Advanced Sampling with eXMC

04_variational_inference.livemd

Igor O'sten

@borodark

eXMC

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Advanced Sampling with eXMC


# CPU only — no GPU required
System.put_env("EXLA_CPU_ONLY", "true")
System.put_env("CUDA_VISIBLE_DEVICES", "")
Mix.install([
  {:exmc, path: Path.expand("../", __DIR__)},
  {:exla, "~> 0.10"},
  {:kino_vega_lite, "~> 0.1"}
])
Application.put_env(:exla, :clients, host: [platform: :host])
Application.put_env(:exla, :default_client, :host)
Nx.default_backend(Nx.BinaryBackend)
Nx.Defn.default_options(compiler: EXLA, client: :host)

Why This Matters

NUTS gives you the exact posterior. It is also slow — 1,000 warmup iterations, 1,000 sampling iterations, minutes to hours on complex models. When you need a quick answer — a first approximation, an initialization point, a model check before committing to full MCMC — you need something faster that gives up accuracy for speed.

ADVI (Automatic Differentiation Variational Inference) fits a Gaussian approximation to the posterior in seconds. The approximation is wrong — posteriors are rarely Gaussian — but it’s wrong in a useful way: the mean is usually close, the variance is usually underestimated, and the result tells you whether the model is identifiable before you spend an hour running NUTS.

This notebook also covers a subtlety that trips up every hierarchical modeler: centered vs non-centered parameterization. When data is sparse, the centered parameterization creates a geometry that NUTS struggles with — a funnel where step size can’t adapt to both the narrow and wide ends. NCP reparameterizes to eliminate the funnel. The choice isn’t about the model — it’s about the sampler’s geometry.

NCP vs Centered Parameterization

eXMC auto-applies Non-Centered Parameterization (NCP) for hierarchical models. NCP helps when data is weak relative to priors, but can hurt when data is strong.

alias Exmc.{Builder, Sampler}
alias VegaLite, as: Vl

# A hierarchical model with strong data
ir = Builder.new_ir()
ir = Builder.rv(ir, "mu_group", Exmc.Dist.Normal, %{mu: Nx.tensor(0.0), sigma: Nx.tensor(10.0)})
ir = Builder.rv(ir, "sigma_group", Exmc.Dist.HalfNormal, %{sigma: Nx.tensor(5.0)}, transform: :log)
ir = Builder.rv(ir, "alpha", Exmc.Dist.Normal, %{mu: "mu_group", sigma: "sigma_group"})
ir = Builder.rv(ir, "y", Exmc.Dist.Normal, %{mu: "alpha", sigma: Nx.tensor(0.5)})
ir = Builder.obs(ir, "y_obs", "y", Nx.tensor([4.1, 3.8, 4.5, 4.2, 3.9, 4.0, 4.3]))

init = %{"mu_group" => 4.0, "sigma_group" => 1.0, "alpha" => 4.0}

# NCP (default)
t0 = System.monotonic_time(:millisecond)
{trace_ncp, _} = Sampler.sample(ir, init, num_samples: 500, seed: 42, num_warmup: 300)
ncp_ms = System.monotonic_time(:millisecond) - t0

# Centered — pass ncp: false
t0 = System.monotonic_time(:millisecond)
{trace_cen, _} = Sampler.sample(ir, init, num_samples: 500, seed: 42, num_warmup: 300, ncp: false)
cen_ms = System.monotonic_time(:millisecond) - t0

IO.puts("NCP:      #{ncp_ms}ms")
IO.puts("Centered: #{cen_ms}ms")

NCP:      4292ms
Centered: 2517ms

:ok

ncp_mu = Nx.to_flat_list(trace_ncp["mu_group"]) |> Enum.filter(&amp;is_number/1)
cen_mu = Nx.to_flat_list(trace_cen["mu_group"]) |> Enum.filter(&amp;is_number/1)

hist_data =
  Enum.map(ncp_mu, fn v -> %{"mu_group" => v, "method" => "NCP"} end) ++
    Enum.map(cen_mu, fn v -> %{"mu_group" => v, "method" => "Centered"} end)

Vl.new(width: 500, height: 250, title: "mu_group: NCP vs Centered")
|> Vl.data_from_values(hist_data)
|> Vl.mark(:area, opacity: 0.5, interpolate: "step")
|> Vl.encode_field(:x, "mu_group", type: :quantitative, bin: [maxbins: 30])
|> Vl.encode(:y, aggregate: :count, stack: nil)
|> Vl.encode_field(:color, "method", type: :nominal)

