Advanced Sampling with eXMC

04_variational_inference.livemd

Igor O'sten

@borodark

exmc

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Advanced Sampling with eXMC

# EMLX (Metal GPU) on macOS Apple Silicon, EXLA (CPU/CUDA) elsewhere
backend_dep =
  case :os.type() do
    {:unix, :darwin} -> {:emlx, "~> 0.2"}
    _ -> {:exla, "~> 0.10"}
  end

Mix.install([
  {:exmc, path: Path.expand("../", __DIR__)},
  backend_dep,
  {:kino_vega_lite, "~> 0.1"}
])

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")


02:34:22.143 [info] XLA service 0x7ed049aa5440 initialized for platform CUDA (this does not guarantee that XLA will be used). Devices:

02:34:22.150 [info]   StreamExecutor device (0): NVIDIA GeForce RTX 3060 Ti, Compute Capability 8.6
NCP:      10591ms
Centered: 6977ms

: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":3.3853274845452304},{"method":"NCP","mu_group":2.9314575176426305},{"method":"NCP","mu_group":2.752780849727776},{"method":"NCP","mu_group":2.80891680622427},{"method":"NCP","mu_group":-1.689127830123651},{"method":"NCP","mu_group":13.364583376906198},{"method":"NCP","mu_group":7.253522218446283},{"method":"NCP","mu_group":6.133159176994455},{"method":"NCP","mu_group":8.760929541765652},{"method":"NCP","mu_group":7.124341378780835},{"method":"NCP","mu_group":7.083657873054079},{"method":"NCP","mu_group":12.305946761442453},{"method":"NCP","mu_group":11.949476998933273},{"method":"NCP","mu_group":14.351060760965565},{"method":"NCP","mu_group":14.771100327323774},{"method":"NCP","mu_group":15.731478931539089},{"method":"NCP","mu_group":13.62662839319244},{"method":"NCP","mu_group":14.601830041546622},{"method":"NCP","mu_group":5.601263796832845},{"method":"NCP","mu_group":6.71811516456852},{"method":"NCP","mu_group":3.388405708476377},{"method":"NCP","mu_group":2.856849081952662},{"method":"NCP","mu_group":3.137663107678943},{"method":"NCP","mu_group":3.3115001430315716},{"method":"NCP","mu_group":3.8372869482532934},{"method":"NCP","mu_group":3.6675661155228045},{"method":"NCP","mu_group":5.220005176190557},{"method":"NCP","mu_group":7.050699857806622},{"method":"NCP","mu_group":9.930540554930376},{"method":"NCP","mu_group":7.943995801804518},{"method":"NCP","mu_group":7.714590144443126},{"method":"NCP","mu_group":7.291851312258916},{"method":"NCP","mu_group":9.983852438483808},{"method":"NCP","mu_group":10.506506166570988},{"method":"NCP","mu_group":4.300372802284989},{"method":"NCP","mu_group":4.228026306666796},{"method":"NCP","mu_group":3.7424969711231992},{"method":"NCP","mu_group":4.197017413214556},{"method":"NCP","mu_group":3.9164598777256967},{"method":"NCP","mu_group":5.12503948889633},{"method":"NCP","mu_group":5.779962305186517},{"method":"NCP","mu_group":5.339239378555735},{"method":"NCP","mu_group":5.778245232376277},{"method":"NCP","mu_group":4.143150805948993},{"method":"NCP","mu_group":4.3585336099914604},{"method":"NCP","mu_group":4.252098588792206},{"method":"NCP","mu_group":4.574666549990223},{"method":"NCP","mu_group":3.0274900255389063},{"method":"NCP","mu_group":1.7987054797770818},{"method":"NCP","mu_group":1.1370637307832572},{"method":"NCP","mu_group":5.716595281237349},{"method":"NCP","mu_group":5.15965640906014},{"method":"NCP","mu_group":4.10899832796125},{"method":"NCP","mu_group":3.4485635736139444},{"method":"NCP","mu_group":3.905577174010291},{"method":"NCP","mu_group":4.710884067189779},{"method":"NCP","mu_group":6.385558104691963},{"