: S => S, dprop: (S, S) => Double = (n: S, o: S) => 1.0 ): ((S, Double)) => (S, Double) = val r = Uniform(0.0,1.0) state => val (x0, ll0) = state val x = rprop(x0) val ll = logPost(x) val a = ll - ll0 + dprop(x0, x) - dprop(x, x0) if (math.log(r.draw()) < a) (x, ll) else (x0, ll0) def malaKernel(lpi: DVD => Double, glpi: DVD => DVD, pre: DVD, dt: Double = 1e-4) = val sdt = math.sqrt(dt) val spre = sqrt(pre) val p = pre.length def advance(beta: DVD): DVD = beta + (0.5*dt)*(pre*:*glpi(beta)) def rprop(beta: DVD): DVD = advance(beta) + sdt*spre.map(Gaussian(0,_).sample()) def dprop(n: DVD, o: DVD): Double = val ao = advance(o) (0 until p).map(i => Gaussian(ao(i), spre(i)*sdt).logPdf(n(i))).sum mhKernel(lpi, rprop, dprop)  See the full runnable script for further details. Haskell mhKernel :: (StatefulGen g m) => (s -> Double) -> (s -> g -> m s) -> (s -> s -> Double) -> g -> (s, Double) -> m (s, Double) mhKernel logPost rprop dprop g (x0, ll0) = do x <- rprop x0 g let ll = logPost(x) let a = ll - ll0 + (dprop x0 x) - (dprop x x0) u <- (genContVar (uniformDistr 0.0 1.0)) g let next = if ((log u) < a) then (x, ll) else (x0, ll0) return next malaKernel :: (StatefulGen g m) => (Vector Double -> Double) -> (Vector Double -> Vector Double) -> Vector Double -> Double -> g -> (Vector Double, Double) -> m (Vector Double, Double) malaKernel lpi glpi pre dt g = let sdt = sqrt dt spre = cmap sqrt pre p = size pre advance beta = beta + (scalar (0.5*dt))*pre*(glpi beta) rprop beta g = do zl <- (replicateM p . genContVar (normalDistr 0.0 1.0)) g let z = fromList zl return$ advance(beta) + (scalar sdt)*spre*z
dprop n o = let
in sum $(\i -> logDensity (normalDistr (ao!i) ((spre!i)*sdt)) (n!i)) <$> [0..(p-1)]
in mhKernel lpi rprop dprop g


See the full runnable script for further details.

Dex

Recall that Dex is differentiable, so we don’t need to provide gradients.

def mhKernel {s} (lpost: s -> Float) (rprop: s -> Key -> s) (dprop: s -> s -> Float)
(sll: (s & Float)) (k: Key) : (s & Float) =
(x0, ll0) = sll
[k1, k2] = split_key k
x = rprop x0 k1
ll = lpost x
a = ll - ll0 + (dprop x0 x) - (dprop x x0)
u = rand k2
select (log u < a) (x, ll) (x0, ll0)

def malaKernel {n} (lpi: (Fin n)=>Float -> Float)
(pre: (Fin n)=>Float) (dt: Float) :
((Fin n)=>Float & Float) -> Key -> ((Fin n)=>Float & Float) =
sdt = sqrt dt
spre = sqrt pre
v = dt .* pre
vinv = map (\ x. 1.0/x) v
def advance (beta: (Fin n)=>Float) : (Fin n)=>Float =
beta + (0.5*dt) .* (pre*(glp beta))
def rprop (beta: (Fin n)=>Float) (k: Key) : (Fin n)=>Float =
(advance beta) + sdt .* (spre*(randn_vec k))
def dprop (new: (Fin n)=>Float) (old: (Fin n)=>Float) : Float =
diff = new - ao
-0.5 * sum ((log v) + diff*diff*vinv)
mhKernel lpi rprop dprop


See the full runnable script for further details.

Next steps

MALA gives us an MCMC algorithm that exploits gradient information to generate "informed" M-H proposals. But it still has a rather "diffusive" character, making it difficult to tune in such a way that large moves are likely to be accepted in challenging high-dimensional situations.

The Langevin dynamics on which MALA is based can be interpreted as the (over-damped) stochastic dynamics of a particle moving in a potential energy field corresponding to minus the log posterior. It turns out that the corresponding deterministic dynamics can be exploited to generate proposals better able to make large moves while still having a high probability of acceptance. This is the idea behind Hamiltonian Monte Carlo (HMC), which we’ll look at next.