{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"method":"NCP","mu_group":11.104824897492815},{"method":"NCP","mu_group":11.614838813017249},{"method":"NCP","mu_group":12.913055371489186},{"method":"NCP","mu_group":12.908894715407335},{"method":"NCP","mu_group":12.398744736877994},{"method":"NCP","mu_group":14.062029296828655},{"method":"NCP","mu_group":11.368677541664102},{"method":"NCP","mu_group":9.170965500002325},{"method":"NCP","mu_group":9.885803921016107},{"method":"NCP","mu_group":10.385368289442885},{"method":"NCP","mu_group":10.350914861846649},{"method":"NCP","mu_group":4.117577495570807},{"method":"NCP","mu_group":4.406929212413293},{"method":"NCP","mu_group":4.844628806621104},{"method":"NCP","mu_group":4.769226371417543},{"method":"NCP","mu_group":4.974128167015523},{"method":"NCP","mu_group":5.191080010408548},{"method":"NCP","mu_group":4.604781897762612},{"method":"NCP","mu_group":4.644259303052485},{"method":"NCP","mu_group":4.430629119571228},{"method":"NCP","mu_group":4.660864167615087},{"method":"NCP","mu_group":4.797876936183953},{"method":"NCP","mu_group":4.552363330975043},{"method":"NCP","mu_group":4.944172001378022},{"method":"NCP","mu_group":4.831970437403856},{"method":"NCP","mu_group":6.4063076594973305},{"method":"NCP","mu_group":7.074204429272316},{"method":"NCP","mu_group":9.704248240596389},{"method":"NCP","mu_group":12.11389274967888},{"method":"NCP","mu_group":-8.88016646527405},{"method":"NCP","mu_group":-0.7307649463723145},{"method":"NCP","mu_group":2.8924711407853843},{"method":"NCP","mu_group":6.967481523627617},{"method":"NCP","mu_group":9.8855550661261},{"method":"NCP","mu_group":4.070395326905633},{"method":"NCP","mu_group":4.104592536643655},{"method":"NCP","mu_group":3.493628936041386},{"method":"NCP","mu_group":14.300567151052467},{"method":"NCP","mu_group":15.050820701239726},{"method":"NCP","mu_group":15.638576979052408},{"method":"NCP","mu_group":5.724006333179231},{"method":"NCP","mu_group":3.059567028214129},{"method":"NCP","mu_group":2.0873129255252554},{"method":"NCP","mu_group":9.683421890319313},{"method":"NCP","mu_group":9.147612940198945},{"method":"NCP","mu_group":13.45087533576026},{"method":"NCP","mu_group":13.946558603497543},{"method":"NCP","mu_group":9.431813827203886},{"method":"NCP","mu_group":9.470275262155251},{"method":"NCP","mu_group":9.648009110849104},{"method":"NCP","mu_group":9.060904157371917},{"method":"NCP","mu_group":8.926620232495395},{"method":"NCP","mu_group":9.109918237617283},{"method":"NCP","mu_group":11.333095928055641},{"method":"NCP","mu_group":13.96764954726172},{"method":"NCP","mu_group":13.906676739083222},{"method":"NCP","mu_group":11.092925805537222},{"method":"NCP","mu_group":13.359392068925892},{"method":"NCP","mu_group":13.730623351162702},{"method":"NCP","mu_group":13.793750903269098},{"method":"NCP","mu_group":2.8784942643740576},{"method":"NCP","mu_group":2.488079972590689},{"method":"NCP","mu_group":2.724288828477571},{"method":"NCP","mu_group":-2.5929348525067906},{"method":"NCP","mu_group":-2.7435974525549067},{"method":"NCP","mu_group":-2.622256688004269},{"method":"NCP","mu_group":-4.550479630520791},{"method":"NCP","mu_group":-0.6736710512972012},{"method":"NCP","mu_group":-0.502609631156146},{"method":"NCP","mu_group":1.926979528796717},{"method":"NCP","mu_group":5.286900988507781},{"method":"NCP","mu_group":5.3535747214611185},{"method":"NCP","mu_group":4.987904089800895},{"method":"NCP","mu_group":5.583893511224969},{"method":"NCP","mu_group":5.012961959442656},{"method":"NCP","mu_group":7.3760483903060905},{"method":"NCP","mu_group":2.7596672526345634},{"method":"NCP","mu_group":1.8638074501957926},{"method":"NCP","mu_group":1.8886311528011441},{"method":"NCP","mu_group":-0.8671138943421665},{"method":"NCP","mu_group":-2.1903794105562966},{"method":"NCP","mu_group":-1.8868783669199112},{"method":"NCP","mu_group":-1.3652958385104879},{"method":"NCP","mu_group":-1.4417896135021115},{"method":"NCP","mu_group":6.065735938558466},{"method":"NCP","mu_group":8.272697197712535},{"method":"NCP","mu_group":5.777881117964609},{"method":"NCP","mu_group":6.7035122045177715},{"method":"NCP","mu_group":6.859506854379276},{"method":"NCP","mu_group":4.99545931360907},{"method":"NCP","mu_group":4.621325016736732},{"method":"NCP","mu_group":4.671061552372004},{"method":"NCP","mu_group":4.49008564566365},{"method":"NCP","mu_group":4.938189990231902},{"method":"NCP","mu_group":4.9240215388950865},{"method":"NCP","mu_group":-3.2667292355884974},{"method":"NCP","mu_group":-3.3185401040740294},{"method":"NCP","mu_group":-3.4037308721486266},{"method":"NCP","mu_group":-5.365986612885316},{"method":"NCP","mu_group":-5.3830837455928116},{"method":"NCP","mu_group":-4.959842858573566},{"method":"NCP","mu_group":9.892291523784497},{"method":"NCP","mu_group":8.08861436292043},{"method":"NCP","mu_group":10.380903917477363},{"method":"NCP","mu_group":4.209711027039809},{"method":"NCP","mu_group":6.01885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NCP vs Centered","width":500}

