Getting Started with eXMC
Section
# 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)
:okWhy This Matters
Your data has 50 observations. The sample mean is 4.2. Is the true mean 4.0 or 4.5? A frequentist confidence interval says “4.0 to 4.4, 95% of the time, in repeated sampling.” That answers a question nobody asked.
The question you asked: “Given the data I have, what should I believe about the parameter?” That’s a posterior distribution. Bayesian inference computes it. eXMC does it on the BEAM — where inference can stream, fault- recover, and distribute across nodes as a consequence of the runtime, not as a feature bolted on.
This notebook starts with the simplest case: one parameter, 50 observations. If the posterior is wrong here, everything built on top of it — hierarchical models, model comparison, streaming updates — is wrong too. We verify against the analytical solution.
Building a Simple Model
eXMC uses an IR (intermediate representation) to define probabilistic models.
Start with Builder.new_ir() and add random variables and observations.
alias Exmc.{Builder, Sampler}
# Build a model: mu ~ Normal(0, 10), observe data at known sigma
ir = Builder.new_ir()
ir = Builder.rv(ir, "mu", Exmc.Dist.Normal, %{mu: Nx.tensor(0.0), sigma: Nx.tensor(10.0)})
ir = Builder.rv(ir, "y", Exmc.Dist.Normal, %{mu: "mu", sigma: Nx.tensor(1.0)})
ir = Builder.obs(ir, "y_obs", "y", Nx.tensor([2.1, 2.5, 1.8, 2.3, 2.7]))
ir%Exmc.IR{
nodes: %{
"mu" => %Exmc.Node{
id: "mu",
op: {:rv, Exmc.Dist.Normal,
%{
mu: #Nx.Tensor<
f32
EXLA.Backend
0.0
>,
sigma: #Nx.Tensor<
f32
EXLA.Backend
10.0
>
}},
deps: [],
shape: nil,
dtype: nil
},
"y" => %Exmc.Node{
id: "y",
op: {:rv, Exmc.Dist.Normal,
%{
mu: "mu",
sigma: #Nx.Tensor<
f32
EXLA.Backend
1.0
>
}},
deps: ["mu"],
shape: nil,
dtype: nil
},
"y_obs" => %Exmc.Node{
id: "y_obs",
op: {:obs, "y",
#Nx.Tensor<
f32[5]
EXLA.Backend
[2.0999999046325684, 2.5, 1.7999999523162842, 2.299999952316284, 2.700000047683716]
>, %{reduce: :sum}},
deps: ["y"],
shape: nil,
dtype: nil
}
},
outputs: [],
ncp_info: %{},
data: nil
}Running NUTS Sampling
The Sampler.sample/3 function runs the No-U-Turn Sampler (NUTS):
{trace, stats} = Sampler.sample(ir, %{"mu" => 2.0}, num_samples: 500, seed: 42, num_warmup: 200)
IO.inspect(Map.keys(trace), label: "Variables")
IO.inspect(Nx.shape(trace["mu"]), label: "Shape")Variables: ["mu"]
Shape: {500}{500}Diagnostics
Check convergence with Diagnostics.summary/1:
summary = Exmc.Diagnostics.summary(trace)
for {name, stats} <- summary do
IO.puts("#{name}: mean=#{Float.round(stats.mean, 3)} std=#{Float.round(stats.std, 3)}")
endmu: mean=2.258 std=0.493[:ok]Visualization with VegaLite
Plot the posterior trace and histogram:
alias VegaLite, as: Vl
mu_samples = Nx.to_flat_list(trace["mu"]) |> Enum.filter(&is_number/1)
data = Enum.with_index(mu_samples, fn val, i -> %{"iteration" => i, "mu" => val} end)
Vl.new(width: 600, height: 200, title: "Trace: mu")
|> Vl.data_from_values(data)
|> Vl.mark(:line)
|> Vl.encode_field(:x, "iteration", type: :quantitative)
