mlxmc 0.2.0

MCMC samplers in Apple MLX

mlxmc

MCMC samplers written in Apple MLX, using its grad / vmap / compile transforms. MLX has no probabilistic-programming library yet (nothing like BlackJAX or NumPyro), so this is a first pass at one.

Status: research code. The samplers are tested (moment recovery, Σ-estimation, affine invariance, the autocorrelation diagnostics, and the convergence diagnostics validated against ArviZ, on both the CPU and Metal backends), but the API is young and likely to change. Provided as-is, no guarantees.

What's here

The package lives under src/mlxmc/; runnable demos and the benchmark study are in examples/.

Module (mlxmc.)	Sampler / tool
ensemble	Affine-invariant ensemble (Goodman & Weare 2010 — the emcee stretch move). Gradient-free, tuning-free. make_sampler, run_ensemble.
hmc	Hamiltonian Monte Carlo, identity mass. grad ∘ vmap batched over chains. make_hmc, run_hmc.
preconditioned	Mass-matrix HMC (M = Σ⁻¹). make_phmc, run_phmc.
warmup	Stan-style warmup: dual-averaging step size + windowed dense mass-matrix estimation. warmup, run_chain.
nuts	NUTS (multinomial; Hoffman & Gelman 2014), vectorized over chains. Reports divergences. make_nuts, run_nuts.
result	Result — the unified return type for every sampler: draws (chain, draw, dim), sample_stats (divergences, tree depth), .summary(), .to_arviz().
diagnostics	Convergence: rank-normalized split-R̂, bulk/tail-ESS, MCSE (Vehtari et al. 2021), validated against ArviZ. Efficiency: integrated autocorrelation time + the cross-sampler ESS/sec metric.
transforms	Constrained-parameter bijectors (Exp, Sigmoid, …) + constrain() — sample bounded/positive parameters without hand-writing the log-Jacobian.
targets	Example log-densities: correlated Gaussian, banana, centered / non-centered funnel, with known moments.

Example (examples/)	What it shows
gaussian_ess.py	Ensemble vs identity-mass HMC vs preconditioned HMC by ESS/sec on the Gaussian.
warmup_validation.py	Warmup recovers the true Σ and matches oracle ESS/sec.
hard_targets.py	Banana + funnel benchmark (lscan / dscan modes).
nuts_funnel.py	NUTS correctness on the Gaussian; funnel mode for the masking-overhead study.
affine_invariance.py	Empirical proof of affine invariance (same RNG → bit-identical acceptance under an affine map).
constrained_model.py	Infer (μ, σ>0) of a Normal with a parameter transform, then Result.summary() in natural space.
plot_hard_targets.py	Renders hard_targets_figure.png (needs the optional viz env).

Why MLX

grad, vmap, jvp/vjp, and compile transfer almost directly from JAX, with JAX-style functional RNG keys (mx.random.split). The wrinkles that shape this code:

• No traced control-flow primitives (no while_loop / scan / cond). MLX is eager execution plus compile of static graphs. Fixed-length unrolled loops (leapfrog, fixed-L HMC) compile fine; data-dependent trajectory length (NUTS) is the hard case — mlxmc.nuts runs every chain to a fixed max_tree_depth and masks finished chains.
• fp32 on the GPU. Apple Metal has no fp64 in hardware (MLX has fp64 only on the CPU backend). See the precision envelope below.

Numerical precision (fp32)

fp32-on-GPU is a permanent constraint, not a bug to fix, so it's worth being explicit about where it's fine and where it isn't:

• Fine (the regime this package targets): smooth, moderate-dimensional targets where Monte Carlo error (~1/√ESS, typically 1–3%) swamps fp32 roundoff (~1e-6). All the examples here live here. Ill-conditioned linear algebra (covariance, Cholesky in warmup) is already kept host-side in numpy fp64; only the leapfrog runs fp32 on the GPU — generalize that pattern for any precision-sensitive sub-step.
• Watch out (validate against fp64 first): large log-likelihood sums (e.g. N≳1e5 data points — roundoff accumulates and corrupts Metropolis acceptance); constrained transforms near a boundary (logit/log of values near their limit); long or stiff trajectories (deep NUTS trees). For these, run the CPU fp64 backend to check, or sum in fp64 host-side.

The defensible niche is fp32-tolerant, memory-bound, differentiable/stochastic work — not fp64-precision HPC.

Install

This is a pixi project (installs the package editable):

pixi install
pixi run python examples/gaussian_ess.py             # ensemble vs HMC vs preconditioned
pixi run python examples/nuts_funnel.py funnel        # several examples have demo modes
pixi run -e viz python examples/plot_hard_targets.py  # plotting needs the optional viz env

Or install into any environment with pip: pip install -e . (needs mlx, so arm64 macOS). Add the plotting extra with pip install -e ".[viz]" (matplotlib).

