Model Comparison with WAIC and LOO
# 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
You built two models. Model A has three parameters. Model B has seven. Model B fits the training data better. Does that mean B is a better model? No. It means B has more knobs to turn.
The question isn’t “which model fits best?” It’s “which model predicts best?” Prediction requires penalizing complexity — a model that memorizes the data it saw will fail on data it hasn’t. WAIC and LOO-CV estimate out-of-sample predictive accuracy without holding data out. They measure how well your model will work tomorrow, not how well it explains today.
Every model selection tool in frequentist statistics (AIC, BIC, cross- validation) approximates this same quantity. WAIC and LOO do it using the full posterior, not a point estimate. They use the information you already computed. The cost is one line of code after sampling.
Two Competing Models
We’ll build two models for the same data and compare them using information criteria.
alias Exmc.{Builder, Sampler, ModelComparison}
# Observed data
data = Nx.tensor([2.1, 2.5, 1.8, 2.3, 2.7, 2.0, 2.4])
# Model 1: Known variance
ir1 = Builder.new_ir()
ir1 = Builder.rv(ir1, "mu", Exmc.Dist.Normal, %{mu: Nx.tensor(0.0), sigma: Nx.tensor(10.0)})
ir1 = Builder.rv(ir1, "y", Exmc.Dist.Normal, %{mu: "mu", sigma: Nx.tensor(1.0)})
ir1 = Builder.obs(ir1, "y_obs", "y", data)
# Model 2: Unknown variance
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", data)
{ir1, ir2}{%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[7]
EXLA.Backend
[2.0999999046325684, 2.5, 1.7999999523162842, 2.299999952316284, 2.700000047683716, 2.0, 2.4000000953674316]
>, %{reduce: :sum}},
deps: ["y"],
shape: nil,
dtype: nil
}
},
outputs: [],
ncp_info: %{},
data: nil
},
%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
},
"sigma" => %Exmc.Node{
id: "sigma",
op: {:rv, Exmc.Dist.HalfNormal,
%{
sigma: #Nx.Tensor<
f32
EXLA.Backend
2.0
>
}, :log},
deps: [],
shape: nil,
dtype: nil
},
"y" => %Exmc.Node{
id: "y",
op: {:rv, Exmc.Dist.Normal, %{mu: "mu", sigma: "sigma"}},
deps: ["mu", "sigma"],
shape: nil,
dtype: nil
},
"y_obs" => %Exmc.Node{
id: "y_obs",
op: {:obs, "y",
#Nx.Tensor<
f32[7]
EXLA.Backend
[2.0999999046325684, 2.5, 1.7999999523162842, 2.299999952316284, 2.700000047683716, 2.0, 2.4000000953674316]
>, %{reduce: :sum}},
deps: ["y"],
shape: nil,
dtype: nil
}
},
outputs: [],
ncp_info: %{},
data: nil
}}Sampling Both Models
{trace1, _} =
Sampler.sample(ir1, %{"mu" => 2.0}, num_samples: 500, seed: 42, num_warmup: 200)
{trace2, _} =
Sampler.sample(ir2, %{"mu" => 2.0, "sigma" => 1.0}, num_samples: 500, seed: 42, num_warmup: 200)
IO.puts("Model 1 vars: #{inspect(Map.keys(trace1))}")
IO.puts("Model 2 vars: #{inspect(Map.keys(trace2))}")Model 1 vars: ["mu"]
Model 2 vars: ["mu", "sigma"]:okPosterior Summaries
for {label, trace} <- [{"Model 1 (fixed var)", trace1}, {"Model 2 (free var)", trace2}] do
summary = Exmc.Diagnostics.summary(trace)
IO.puts("\n#{label}:")
for {name, s} <- Enum.sort(summary) do
IO.puts(" #{name}: mean=#{Float.round(s.mean, 3)} std=#{Float.round(s.std, 3)}")
end
end
:ok
Model 1 (fixed var):
mu: mean=2.229 std=0.42
Model 2 (free var):
mu: mean=2.26 std=0.147
sigma: mean=0.389 std=0.166:okComputing WAIC
WAIC (Widely Applicable Information Criterion) estimates out-of-sample prediction accuracy using the log pointwise predictive density.