|> Vl.encode_field(:y, "mu", type: :quantitative){"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"iteration":0,"mu":2.7395903314555086},{"iteration":1,"mu":1.9186682379867364},{"iteration":2,"mu":2.456118880478635},{"iteration":3,"mu":1.791696600143163},{"iteration":4,"mu":1.9788682718893946},{"iteration":5,"mu":1.7678027397808247},{"iteration":6,"mu":1.765500222086352},{"iteration":7,"mu":2.367628902179946},{"iteration":8,"mu":2.683973084427417},{"iteration":9,"mu":1.9822298945557153},{"iteration":10,"mu":2.5190659338713286},{"iteration":11,"mu":2.4525415025477106},{"iteration":12,"mu":2.0715552390734455},{"iteration":13,"mu":2.726884631370834},{"iteration":14,"mu":2.6129501954651344},{"iteration":15,"mu":2.7414925045152105},{"iteration":16,"mu":2.8534010388433857},{"iteration":17,"mu":2.371735188141976},{"iteration":18,"mu":1.851162571188105},{"iteration":19,"mu":1.7269341616078493},{"iteration":20,"mu":1.5484274673646088},{"iteration":21,"mu":1.7333840037010493},{"iteration":22,"mu":1.6029358879538635},{"iteration":23,"mu":1.3444139652723432},{"iteration":24,"mu":1.8552692424711255},{"iteration":25,"mu":2.6039939077486567},{"iteration":26,"mu":2.781029854178916},{"iteration":27,"mu":2.567095130936004},{"iteration":28,"mu":2.537474763375447},{"iteration":29,"mu":1.8960437934812673},{"iteration":30,"mu":1.4783898747455015},{"iteration":31,"mu":2.173209226229865},{"iteration":32,"mu":2.432613745405292},{"iteration":33,"mu":2.048914290429481},{"iteration":34,"mu":1.984280495965271},{"iteration":35,"mu":2.0754586050655113},{"iteration":36,"mu":2.137131260006997},{"iteration":37,"mu":2.5015266135426346},{"iteration":38,"mu":2.4047359347503057},{"iteration":39,"mu":2.4712286195946325},{"iteration":40,"mu":2.4585787573982727},{"iteration":41,"mu":1.8232514213860023},{"iteration":42,"mu":2.2763223805293338},{"iteration":43,"mu":1.8994344060555435},{"iteration":44,"mu":1.9736379824993349},{"iteration":45,"mu":1.8968177136392135},{"iteration":46,"mu":2.06164521413708},{"iteration":47,"mu":2.398157477523028},{"iteration":48,"mu":2.361088735895688},{"iteration":49,"mu":2.3781934562051137},{"iteration":50,"mu":3.067561681236847},{"iteration":51,"mu":2.9547781725220963},{"iteration":52,"mu":1.9547950296447907},{"iteration":53,"mu":2.685530632486233},{"iteration":54,"mu":1.7329278221217128},{"iteration":55,"mu":1.7329278221217128},{"iteration":56,"mu":1.547514198578763},{"iteration":57,"mu":1.2398428805653472},{"iteration":58,"mu":1.7189261062269288},{"iteration":59,"mu":1.698706162115631},{"iteration":60,"mu":1.738072355858925},{"iteration":61,"mu":1.908480981980597},{"iteration":62,"mu":2.0315264216773175},{"iteration":63,"mu":2.887236742684042},{"iteration":64,"mu":1.6889605143287738},{"iteration":65,"mu":2.6652711499579054},{"iteration":66,"mu":2.6994107026684353},{"iteration":67,"mu":1.8527323776046456},{"iteration":68,"mu":1.8868993962519183},{"iteration":69,"mu":2.8368523220386748},{"iteration":70,"mu":3.0515428672042924},{"iteration":71,"mu":2.8215083684367515},{"iteration":72,"mu":2.9757029906211607},{"iteration":73,"mu":1.7390913857818964},{"iteration":74,"mu":1.9762044815743096},{"iteration":75,"mu":1.5261944225330781},{"iteration":76,"mu":2.4127358364297278},{"iteration":77,"mu":2.646887383996986},{"iteration":78,"mu":2.014889399804971},{"iteration":79,"mu":2.85853493384415},{"iteration":80,"mu":2.3999577467820297},{"iteration":81,"mu":2.3862535766104904},{"iteration