Usage

Every sampler takes a single-point log-density logp(x) -> scalar for x of shape (D,); batching over walkers/chains is handled internally with vmap. Positions are MLX arrays of shape (n_chains, D).

import mlx.core as mx
import numpy as np

# Target: a strongly correlated 2-D Gaussian (corr 0.9, 25:1 variance ratio).
# mlxmc.targets ships this one (as `gaussian_logp`) plus banana / funnel.
mu = mx.array([1.0, -2.0])
Sig_inv = mx.array(np.linalg.inv([[25.0, 4.5], [4.5, 1.0]]))

def logp(x):                              # x: (D,) -> scalar
    d = x - mu
    return -0.5 * (d @ Sig_inv @ d)

key = mx.random.key(0)

Gradient-free ensemble — no tuning, handles the ill-conditioning for free:

from mlxmc import run_ensemble

key, k = mx.random.split(key)
ensemble = mx.random.normal(shape=(2000, 2), key=k) * 5.0     # (n_walkers, D)
result = run_ensemble(logp, ensemble, n_steps=3000, burn=1000, key=key)
print(result.accept_frac)
result.summary()                                             # mean / sd / mcse / ESS / R̂

HMC, hand-tuned, and NUTS after Stan-style warmup (same logp):

from mlxmc import run_hmc, warmup, run_nuts

key, k = mx.random.split(key)
q0 = mx.random.normal(shape=(1000, 2), key=k) * 5.0           # (n_chains, D)

result = run_hmc(logp, q0, n_steps=1500, burn=500, eps=0.15, n_leap=40, key=key)

# Warmup adapts (eps, dense M); NUTS then adapts trajectory length itself.
q_last, eps, Minv = warmup(logp, q0, n_warmup=600, n_leap=8, key=key)
result = run_nuts(logp, q_last, n_samples=1500, eps=eps, Minv_np=Minv, key=key)
print("divergences:", result.n_divergent)                    # NUTS surfaces these now
result.summary()

Every sampler returns a Result. result.samples is (chain, draw, dim), result.flat pools to (chain·draw, dim), result.accept_frac is the acceptance, and result.sample_stats carries per-draw diverging / tree_depth (NUTS). result.summary() prints rank-normalized split-R̂, bulk/tail-ESS, and MCSE per parameter; result.to_arviz() hands off to ArviZ (pip install 'mlxmc[arviz]') for the full plotting/diagnostic suite. The warmup helpers (warmup, nuts_warmup) instead return the tuning triple (q_last, eps, Minv).

Constrained parameters — sample positive / bounded parameters without writing a Jacobian by hand (see examples/constrained_model.py):

from mlxmc import Transform, Identity, Exp, constrain, run_nuts

def logp_constrained(theta):          # theta = (mu, sigma > 0)
    mu, sigma = theta[0], theta[1]
    return -0.5 * (((data - mu) / sigma) ** 2).sum() - data.shape[0] * mx.log(sigma)

# mu free, sigma positive; constrain() adds the log-Jacobian so you sample in R^2.
logp_u, transform = constrain(logp_constrained, Transform([Identity(), Exp()]))
result = run_nuts(logp_u, u0, n_samples=1000, eps=eps, Minv_np=Minv, key=key)
result.transform = transform          # map draws back to natural (mu, sigma) space
result.summary()                      # reported in constrained space
natural_draws = result.constrained()

Findings

Validated on a corr-0.9, 25:1-variance Gaussian and on banana / funnel targets; every number below is reproducible with the scripts in examples/:

• Affine-invariant ensemble is the robust low-D default: gradient-free, tuning-free, handles ill-conditioning for free (acceptance is bit-identical under an affine map). But weaker per-step mixing and it degrades with dimension.
• HMC needs gradients and a tuned eps/L, but mixes far better (τ≈2 vs ≈26). A warmup-adapted dense mass matrix recovers the true Σ to <1% Frobenius error and buys ~7–11× the ESS/sec — HMC's version of affine invariance, earned rather than supplied.
• Fixed-L HMC has a trajectory resonance: on near-Gaussian targets, when eps·L lands near a multiple of 2π the trajectory returns to its start and mixing collapses. Jittering eps per trajectory cures it; NUTS's adaptive trajectory length is the principled fix.
• NUTS is validated exact on the Gaussian (recovered covariance 24.97 vs 25) and auto-tunes trajectory length, but vectorized NUTS pays a real masking cost when trajectory lengths are heterogeneous — with no while_loop, every chain runs to the deepest chain's tree depth, up to a ~30× wall-time penalty at the funnel mouth versus the same target reparametrized.
• Geometry matters more than the sampler: on the centered funnel the gradient-free ensemble beats a global-metric HMC, because a constant mass matrix is wrong everywhere when the scale is position-dependent; a non-centered reparametrization removes the geometry and makes HMC unbiased again.
• ESS/sec is the honest efficiency metric — acceptance fraction is a misleading proxy.

Development

pixi run test                                   # full suite on the default device
MLXMC_TEST_DEVICE=cpu pixi run test             # force the CPU backend
MLXMC_TEST_DEVICE=gpu pixi run test             # force the Metal GPU
pixi run -e dev test                            # incl. the ArviZ-validated diagnostics tests

The suite (tests/) checks moment recovery for every sampler, warmup's Σ estimate, the affine-invariance identity, the autocorrelation-time diagnostics, divergence reporting, the constrained-parameter transforms, and the convergence diagnostics (R̂ / bulk-tail-ESS / MCSE) against ArviZ. The ArviZ comparison needs arviz + scipy, which live in the optional dev pixi environment (run those via pixi run -e dev test); they're skipped in the lean default env. A GitHub Actions workflow (.github/workflows/tests.yml) runs the CPU + GPU matrix on an Apple-silicon runner for pull requests to main (and on manual dispatch from the Actions tab). Direct pushes to main don't trigger it, which keeps the (10x-billed) macOS runner minutes down.

References

• Goodman & Weare (2010), Ensemble samplers with affine invariance.
• Hoffman & Gelman (2014), The No-U-Turn Sampler.
• Betancourt (2017), A Conceptual Introduction to Hamiltonian Monte Carlo.

License

BSD-3-Clause.