Part 4: Gradients and the Langevin algorithm

Introduction

This is the fourth part in a series of posts on MCMC-based Bayesian inference for a logistic regression model. If you are new to this series, please go back to Part 1.

In the previous post we saw how the Metropolis algorithm could be used to generate a Markov chain targeting our posterior distribution. In high dimensions the diffusive nature of the Metropolis random walk proposal becomes increasingly inefficient. It is therefore natural to try and develop algorithms that use additional information about the target distribution. In the case of a differentiable log posterior target, a natural first step in this direction is to try and make use of gradient information.

Gradient of a logistic regression model

There are various ways to derive the gradient of our logistic regression model, but it might be simplest to start from the first form of the log likelihood that we deduced in Part 2:

$\displaystyle l(\beta;y) = y^\textsf{T}X\beta - \mathbf{1}^\textsf{T}\log(\mathbf{1}+\exp[X\beta])$

We can write this out in component form as

$\displaystyle l(\beta;y) = \sum_j\sum_j y_iX_{ij}\beta_j - \sum_i\log\left(1+\exp\left[\sum_jX_{ij}\beta_j\right]\right).$

Differentiating wrt $\beta_k$ gives

$\displaystyle \frac{\partial l}{\partial \beta_k} = \sum_i y_iX_{ik} - \sum_i \frac{\exp\left[\sum_j X_{ij}\beta_j\right]X_{ik}}{1+\exp\left[\sum_j X_{ij}\beta_j\right]}.$

It’s then reasonably clear that stitching all of the partial derivatives together will give the gradient vector

$\displaystyle \nabla l = X^\textsf{T}\left[ y - \frac{\mathbf{1}}{\mathbf{1}+\exp[-X\beta]} \right].$

This is the gradient of the log likelihood, but we also need the gradient of the log prior. Since we are assuming independent $\beta_i \sim N(0,v_i)$ priors, it is easy to see that the gradient of the log prior is just $-\beta\circ v^{-1}$. It is the sum of these two terms that gives the gradient of the log posterior.

R

In R we can implement our gradient function as

glp = function(beta) {
glpr = -beta/(pscale*pscale)
gll = as.vector(t(X) %*% (y - 1/(1 + exp(-X %*% beta))))
glpr + gll
}


Python

In Python we could use

def glp(beta):
glpr = -beta/(pscale*pscale)
gll = (X.T).dot(y - 1/(1 + np.exp(-X.dot(beta))))
return (glpr + gll)


We don’t really need a JAX version, since JAX can auto-diff the log posterior for us.

Scala

  def glp(beta: DVD): DVD =
val glpr = -beta /:/ pvar
val gll = (X.t)*(y - ones/:/(ones + exp(-X*beta)))
glpr + gll



Using hmatrix we could use something like

glp :: Matrix Double -> Vector Double -> Vector Double -> Vector Double
glp x y b = let
glpr = -b / (fromList [100.0, 1, 1, 1, 1, 1, 1, 1])
gll = (tr x) #> (y - (scalar 1)/((scalar 1) + (cmap exp (-x #> b))))
in glpr + gll


There’s something interesting to say about Haskell and auto-diff, but getting into this now will be too much of a distraction. I may come back to it in some future post.

Dex

Dex is differentiable, so we don’t need a gradient function – we can just use grad lpost. However, for interest and comparison purposes we could nevertheless implement it directly with something like

prscale = map (\ x. 1.0/x) pscale

def glp (b: (Fin 8)=>Float) : (Fin 8)=>Float =
glpr = -b*prscale*prscale
gll = (transpose x) **. (y - (map (\eta. 1.0/(1.0 + eta)) (exp (-x **. b))))
glpr + gll


Langevin diffusions

Now that we have a way of computing the gradient of the log of our target density we need some MCMC algorithms that can make good use of it. In this post we will look at a simple approximate MCMC algorithm derived from an overdamped Langevin diffusion model. In subsequent posts we’ll look at more sophisticated, exact MCMC algorithms.

The multivariate stochastic differential equation (SDE)

$\displaystyle dX_t = \frac{1}{2}\nabla\log\pi(X_t)dt + dW_t$

has $\pi(\cdot)$ as its equilibrium distribution. Informally, an SDE of this form is a continuous time process with infinitesimal transition kernel

$\displaystyle X_{t+dt}|(X_t=x_t) \sim N\left(x_t+\frac{1}{2}\nabla\log\pi(x_t)dt,\mathbf{I}dt\right).$

There are various more-or-less formal ways to see that $\pi(\cdot)$ is stationary. A good way is to check it satisfies the Fokker–Planck equation with zero LHS. A less formal approach would be to see that the infinitesimal transition kernel for the process satisfies detailed balance with $\pi(\cdot)$.