First compute pointwise log-likelihoods, then pass to waic/1:
ll1 = ModelComparison.pointwise_log_likelihood(ir1, trace1)
ll2 = ModelComparison.pointwise_log_likelihood(ir2, trace2)
waic1 = ModelComparison.waic(ll1)
waic2 = ModelComparison.waic(ll2)
IO.puts("Model 1 WAIC: #{Float.round(waic1.waic, 2)} (se: #{Float.round(waic1.se, 2)}, p_waic: #{Float.round(waic1.p_waic, 2)})")
IO.puts("Model 2 WAIC: #{Float.round(waic2.waic, 2)} (se: #{Float.round(waic2.se, 2)}, p_waic: #{Float.round(waic2.p_waic, 2)})")Model 1 WAIC: 14.87 (se: 0.14, p_waic: 0.17)
Model 2 WAIC: 6.63 (se: 1.26, p_waic: 1.37):okComputing LOO
LOO-CV (Leave-One-Out Cross-Validation) via Pareto-smoothed importance sampling:
loo1 = ModelComparison.loo(ll1)
loo2 = ModelComparison.loo(ll2)
IO.puts("Model 1 LOO: #{Float.round(loo1.loo, 2)} (se: #{Float.round(loo1.se, 2)}, p_loo: #{Float.round(loo1.p_loo, 2)})")
IO.puts("Model 2 LOO: #{Float.round(loo2.loo, 2)} (se: #{Float.round(loo2.se, 2)}, p_loo: #{Float.round(loo2.p_loo, 2)})")Model 1 LOO: 14.88 (se: 0.14, p_loo: 0.18)
Model 2 LOO: 6.71 (se: 1.28, p_loo: 1.41):okModel Comparison Table
Use compare/1 to rank models. Pass {label, result} tuples:
comparison = ModelComparison.compare([
{"Fixed variance", waic1},
{"Free variance", waic2}
])
for row <- comparison do
d_str = if row.d_elpd == 0.0, do: "(best)", else: "d_elpd=#{Float.round(row.d_elpd, 2)}"
IO.puts("#{row.label}: IC=#{Float.round(row.ic, 2)} #{d_str}")
endFree variance: IC=6.63 (best)
Fixed variance: IC=14.87 d_elpd=-4.12[:ok, :ok]Posterior Comparison Plot
alias VegaLite, as: Vl
mu1 = Nx.to_flat_list(trace1["mu"]) |> Enum.filter(&is_number/1)
mu2 = Nx.to_flat_list(trace2["mu"]) |> Enum.filter(&is_number/1)
hist_data =
Enum.map(mu1, fn v -> %{"mu" => v, "model" => "Fixed variance"} end) ++
Enum.map(mu2, fn v -> %{"mu" => v, "model" => "Free variance"} end)
Vl.new(width: 500, height: 250, title: "Posterior mu: Model 1 vs Model 2")
|> Vl.data_from_values(hist_data)
|> Vl.mark(:area, opacity: 0.5, interpolate: "step")
|> Vl.encode_field(:x, "mu", type: :quantitative, bin: [maxbins: 40])
|> Vl.encode(:y, aggregate: :count, stack: nil)
|> Vl.encode_field(:color, "model", type: :nominal){"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"model":"Fixed variance","mu":2.77122498462124},{"model":"Fixed variance","mu":1.7861653622622047},{"model":"Fixed variance","mu":2.638874506545955},{"model":"Fixed variance","mu":2.7835043076414077},{"model":"Fixed variance","mu":2.01899748355281},{"model":"Fixed variance","mu":1.8331301824183035},{"model":"Fixed variance","mu":1.8210017231429458},{"model":"Fixed variance","mu":2.304955874673874},{"model":"Fixed variance","mu":2.6324733886964697},{"model":"Fixed variance","mu":1.928291598176321},{"model":"Fixed variance","mu":2.5568772953889316},{"model":"Fixed variance","mu":2.4529628467520417},{"model":"Fixed variance","mu":2.0581069969719756},{"model":"Fixed variance","mu":2.6087541418145457},{"model":"Fixed variance","mu":2.591868608551335},{"model":"Fixed variance","mu":2.6735926800263283},{"model":"Fixed variance","mu":2.757451087653869},{"model":"Fixed variance","mu":2.3646149645314978},{"model":"Fixed variance","mu":1.9207546774449475},{"model":"Fixed variance","mu":1.7998596069989947},{"model":"Fixed variance","mu":1.6412721696118884},{"model":"Fixed variance","mu":1.7845019137935876},{"model":"Fixed variance","mu":1.6767371128225999},{"model":"Fixed variance","mu":1.4605289935043653},{"model":"Fixed variance","mu":1.8746243733614163},{"model":"Fixed variance","mu":2.372671679868422},{"model":"Fixed variance","mu":1.908993038200721},{"model":"Fixed variance","mu":2.0356222806127704},{"model":"Fixed variance","mu":2.0416473101842145},{"model":"Fixed variance","mu":1.9942783999157163},{"model":"Fixed variance","mu":2.653049471286615},{"model":"Fixed variance","mu":2.996894414104262},{"model":"Fixed variance","mu":1.6428210049690193},{"model":"Fixed variance","mu":1.847199378195786},{"model":"Fixed variance","mu":2.1070956007499633},{"model":"Fixed variance","mu":2.0050145648601485},{"model":"Fixed variance","mu":2.096253819486138},{"model":"Fixed variance","mu":2.45341641182168},{"model":"Fixed variance","mu":2.371689579371447},{"model":"Fixed variance","mu":2.4283765622766356},{"model":"Fixed variance","mu":2.4210354893722026},{"model":"Fixed variance","mu":1.8619743587653648},{"model":"Fixed variance","mu":2.2329148168977286},{"model":"Fixed variance","mu":1.9337298371497331},{"model":"Fixed variance","mu":1.9918933269639103},{"model":"Fixed variance","mu":1.928792549823687},{"model":"Fixed variance","mu":2.0631177109678935},{"model":"Fixed variance","mu":2.402545929090236},{"model":"Fixed variance","mu":2.3493908489768374},{"model":"Fixed variance","mu":2.352145436026246},{"model":"Fixed variance","mu":2.9473900634241113},{"model":"Fixed variance","mu":2.8506423834175094},{"model":"Fixed variance","mu":2.001340190658578},{"model":"Fixed variance","mu":2.583273041579446},{"model":"Fixed 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- Lower WAIC/LOO is better (less information loss)
- d_elpd shows the difference from the best model
- SE indicates uncertainty in the estimate
- p_waic/p_loo is the effective number of parameters
- Large p_loo with Pareto k warnings suggests model misspecification