":82,"mu":2.349532681963688},{"iteration":83,"mu":2.209061292897743},{"iteration":84,"mu":2.020859455895414},{"iteration":85,"mu":2.0450851362084417},{"iteration":86,"mu":3.0586408961124643},{"iteration":87,"mu":2.2689358246134503},{"iteration":88,"mu":2.14637874035517},{"iteration":89,"mu":2.154531939604255},{"iteration":90,"mu":1.3946141609956724},{"iteration":91,"mu":2.779407082246341},{"iteration":92,"mu":3.032143469944849},{"iteration":93,"mu":1.7192631639979277},{"iteration":94,"mu":2.6727496375171045},{"iteration":95,"mu":2.02684443678199},{"iteration":96,"mu":1.7811765904203387},{"iteration":97,"mu":2.575009407569447},{"iteration":98,"mu":2.6258149282091248},{"iteration":99,"mu":1.9472699931169766},{"iteration":100,"mu":2.5804411573435018},{"iteration":101,"mu":2.2727852125527037},{"iteration":102,"mu":2.4923160646551885},{"iteration":103,"mu":2.713516594206853},{"iteration":104,"mu":2.398884909751191},{"iteration":105,"mu":2.435480933247459},{"iteration":106,"mu":2.5202948501744724},{"iteration":107,"mu":1.9820096927470585},{"iteration":108,"mu":2.2182099937832764},{"iteration":109,"mu":2.14790220990723},{"iteration":110,"mu":1.829886350206377},{"iteration":111,"mu":2.8297245453633004},{"iteration":112,"mu":2.8297245453633004},{"iteration":113,"mu":2.854540491933574},{"iteration":114,"mu":3.0195140019265883},{"iteration":115,"mu":1.858781342840136},{"iteration":116,"mu":3.3322308818623108},{"iteration":117,"mu":3.1055025718202325},{"iteration":118,"mu":2.0454950382722323},{"iteration":119,"mu":2.2634226084283893},{"iteration":120,"mu":1.874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mu","width":600}Vl.new(width: 600, height: 200, title: "Posterior: mu")
|> Vl.data_from_values(data)
|> Vl.mark(:bar)
|> Vl.encode_field(:x, "mu", type: :quantitative, bin: [maxbins: 30])
|> Vl.encode(:y, aggregate: 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mu","width":600}A Model with Free Parameters
When multiple parameters are free, NUTS explores the joint posterior:
ir2 = Builder.new_ir()
ir2 = Builder.rv(ir2, "mu", Exmc.Dist.Normal, %{mu: Nx.tensor(0.0), sigma: Nx.tensor(10.0)})
ir2 = Builder.rv(ir2, "sigma", Exmc.Dist.HalfNormal, %{sigma: Nx.tensor(2.0)}, transform: :log)
ir2 = Builder.rv(ir2, "y", Exmc.Dist.Normal, %{mu: "mu", sigma: "sigma"})
ir2 = Builder.obs(ir2, "y_obs", "y", Nx.tensor([2.1, 2.5, 1.8, 2.3, 2.7]))
{trace2, stats2} =
Sampler.sample(ir2, %{"mu" => 2.0, "sigma" => 1.0},
num_samples: 500,
seed: 123,
num_warmup: 300
)
summary2 = Exmc.Diagnostics.summary(trace2)
for {name, s} <- summary2 do
IO.puts("#{name}: mean=#{Float.round(s.mean, 3)} std=#{Float.round(s.std, 3)}")
endmu: mean=2.259 std=0.26
sigma: mean=0.535 std=0.342[:ok, :ok]mu2 = Nx.to_flat_list(trace2["mu"]) |> Enum.filter(&is_number/1)
sigma2 = Nx.to_flat_list(trace2["sigma"]) |> Enum.filter(&is_number/1)
scatter_data =
Enum.zip(mu2, sigma2)
|> Enum.map(fn {m, s} -> %{"mu" => m, "sigma" => s} end)
Vl.new(width: 400, height: 400, title: "Joint posterior: mu vs sigma")
|> Vl.data_from_values(scatter_data)
|> Vl.mark(:point, opacity: 0.3, size: 10)
|> Vl.encode_field(:x, "mu", type: :quantitative)
|> Vl.encode_field(:y, "sigma", type: 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