method":"NCP","mu_group":7.683559251989061},{"method":"NCP","mu_group":7.849943456032188},{"method":"NCP","mu_group":4.707223957917202},{"method":"NCP","mu_group":0.6501788044255987},{"method":"NCP","mu_group":0.4869767678695094},{"method":"NCP","mu_group":0.7852532320987566},{"method":"NCP","mu_group":-2.347222378097067},{"method":"NCP","mu_group":-2.347222378097067},{"method":"NCP","mu_group":-2.411852761995142},{"method":"NCP","mu_group":2.5371656015244666},{"method":"NCP","mu_group":2.3420766327005063},{"method":"NCP","mu_group":4.264468305399497},{"method":"NCP","mu_group":4.399560518163145},{"method":"NCP","mu_group":4.142150163510763},{"method":"NCP","mu_group":4.204057948853874},{"method":"NCP","mu_group":3.681122381927757},{"method":"NCP","mu_group":4.661335753011503},{"method":"NCP","mu_group":4.1868664166815615},{"method":"NCP","mu_group":4.4798739192157635},{"method":"NCP","mu_group":3.9242924181056735},{"method":"NCP","mu_group":3.674570007295053},{"method":"NCP","mu_group":3.914178579368468},{"method":"NCP","mu_group":1.4806762652167833},{"method":"NCP","mu_group":0.2325762226921422},{"method":"NCP","mu_group":-0.41327306753779675},{"method":"NCP","mu_group":3.8464712647993666},{"method":"NCP","mu_group":2.5777331726177994},{"method":"NCP","mu_group":7.133633726828459},{"method":"NCP","mu_group":-0.7082377316704191},{"method":"NCP","mu_group":-0.726734752876386},{"method":"NCP","mu_group":5.4710825109406915},{"method":"NCP","mu_group":5.830463967658249},{"method":"NCP","mu_group":4.062936520829588},{"method":"NCP","mu_group":4.569899829006232},{"method":"NCP","mu_group":4.163130938328204},{"method":"NCP","mu_group":4.014741003659072},{"method":"NCP","mu_group":3.6484399294520453},{"method":"NCP","mu_group":4.101415420758376},{"method":"NCP","mu_group":-5.83890531763731},{"method":"NCP","mu_group":-5.83890531763731},{"method":"NCP","mu_group":1.5077396419888252},{"method":"NCP","mu_group":-1.7410742260427894},{"method":"NCP","mu_group":-1.6364767649239333},{"method":"NCP","mu_group":0.7713636603019168},{"method":"NCP","mu_group":7.647936127677583},{"method":"NCP","mu_group":8.428715739191048},{"method":"NCP","mu_group":10.242720461784694},{"method":"NCP","mu_group":10.418138424243521},{"method":"NCP","mu_group":13.53940085240565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u_group":3.2222664788518767},{"method":"Centered","mu_group":3.6423558004309595},{"method":"Centered","mu_group":3.9500275413724726},{"method":"Centered","mu_group":2.762163231213596},{"method":"Centered","mu_group":5.092364674913565},{"method":"Centered","mu_group":4.936423078456985},{"method":"Centered","mu_group":6.1253455305697955},{"method":"Centered","mu_group":7.07844535828211},{"method":"Centered","mu_group":7.5692683349810235},{"method":"Centered","mu_group":9.105429659163361},{"method":"Centered","mu_group":1.4958703156926827},{"method":"Centered","mu_group":3.5759132250683994},{"method":"Centered","mu_group":5.755864445039919},{"method":"Centered","mu_group":1.7570187694166748},{"method":"Centered","mu_group":1.8014191544438618},{"method":"Centered","mu_group":3.3444144319080173},{"method":"Centered","mu_group":1.5662740991485298},{"method":"Centered","mu_group":6.515483051184819},{"method":"Centered","mu_group":1.2053962302867887},{"method":"Centered","mu_group":0.5065478302963575},{"method":"Centered","mu_group":2.1528823243893096},{"method":"Centered","mu_group":6.38743571430179},{"method":"Centered