Similar arguments show that for any fixed positive definite matrix $A$, the SDE

$\displaystyle dX_t = \frac{1}{2}A\nabla\log\pi(X_t)dt + A^{1/2}dW_t$

also has $\pi(\cdot)$ as a stationary distribution. It is quite common to choose a diagonal matrix $A$ to put the components of $X_t$ on a common scale.

Simulating exact sample paths from SDEs such as the overdamped Langevin diffusion model is typically difficult (though not necessarily impossible), so we instead want something simple and tractable as the basis of our MCMC algorithms. Here we will just simulate from the Euler–Maruyama approximation of the process by choosing a small but finite time step $\Delta t$ and using the transition kernel

$\displaystyle X_{t+\Delta t}|(X_t=x_t) \sim N\left(x_t+\frac{1}{2}A\nabla\log\pi(x_t)\Delta t, A\Delta t\right)$

as the basis of our MCMC method. For sufficiently small $\Delta t$ this should accurately approximate the Langevin dynamics, leading to an equilibrium distribution very close to $\pi(\cdot)$. That said, we would like to choose $\Delta t$ as large as we can get away with, since that will lead to a more rapidly mixing MCMC chain. Below are some implementations of this kernel for a diagonal pre-conditioning matrix.

Implementation

R

We can create a kernel for the unadjusted Langevin algorithm in R with the following function.

ulKernel = function(glpi, dt = 1e-4, pre = 1) {
sdt = sqrt(dt)
spre = sqrt(pre)
advance = function(x) x + 0.5*pre*glpi(x)*dt
}


Here, we can pass in pre, which is expected to be a vector representing the diagonal of the pre-conditioning matrix, $A$. We can then use this kernel to generate an MCMC chain as we have seen previously. See the full runnable script for further details.

Python

def ulKernel(glpi, dt = 1e-4, pre = 1):
p = len(init)
sdt = np.sqrt(dt)
spre = np.sqrt(pre)
advance = lambda x: x + 0.5*pre*glpi(x)*dt
def kernel(x):
return kernel


See the full runnable script for further details.

JAX

def ulKernel(lpi, dt = 1e-4, pre = 1):
p = len(init)
sdt = jnp.sqrt(dt)
spre = jnp.sqrt(pre)
advance = jit(lambda x: x + 0.5*pre*glpi(x)*dt)
@jit
def kernel(key, x):
return kernel


Note how for JAX we can just pass in the log posterior, and the gradient function can be obtained by automatic differentiation. See the full runnable script for further details.

Scala

def ulKernel(glp: DVD => DVD, pre: DVD, dt: Double): DVD => DVD =
val sdt = math.sqrt(dt)
val spre = sqrt(pre)
beta + (0.5*dt)*(pre*:*glp(beta))


See the full runnable script for further details.