Multi-Chain with R-hat

Running 4 chains and checking convergence:

{traces, stats_list} =
  Sampler.sample_chains(ir, 4,
    init_values: init,
    num_samples: 300,
    seed: 42,
    num_warmup: 200,
    ncp: false
  )

IO.puts("Chains: #{length(traces)}")

for name <- Map.keys(hd(traces)) |> Enum.sort() do
  chains_for_var = Enum.map(traces, fn t -> t[name] end)
  rhat = Exmc.Diagnostics.rhat(chains_for_var)
  ess = Exmc.Diagnostics.ess(hd(traces)[name])
  IO.puts("#{name}: R-hat=#{Float.round(rhat, 3)}, ESS=#{Float.round(ess, 1)}")
end

Chains: 4
alpha: R-hat=1.033, ESS=12.5
mu_group: R-hat=1.005, ESS=151.7
sigma_group: R-hat=1.044, ESS=19.7

[:ok, :ok, :ok]

chain_data =
  traces
  |> Enum.with_index()
  |> Enum.flat_map(fn {t, c} ->
    t["mu_group"]
    |> Nx.to_flat_list()
    |> Enum.filter(&amp;is_number/1)
    |> Enum.with_index(fn val, i ->
      %{"iteration" => i, "value" => val, "chain" => "chain #{c}"}
    end)
  end)

Vl.new(width: 600, height: 200, title: "mu_group — 4 chains")
|> Vl.data_from_values(chain_data)
|> Vl.mark(:line, opacity: 0.6)
|> Vl.encode_field(:x, "iteration", type: :quantitative)
|> Vl.encode_field(:y, "value", type: :quantitative)
|> Vl.encode_field(:color, "chain", type: :nominal)

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New Distributions

eXMC includes distributions beyond the standard Normal/HalfNormal:

# Lognormal prior for a rate parameter
ir_ln = Builder.new_ir()
ir_ln = Builder.rv(ir_ln, "rate", Exmc.Dist.Lognormal, %{mu: Nx.tensor(0.0), sigma: Nx.tensor(0.5)})
ir_ln = Builder.rv(ir_ln, "y", Exmc.Dist.Normal, %{mu: "rate", sigma: Nx.tensor(0.3)})
ir_ln = Builder.obs(ir_ln, "y_obs", "y", Nx.tensor([1.2, 1.4, 1.1, 1.3, 1.5]))

{trace_ln, _} = Sampler.sample(ir_ln, %{"rate" => 1.3}, num_samples: 500, seed: 42, num_warmup: 200)

rate_samples = Nx.to_flat_list(trace_ln["rate"]) |> Enum.filter(&amp;is_number/1)
rate_data = Enum.with_index(rate_samples, fn v, i -> %{"iteration" => i, "rate" => v} end)

Vl.new(width: 600, height: 200, title: "Lognormal prior: rate parameter")
|> Vl.data_from_values(rate_data)
|> Vl.mark(:bar)
|> Vl.encode_field(:x, "rate", type: :quantitative, bin: [maxbins: 30])
|> Vl.encode(:y, aggregate: :count)