","mu_group":3.7394016957931733},{"method":"Centered","mu_group":2.0711102818011136},{"method":"Centered","mu_group":7.186899749159861},{"method":"Centered","mu_group":-1.3905830059261801},{"method":"Centered","mu_group":5.9699018024689545},{"method":"Centered","mu_group":5.358060018267369},{"method":"Centered","mu_group":2.1473579390973443},{"method":"Centered","mu_group":4.290372765812953},{"method":"Centered","mu_group":-0.6146732364152585},{"method":"Centered","mu_group":-0.7321912812556357},{"method":"Centered","mu_group":-1.193571904727202},{"method":"Centered","mu_group":-0.7638744268525588},{"method":"Centered","mu_group":-0.5673734322093044},{"method":"Centered","mu_group":-0.3654449951195351},{"method":"Centered","mu_group":7.755522452763963},{"method":"Centered","mu_group":7.012988055466045},{"method":"Centered","mu_group":7.289556996585079},{"method":"Centered","mu_group":1.5935893940926777},{"method":"Centered","mu_group":-0.21393034673791306},{"method":"Centered","mu_group":2.929033864744089},{"method":"Centered","mu_group":5.025099294998409},{"method":"Centered","mu_group":6.744760535087793},{"method":"Centered","mu_group":9.554385137763013},{"method":"Centered","mu_group":2.2270547152577675},{"method":"Centered","mu_group":3.2265943798006087},{"method":"Centered","mu_group":4.126375942566769},{"method":"Centered","mu_group":3.485215194171779},{"method":"Centered","mu_group":2.90559065804448},{"method":"Centered","mu_group":2.378160746465208},{"method":"Centered","mu_group":4.371134466490259},{"method":"Centered","mu_group":4.511055129682293},{"method":"Centered","mu_group":3.6703517050351415},{"method":"Centered","mu_group":5.754033364265426},{"method":"Centered","mu_group":5.330260405955895},{"method":"Centered","mu_group":3.2197495657579553},{"method":"Centered","mu_group":3.2197495657579553},{"method":"Centered","mu_group":2.501766687757964},{"method":"Centered","mu_group":-6.08118421779877},{"method":"Centered","mu_group":9.9388120612148},{"method":"Centered","mu_group":9.741750659753798},{"method":"Centered","mu_group":9.357510851727174},{"method":"Centered","mu_group":4.4380298276538195},{"method":"Centered","mu_group":2.862682521772545},{"method":"Centered","mu_group":3.8263853403822607},{"method":"Centered","mu_group":3.688940055041562},{"method":"Centered","mu_group":3.688940055041562},{"method":"Centered","mu_group":3.93371601071942},{"method":"Centered","mu_group":4.055055121402864},{"method":"Centered","mu_group":3.7566272420473643},{"method":"Centered","mu_group":4.115169543789373},{"method":"Centered","mu_group":4.115169543789373},{"method":"Centered","mu_group":4.115169543789373},{"method":"Centered","mu_group":4.115169543789373},{"method":"Centered","mu_group":4.115169543789373},{"method":"Centered","mu_group":4.115169543789373},{"method":"Centered","mu_group":4.115169543789373},{"method":"Centered","mu_group":4.115169543789373},{"method":"Centered","mu_group":4.115169543789373},{"method":"Centered","mu_group":4.115169543789373},{"method":"Centered","mu_group":3.9918384389642556}]},"encoding":{"color":{"field":"method","type":"nominal"},"x":{"bin":{"maxbins":30},"field":"mu_group","type":"quantitative"},"y":{"aggregate":"count","stack":null}},"height":250,"mark":{"interpolate":"step","opacity":0.5,"type":"area"},"title":"mu_group: 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=1.009, ESS=49.0
mu_group: R-hat=1.019, ESS=28.2
sigma_group: R-hat=1.007, ESS=38.6