ulKernel :: (StatefulGen g m) =>
(Vector Double -> Vector Double) -> Vector Double -> Double -> g ->
Vector Double -> m (Vector Double)
ulKernel glpi pre dt g beta = do
let sdt = sqrt dt
let spre = cmap sqrt pre
let p = size pre
let advance beta = beta + (scalar (0.5*dt))*pre*(glpi beta)
zl <- (replicateM p . genContVar (normalDistr 0.0 1.0)) g
let z = fromList zl
ll = pair$ll } mat[i, ] = x } if (verb) message("Done.") mat }  Then, in the context of our running logistic regression example, and using the log-posterior from the previous post, we can construct our kernel and run it as follows. pre = c(10.0,1,1,1,1,1,5,1) out = mcmc(init, mhKernel(lpost, function(x) x + pre*rnorm(p, 0, 0.02)), thin=1000)  Note the use of a symmetric proposal, so the proposal density is not required. Also note the use of a larger proposal variance for the intercept term and the second last covariate. See the full runnable script for further details. Python We can do something very similar to R in Python using NumPy. Our HoF for constructing a M-H kernel is def mhKernel(lpost, rprop, dprop = lambda new, old: 1.): def kernel(x, ll): prop = rprop(x) lp = lpost(prop) a = lp - ll + dprop(x, prop) - dprop(prop, x) if (np.log(np.random.rand()) < a): x = prop ll = lp return x, ll return kernel  Our Markov chain runner function is def mcmc(init, kernel, thin = 10, iters = 10000, verb = True): p = len(init) ll = -np.inf mat = np.zeros((iters, p)) x = init if (verb): print(str(iters) + " iterations") for i in range(iters): if (verb): print(str(i), end=" ", flush=True) for j in range(thin): x, ll = kernel(x, ll) mat[i,:] = x if (verb): print("\nDone.", flush=True) return mat  We can use this code in the context of our logistic regression example as follows. pre = np.array([10.,1.,1.,1.,1.,1.,5.,1.]) def rprop(beta): return beta + 0.02*pre*np.random.randn(p) out = mcmc(init, mhKernel(lpost, rprop), thin=1000)  See the full runnable script for further details. JAX The above R and Python scripts are fine, but both languages are rather slow for this kind of workload. Fortunately it’s rather straightforward to convert the Python code to JAX to obtain quite amazing speed-up. We can write our M-H kernel as def mhKernel(lpost, rprop, dprop = jit(lambda new, old: 1.)): @jit def kernel(key, x, ll): key0, key1 = jax.random.split(key) prop = rprop(key0, x) lp = lpost(prop) a = lp - ll + dprop(x, prop) - dprop(prop, x) accept = (jnp.log(jax.random.uniform(key1)) < a) return jnp.where(accept, prop, x), jnp.where(accept, lp, ll) return kernel  and our MCMC runner function as def mcmc(init, kernel, thin = 10, iters = 10000): key = jax.random.PRNGKey(42) keys = jax.random.split(key, iters) @jit def step(s, k): [x, ll] = s x, ll = kernel(k, x, ll) s = [x, ll] return s, s @jit def iter(s, k): keys = jax.random.split(k, thin) _, states = jax.lax.scan(step, s, keys) final = [states[0][thin-1], states[1][thin-1]] return final, final ll = -np.inf x = init _, states = jax.lax.scan(iter, [x, ll], keys) return states[0]  There are really only two slightly tricky things about this code. The first relates to the way JAX handles pseudo-random numbers. Since JAX is a pure functional eDSL, it can’t be used in conjunction with the typical pseudo-random number generators often used in imperative programming languages which rely on a global mutable state. This can be dealt with reasonably straightforwardly by explicitly passing around the random number state. There is a standard way of doing this that has been common practice in functional programming languages for decades. However, this standard approach is very sequential, and so doesn’t work so well in a parallel context. JAX therefore uses a splittable random number generator, where new states are created by splitting the current state into two (or more). We’ll come back to this when we get to the Haskell examples. The second thing that might be unfamiliar to imperative programmers is the use of the scan operation (jax.lax.scan) to generate the Markov chain rather than a "for" loop. But scans are standard operations in most functional programming languages. We can then call this code for our logistic regression example with pre = jnp.array([10.,1.,1.,1.,1.,1.,5.,1.]).astype(jnp.float32) @jit def rprop(key, beta): return beta + 0.02*pre*jax.random.normal(key, [p]) out = mcmc(init, mhKernel(lpost, rprop), thin=1000)  See the full runnable script for further details. Scala In Scala we can use a similar approach to that already seen for defining a HoF to return a M-H kernel. def mhKernel[S]( logPost: S => Double, rprop: S => S, dprop: (S, S) => Double = (n: S, o: S) => 1.0 ): ((S, Double)) => (S, Double) = val r = Uniform(0.0,1.0) state => val (x0, ll0) = state val x = rprop(x0) val ll = logPost(x) val a = ll - ll0 + dprop(x0, x) - dprop(x, x0) if (math.log(r.draw()) < a) (x, ll) else (x0, ll0)  Note that Scala’s static typing does not prevent us from defining a function that is polymorphic in the type of the chain state, which we here call S. Also note that we are adopting a pragmatic approach to random number generation, exploiting the fact that Scala is not a pure functional language, using a mutable generator, and omitting to capture the non-determinism of the rprop function (and the returned kernel) in its type signature. In Scala this is a choice, and we could adopt a purer approach if preferred. We’ll see what such an approach will look like in Haskell, coming up next. Now that we have the kernel, we don’t need to write an explicit runner function since Scala has good support for streaming data. There are many more-or-less sophisticated ways that we can work with data streams in Scala, and the choice depends partly on how pure one is being about tracking effects (such as non-determinism), but here I’ll just use the simple LazyList from the standard library for unfolding the kernel into an infinite MCMC chain before thinning and truncating appropriately.  val pre = DenseVector(10.0,1.0,1.0,1.0,1.0,1.0,5.0,1.0) def rprop(beta: DVD): DVD = beta + pre *:* (DenseVector(Gaussian(0.0,0.02).sample(p).toArray)) val kern = mhKernel(lpost, rprop) val s = LazyList.iterate((init, -Inf))(kern) map (_._1) val out = s.drop(150).thin(1000).take(10000)  See the full runnable script for further details. Haskell Since Haskell is a pure functional language, we need to have some convention regarding pseudo-random number generation. Haskell supports several styles. The most commonly adopted approach wraps a mutable generator up in a monad. The typical alternative is to use a pure functional generator and either explicitly thread the state through code or hide this in a monad similar to the standard approach. However, Haskell also supports the use of splittable generators, so we can consider all three approaches for comparative purposes. The approach taken does affect the code and the type signatures, and even the streaming data abstractions most appropriate for chain generation. Starting with a HoF for producing a Metropolis kernel, an approach using the standard monadic generators could like like mKernel :: (StatefulGen g m) => (s -> Double) -> (s -> g -> m s) -> g -> (s, Double) -> m (s, Double) mKernel logPost rprop g (x0, ll0) = do x <- rprop x0 g let ll = logPost(x) let a = ll - ll0 u <- (genContVar (uniformDistr 0.0 1.0)) g let next = if ((log u) < a) then (x, ll) else (x0, ll0) return next  Note how non-determinism is captured in the type signatures by the monad m. The explicit pure approach is to thread the generator through non-deterministic functions. mKernelP :: (RandomGen g) => (s -> Double) -> (s -> g -> (s, g)) -> g -> (s, Double) -> ((s, Double), g) mKernelP logPost rprop g (x0, ll0) = let (x, g1) = rprop x0 g ll = logPost(x) a = ll - ll0 (u, g2) = uniformR (0, 1) g1 next = if ((log u) < a) then (x, ll) else (x0, ll0) in (next, g2)  Here the updated random number generator state is returned from each non-deterministic function for passing on to subsequent non-deterministic functions. This explicit sequencing of operations makes it possible to wrap the generator state in a state monad giving code very similar to the stateful monadic generator approach, but as already discussed, the sequential nature of this approach makes it unattractive in parallel and concurrent settings. Fortunately the standard Haskell pure generator is splittable, meaning that we can adopt a splitting approach similar to JAX if we prefer, since this is much more parallel-friendly. mKernelP :: (RandomGen g) => (s -> Double) -> (s -> g -> s) -> g -> (s, Double) -> (s, Double) mKernelP logPost rprop g (x0, ll0) = let (g1, g2) = split g x = rprop x0 g1 ll = logPost(x) a = ll - ll0 u = unif g2 next = if ((log u) < a) then (x, ll) else (x0, ll0) in next  Here non-determinism is signalled by passing a generator state (often called a "key" in the context of splittable generators) into a function. Functions receiving a key are responsible for splitting it to ensure that no key is ever used more than once. Once we have a kernel, we need to unfold our Markov chain. When using the monadic generator approach, it is most natural to unfold using a monadic stream mcmc :: (StatefulGen g m) => Int -> Int -> s -> (g -> s -> m s) -> g -> MS.Stream m s