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rate":1.31669203648752},{"iteration":458,"rate":1.2597033824659323},{"iteration":459,"rate":1.3534747776758693},{"iteration":460,"rate":1.3543107686261298},{"iteration":461,"rate":1.3588491677165326},{"iteration":462,"rate":1.3958514797936497},{"iteration":463,"rate":1.3958514797936497},{"iteration":464,"rate":1.1179585625687052},{"iteration":465,"rate":1.3641103867152349},{"iteration":466,"rate":1.7057037844294864},{"iteration":467,"rate":1.3493627382099682},{"iteration":468,"rate":1.079471121002728},{"iteration":469,"rate":1.4736501243562075},{"iteration":470,"rate":1.4736501243562075},{"iteration":471,"rate":1.3773751325349233},{"iteration":472,"rate":1.3773751325349233},{"iteration":473,"rate":1.38780909216737},{"iteration":474,"rate":1.1797948390602178},{"iteration":475,"rate":1.442832281167539},{"iteration":476,"rate":1.4041591101600686},{"iteration":477,"rate":1.4169899028815331},{"iteration":478,"rate":1.2616635692749283},{"iteration":479,"rate":1.4999967126704528},{"iteration":480,"rate":1.4237562600255798},{"iteration":481,"rate":1.2713539979507626},{"iteration":482,"rate":1.1639070081766585},{"iteration":483,"rate":1.18142849614336},{"iteration":484,"rate":1.2226825129189784},{"iteration":485,"rate":1.3318648259650658},{"iteration":486,"rate":1.1979682108601624},{"iteration":487,"rate":1.1928620021968854},{"iteration":488,"rate":1.3169168005968053},{"iteration":489,"rate":1.0863832048945643},{"iteration":490,"rate":1.2747349886799408},{"iteration":491,"rate":1.1823192966913176},{"iteration":492,"rate":1.3227324807825436},{"iteration":493,"rate":1.3823221705274444},{"iteration":494,"rate":1.4277448215686046},{"iteration":495,"rate":1.1679491933777062},{"iteration":496,"rate":1.1091328741740862},{"iteration":497,"rate":1.401141613775337},{"iteration":498,"rate":1.3771848976814554},{"iteration":499,"rate":1.3428259040388844}]},"encoding":{"x":{"bin":{"maxbins":30},"field":"rate","type":"quantitative"},"y":{"aggregate":"count"}},"height":200,"mark":"bar","title":"Lognormal prior: rate parameter","width":600}

Sampling Diagnostics

Inspect step size, divergences, and tree depth from sampling stats:

{_trace_diag, stats_diag} =
  Sampler.sample(ir, init, num_samples: 500, seed: 42, num_warmup: 300, ncp: false)

IO.puts("Step size: #{Float.round(stats_diag.step_size, 4)}")
IO.puts("Divergences: #{stats_diag.divergences}")

mean_tree_depth =
  stats_diag.sample_stats
  |> Enum.map(&amp; &amp;1.tree_depth)
  |> then(fn ds -> Enum.sum(ds) / length(ds) end)

mean_accept_prob =
  stats_diag.sample_stats
  |> Enum.map(&amp; &amp;1.accept_prob)
  |> then(fn ps -> Enum.sum(ps) / length(ps) end)

IO.puts("Mean tree depth: #{Float.round(mean_tree_depth, 2)}")
IO.puts("Mean accept prob: #{Float.round(mean_accept_prob, 3)}")

Step size: 0.2545
Divergences: 85
Mean tree depth: 2.89
Mean accept prob: 0.728

:ok

Roadmap

Features planned for future notebooks:

ADVI — Automatic Differentiation Variational Inference
Pathfinder — Fast L-BFGS-based approximate inference
SMC — Sequential Monte Carlo (gradient-free)
GPU Acceleration — device: :cuda for NVIDIA GPUs

Other notebooks:

Michal Slaski
@michalslaski

livebook_examples

Salary predictions

salary_prediction.livemd

data-science advanced exla axon nx

2022-8-18
Dr. Christian Geuer-Pollmann
@chgeuer

livebook_on_azure

Christian's first LiveBook test

notebook1.livemd

data-science advanced tutorial axon exla nx

2022-8-18
@andyl

elix_util

MNIST

mnist.livemd

data-science advanced tutorial req axon exla nx

2022-8-18
@TomBers

livebookNotes

Trying Nx

NX.livemd

advanced data-science exla axon nx

2022-8-18
Daniel Jaouen
@danieljaouen

livebook-axon-test

Predicting Stock Prices with Elixir & Axon

testing-axon-and-vegalite.livemd

advanced data-science testing nx axon vega_lite explorer kino kino_vega_lite scholar jason httpoison exla

2024-1-22
Philip Sampaio
@philss

brazil-in-notebooks

The victims of the COVID-19

the-victims-of-the-covid-19.livemd

data-science advanced tutorial req explorer kino_vega_lite

2023-5-1
@aforward-oss

empex_recommend

Pizzeria

pizzeria.livemd

data-science advanced tutorial scholar explorer exla req kino_vega_lite kino kino_explorer

2023-12-4

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