[: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)

{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"chain":"chain 0","iteration":0,"value":5.90748489650505},{"chain":"chain 0","iteration":1,"value":7.2584372891865865},{"chain":"chain 0","iteration":2,"value":7.538719329295416},{"chain":"chain 0","iteration":3,"value":5.1711421423160235},{"chain":"chain 0","iteration":4,"value":10.316546965238707},{"chain":"chain 0","iteration":5,"value":5.618729686134955},{"chain":"chain 0","iteration":6,"value":6.092868030051468},{"chain":"chain 0","iteration":7,"value":3.584380979967012},{"chain":"chain 0","iteration":8,"value":4.633431521925753},{"chain":"chain 0","iteration":9,"value":3.6059611367541407},{"chain":"chain 0","iteration":10,"value":4.405364120325615},{"chain":"chain 0","iteration":11,"value":3.802737906591314},{"chain":"chain 0","iteration":12,"value":3.861611445899266},{"chain":"chain 0","iteration":13,"value":4.162000307007551},{"chain":"chain 0","iteration":14,"value":4.162000307007551},{"chain":"chain 0","iteration":15,"value":4.162000307007551},{"chain":"chain 0","iteration":16,"value":4.162000307007551},{"chain":"chain 0","iteration":17,"value":4.162000307007551},{"chain":"chain 0","iteration":18,"value":4.162000307007551},{"chain":"chain 0","iteration":19,"value":3.470574851735485},{"chain":"chain 0","iteration":20,"value":2.6854169575829014},{"chain":"chain 0","iteration":21,"value":4.816134602398014},{"chain":"chain 0","iteration":22,"value":-3.456339096265684},{"chain":"chain 0","iteration":23,"value":-4.008389570341486},{"chain":"chain 0","iteration":24,"value":0.3273055739550299},{"chain":"chain 0","iteration":25,"value":-0.6093144801767936},{"chain":"chain 0","iteration":26,"value":-0.10942767494187638},{"chain":"chain 0","iteration":27,"value":6.790986886133467},{"chain":"chain 0","iteration":28,"value":2.673716393314284},{"chain":"chain 0","iteration":29,"value":0.592230257967256},{"chain":"chain 0","iteration":30,"value":1.1506551472391564},{"chain":"chain 0","iteration":31,"value":-0.2581900958233762},{"chain":"chain 0","iteration":32,"value":4.813800877691755},{"chain":"chain 0","iteration":33,"value":3.5048296117081974},{"chain":"chain 0","iteration":34,"value":7.37766037573081},{"chain":"chain 0","iteration":35,"value":3.6827610086627556},{"chain":"chain 0","iteration":36,"value":4.770991086760053},{"chain":"chain 0","iteration":37,"value":4.557144580689741},{"chain":"chain 0","iteration":38,"value":3.0030105342051168},{"chain":"chain 0","iteration":39,"value":3.6733831423162373},{"chain":"chain 0","iteration":40,"value":3.668564348506019},{"chain":"chain 0","iteration":41,"value":3.2591110564060504},{"chain":"chain 0","iteration":42,"value":2.513697917576585},{"chain":"chain 0","iteration":43,"value":4.710952603258024},{"chain":"chain 0","iteration":44,"value":5.672944559238137},{"chain":"chain 0","iteration":45,"value":5.405793347511362},{"chain":"chain 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3","iteration":299,"value":12.003760830168773}]},"encoding":{"color":{"field":"chain","type":"nominal"},"x":{"field":"iteration","type":"quantitative"},"y":{"field":"value","type":"quantitative"}},"height":200,"mark":{"opacity":0.6,"type":"line"},"title":"mu_group — 4 chains","width":600}

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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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: 0.1512
Divergences: 51
Mean tree depth: 3.52
Mean accept prob: 0.83

: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:

Jeremy Brayton
@w0rd-driven

livebook_notebooks

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req--job_fields.livemd

tutorial intermediate kino kino_lab kino_vega_lite kino_db req req_easyhtml

2022-8-18
Livebook
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Components

components.livemd

tutorial intermediate vega_lite kino_vega_lite

2022-8-18
Ryan Curtin
@ryancurtin

titanic-machine-learning

Titanic Machine Learning Project

titanic-machine-learning.livemd

tutorial advanced data-science axon exla nx explorer vega_lite kino_vega_lite jason analysis_prep

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@hugobarauna

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livebook_google_big_query.livemd

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@aforward-oss

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kino.livemd

tutorial advanced data-science nx scidata explorer kino

2023-12-4
@DockYard-Academy

curriculum

Todo List

deprecated_todo_list.livemd

tutorial intermediate data-structures jason kino youtube hidden_cell

2023-1-23
Daniel Kukula
@dkuku

livebooks

Untitled notebook

ortex_yolo.livemd

advanced data-science ortex image kino

2024-4-23

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