mcmc it th x0 kern g = MS.iterateNM it (stepN th (kern g)) x0 stepN :: (Monad m) => Int -> (a -> m a) -> (a -> m a) stepN n fa = if (n == 1) then fa else (\x -> (fa x) >>= (stepN (n-1) fa))  whereas for the explicit approaches it is more natural to unfold into a regular infinite data stream. So, for the explicit sequential approach we could use mcmcP :: (RandomGen g) => s -> (g -> s -> (s, g)) -> g -> DS.Stream s mcmcP x0 kern g = DS.unfold stepUf (x0, g) where stepUf xg = let (x1, g1) = kern (snd xg) (fst xg) in (x1, (x1, g1))  and with the splittable approach we could use mcmcP :: (RandomGen g) => s -> (g -> s -> s) -> g -> DS.Stream s mcmcP x0 kern g = DS.unfold stepUf (x0, g) where stepUf xg = let (x1, g1) = xg x2 = kern g1 x1 (g2, _) = split g1 in (x2, (x2, g2))  Calling these functions for our logistic regression example is similar to what we have seen before, but again there are minor syntactic differences depending on the approach. For further details see the full runnable scripts for the monadic approach, the pure sequential approach, and the splittable approach. Dex Dex is a pure functional language and uses a splittable random number generator, so the style we use is similar to JAX (or Haskell using a splittable generator). We can generate a Metropolis kernel with def mKernel {s} (lpost: s -> Float) (rprop: Key -> s -> s) : Key -> (s & Float) -> (s & Float) = def kern (k: Key) (sll: (s & Float)) : (s & Float) = (x0, ll0) = sll [k1, k2] = split_key k x = rprop k1 x0 ll = lpost x a = ll - ll0 u = rand k2 select (log u < a) (x, ll) (x0, ll0) kern  We can then unfold our Markov chain with def markov_chain {s} (k: Key) (init: s) (kern: Key -> s -> s) (its: Nat) : Fin its => s = with_state init \st. for i:(Fin its). x = kern (ixkey k i) (get st) st := x x  Here we combine Dex’s state effect with a for loop to unfold the stream. See the full runnable script for further details. Next steps As previously discussed, none of these codes are optimised, so care should be taken not to over-interpret running times. However, JAX and Dex are noticeably faster than the alternatives, even running on a single CPU core. Another interesting feature of both JAX and Dex is that they are differentiable. This makes it very easy to develop algorithms using gradient information. In subsequent posts we will think about the gradient of our example log-posterior and how we can use gradient information to develop "better" sampling algorithms. The complete runnable scripts are all available from this public github repo. Bayesian inference for a logistic regression model (Part 2) Part 2: The log posterior Introduction This is the second part in a series of posts on MCMC-based Bayesian inference for a logistic regression model. If you are new to this series, please go back to Part 1. In the previous post we looked at the basic modelling concepts, and how to fit the model using a variety of PPLs. In this post we will prepare for doing MCMC by considering the problem of computing the unnormalised log posterior for the model. We will then see how this posterior can be implemented in several different languages and libraries. Derivation Basic structure In Bayesian inference the posterior distribution is just the conditional distribution of the model parameters given the data, and therefore proportional to the joint distribution of the model and data. We often write this as $\displaystyle \pi(\theta|y) \propto \pi(\theta,y) = \pi(\theta)\pi(y|\theta).$ Taking logs we have $\displaystyle \log \pi(\theta, y) = \log \pi(\theta) + \log \pi(y|\theta).$ So (up to an additive constant) the log posterior is just the sum of the log prior and log likelihood. There are many good (numerical) reasons why we try to work exclusively with the log posterior and try to avoid ever evaluating the raw posterior density. For our example logistic regression model, the parameter vector $\theta$ is just the vector of regression coefficients, $\beta$. We assumed independent mean zero normal priors for the components of this vector, so the log prior is just the sum of logs of normal densities. Many scientific libraries will have built-in functions for returning the log-pdf of standard distributions, but if an explicit form is required, the log of the density of a $N(0,\sigma^2)$ at $x$ is just $\displaystyle -\log(2\pi)/2 - \log|\sigma| - x^2/(2\sigma^2),$ and the initial constant term normalising the density can often be dropped. Log-likelihood (first attempt) Information from the data comes into the log posterior via the log-likelihood. The typical way to derive the likelihood for problems of this type is to assume the usual binary encoding of the data (success 1, failure 0). Then, for a Bernoulli observation with probability $p_i,\ i=1,\ldots,n$, the likelihood associated with observation $y_i$ is $\displaystyle f(y_i|p_i) = \left[ \hphantom{1-}p_i \quad :\ y_i=1 \atop 1-p_i \quad :\ y_i=0 \right. \quad = \quad p_i^{y_i}(1-p_i)^{1-y_i}.$ Taking logs and then switching to parameter $\eta_i=\text{logit}(p_i)$ we have $\displaystyle \log f(y_i|\eta_i) = y_i\eta_i - \log(1+e^{\eta_i}),$ and summing over $n$ observations gives the log likelihood $\displaystyle \log\pi(y|\eta) \equiv \ell(\eta;y) = y\cdot \eta - \mathbf{1}\cdot\log(\mathbf{1}+\exp{\eta}).$ In the context of logistic regression, $\eta$ is the linear predictor, so $\eta=X\beta$, giving $\displaystyle \ell(\beta;y) = y^\textsf{T}X\beta - \mathbf{1}^\textsf{T}\log(\mathbf{1}+\exp[X\beta]).$ This is a perfectly good way of expressing the log-likelihood, and we will come back to it later when we want the gradient of the log-likelihood, but it turns out that there is a similar-but-different way of deriving it that results in an expression that is equivalent but slightly cheaper to evaluate. Log-likelihood (second attempt) For our second attempt, we will assume that the data is coded in a different way. Instead of the usual binary encoding, we will assume that the observation $\tilde y_i$ is 1 for success and -1 for failure. This isn’t really a problem, since the two encodings are related by $\tilde y_i = 2y_i-1$. This new encoding is convenient in the context of a logit parameterisation since then $\displaystyle f(y_i|\eta_i) = \left[ p_i \ :\ \tilde y_i=1\atop 1-p_i\ :\ \tilde y_i=-1 \right. \ = \ \left[ (1+e^{-\eta_i})^{-1} \ :\ \tilde y_i=1\atop (1+e^{\eta_i})^{-1} \ :\ \tilde y_i=-1 \right. \ = \ (1+e^{-\tilde y_i\eta_i})^{-1} ,$ and hence $\displaystyle \log f(y_i|\eta_i) = -\log(1+e^{-\tilde y_i\eta_i}).$ Summing over observations gives $\displaystyle \ell(\eta;\tilde y) = -\mathbf{1}\cdot \log(\mathbf{1}+\exp[-\tilde y\circ \eta]),$ where $\circ$ denotes the Hadamard product. Substituting $\eta=X\beta$ gives the log-likelihood $\displaystyle \ell(\beta;\tilde y) = -\mathbf{1}^\textsf{T} \log(\mathbf{1}+\exp[-\tilde y\circ X\beta]).$ This likelihood is a bit cheaper to evaluate that the one previously derived. If we prefer to write in terms of the original data encoding, we can obviously do so as $\displaystyle \ell(\beta; y) = -\mathbf{1}^\textsf{T} \log(\mathbf{1}+\exp[-(2y-\mathbf{1})\circ (X\beta)]),$ and in practice, it is this version that is typically used. To be clear, as an algebraic function of $\beta$ and $y$ the two functions are different. But they coincide for binary vectors $y$ which is all that matters. Implementation R In R we can create functions for evaluating the log-likelihood, log-prior and log-posterior as follows (assuming that X and y are in scope). ll = function(beta) sum(-log(1 + exp(-(2*y - 1)*(X %*% beta)))) lprior = function(beta) dnorm(beta[1], 0, 10, log=TRUE) + sum(dnorm(beta[-1], 0, 1, log=TRUE)) lpost = function(beta) ll(beta) + lprior(beta)  Python In Python (with NumPy and SciPy) we can define equivalent functions with def ll(beta): return np.sum(-np.log(1 + np.exp(-(2*y - 1)*(X.dot(beta))))) def lprior(beta): return (sp.stats.norm.logpdf(beta[0], loc=0, scale=10) + np.sum(sp.stats.norm.logpdf(beta[range(1,p)], loc=0, scale=1))) def lpost(beta): return ll(beta) + lprior(beta)  JAX Python, like R, is a dynamic language, and relatively slow for MCMC algorithms. JAX is a tensor computation framework for Python that embeds a pure functional differentiable array processing language inside Python. JAX can JIT-compile high-performance code for both CPU and GPU, and has good support for parallelism. It is rapidly becoming the preferred way to develop high-performance sampling algorithms within the Python ecosystem. We can encode our log-posterior in JAX as follows. @jit def ll(beta): return jnp.sum(-jnp.log(1 + jnp.exp(-(2*y - 1)*jnp.dot(X, beta)))) @jit def lprior(beta): return (jsp.stats.norm.logpdf(beta[0], loc=0, scale=10) + jnp.sum(jsp.stats.norm.logpdf(beta[jnp.array(range(1,p))], loc=0, scale=1))) @jit def lpost(beta): return ll(beta) + lprior(beta)  Scala JAX is a pure functional programming language embedded in Python. Pure functional programming languages are intrinsically more scalable and compositional than imperative languages such as R and Python, and are much better suited to exploit concurrency and parallelism. I’ve given a bunch of talks about this recently, so if you are interested in this, perhaps start with the materials for my Laplace’s Demon talk. Scala and Haskell are arguably the current best popular general purpose functional programming languages, so it is possibly interesting to consider the use of these languages for the development of scalable statistical inference codes. Since both languages are statically typed compiled functional languages with powerful type systems, they can be highly performant. However, neither is optimised for numerical (tensor) computation, so you should not expect that they will have performance comparable with optimised tensor computation frameworks such as JAX. We can encode our log-posterior in Scala (with Breeze) as follows:  def ll(beta: DVD): Double = sum(-log(ones + exp(-1.0*(2.0*y - ones)*:*(X * beta)))) def lprior(beta: DVD): Double = Gaussian(0,10).logPdf(beta(0)) + sum(beta(1 until p).map(Gaussian(0,1).logPdf(_))) def lpost(beta: DVD): Double = ll(beta) + lprior(beta)  Spark Apache Spark is a Scala library for distributed "big data" processing on clusters of machines. Despite fundamental differences, there is a sense in which Spark for Scala is a bit analogous to JAX for Python: both Spark and JAX are concerned with scalability, but they are targeting rather different aspects of scalability: JAX is concerned with getting regular sized data processing algorithms to run very fast (on GPUs), whereas Spark is concerned with running huge data processing tasks quickly by distributing work over clusters of machines. Despite obvious differences, the fundamental pure functional computational model adopted by both systems is interestingly similar: both systems are based on lazy transformations of immutable data structures using pure functions. This is a fundamental pattern for scalable data processing transcending any particular language, library or framework. We can encode our log posterior in Spark as follows.  def ll(beta: DVD): Double = df.map{row => val y = row.getAs[Double](0) val x = BDV.vertcat(BDV(1.0),toBDV(row.getAs[DenseVector](1))) -math.log(1.0 + math.exp(-1.0*(2.0*y-1.0)*(x.dot(beta))))}.reduce(_+_) def lprior(beta: DVD): Double = Gaussian(0,10).logPdf(beta(0)) + sum(beta(1 until p).map(Gaussian(0,1).logPdf(_))) def lpost(beta: DVD): Double = ll(beta) + lprior(beta)  Haskell Haskell is an old, lazy pure functional programming language with an advanced type system, and remains the preferred language for the majority of functional programming language researchers. Hmatrix is the standard high performance numerical linear algebra library for Haskell, so we can use it to encode our log-posterior as follows. ll :: Matrix Double -> Vector Double -> Vector Double -> Double ll x y b = (negate) (vsum (cmap log ( (scalar 1) + (cmap exp (cmap (negate) ( (((scalar 2) * y) - (scalar 1)) * (x #> b) ) ))))) pscale :: [Double] -- prior standard deviations pscale = [10.0, 1, 1, 1, 1, 1, 1, 1] lprior :: Vector Double -> Double lprior b = sum$  (\x -> logDensity (normalDistr 0.0 (snd x)) (fst x)) <$> (zip (toList b) pscale) lpost :: Matrix Double -> Vector Double -> Vector Double -> Double lpost x y b = (ll x y b) + (lprior b)  Again, a reminder that, here and elsewhere, there are various optimisations could be done that I’m not bothering with. This is all just proof-of-concept code. Dex JAX proves that a pure functional DSL for tensor computation can be extremely powerful and useful. But embedding such a language in a dynamic imperative language like Python has a number of drawbacks. Dex is an experimental statically typed stand-alone DSL for differentiable array and tensor programming that attempts to combine some of the correctness and composability benefits of powerful statically typed functional languages like Scala and Haskell with the performance benefits of tensor computation systems like JAX. It is currently rather early its development, but seems very interesting, and is already quite useable. We can encode our log-posterior in Dex as follows. def ll (b: (Fin 8)=>Float) : Float = neg$  sum (log (map (\ x. (exp x) + 1) ((map (\ yi. 1 - 2*yi) y)*(x **. b))))

pscale = [10.0, 1, 1, 1, 1, 1, 1, 1] -- prior SDs
prscale = map (\ x. 1.0/x) pscale

def lprior (b: (Fin 8)=>Float) : Float =
bs = b*prscale
neg \$  sum ((log pscale) + (0.5 .* (bs*bs)))

def lpost (b: (Fin 8)=>Float) : Float =
(ll b) + (lprior b)


Next steps

Now that we have a way of evaluating the log posterior, we can think about constructing Markov chains having the posterior as their equilibrium distribution. In the next post we will look at one of the simplest ways of doing this: the Metropolis algorithm.

Complete runnable scripts are available from this public github repo.