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Model Comparison with WAIC and LOO

notebooks/03_model_comparison.livemd

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Igor O'sten
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Model Comparison with WAIC and LOO

# 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"}
])

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
            0.0
          >,
          sigma: #Nx.Tensor<
            f32
            10.0
          >
        }},
       deps: [],
       shape: nil,
       dtype: nil
     },
     "y" => %Exmc.Node{
       id: "y",
       op: {:rv, Exmc.Dist.Normal,
        %{
          mu: "mu",
          sigma: #Nx.Tensor<
            f32
            1.0
          >
        }},
       deps: ["mu"],
       shape: nil,
       dtype: nil
     },
     "y_obs" => %Exmc.Node{
       id: "y_obs",
       op: {:obs, "y",
        #Nx.Tensor<
          f32[7]
          [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: %{}
 },
 %Exmc.IR{
   nodes: %{
     "mu" => %Exmc.Node{
       id: "mu",
       op: {:rv, Exmc.Dist.Normal,
        %{
          mu: #Nx.Tensor<
            f32
            0.0
          >,
          sigma: #Nx.Tensor<
            f32
            10.0
          >
        }},
       deps: [],
       shape: nil,
       dtype: nil
     },
     "sigma" => %Exmc.Node{
       id: "sigma",
       op: {:rv, Exmc.Dist.HalfNormal,
        %{
          sigma: #Nx.Tensor<
            f32
            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]
          [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: %{}
 }}

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))}")

02:32:26.343 [info] XLA service 0x7bd65da5e510 initialized for platform CUDA (this does not guarantee that XLA will be used). Devices:

02:32:26.344 [info]   StreamExecutor device (0): NVIDIA GeForce RTX 3060 Ti, Compute Capability 8.6
Model 1 vars: ["mu"]
Model 2 vars: ["mu", "sigma"]
:ok

Posterior 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=2.227 std=0.419

Model 2 (free var):
  mu: mean=2.26 std=0.155
  sigma: mean=0.39 std=0.156
:ok

Computing 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: 6.79 (se: 1.25, p_waic: 1.42)
:ok

Computing 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: 14.87 (se: 0.14, p_loo: 0.17)
Model 2 LOO: 6.94 (se: 1.3, p_loo: 1.49)
:ok

Model 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}")
end
Free variance: IC=6.79 (best)
Fixed variance: IC=14.87 d_elpd=-4.04
[:ok, :ok]

Posterior Comparison Plot

alias VegaLite, as: Vl

mu1 = Nx.to_flat_list(trace1["mu"]) |> Enum.filter(&amp;is_number/1)
mu2 = Nx.to_flat_list(trace2["mu"]) |> Enum.filter(&amp;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":1.9554326365280899},{"model":"Fixed variance","mu":1.7665232499314862},{"model":"Fixed variance","mu":2.6542420607217134},{"model":"Fixed variance","mu":2.790543166716755},{"model":"Fixed variance","mu":2.013242492161991},{"model":"Fixed variance","mu":1.830366327399831},{"model":"Fixed variance","mu":1.8200464072541882},{"model":"Fixed variance","mu":2.3072111881497106},{"model":"Fixed variance","mu":2.6302199386593426},{"model":"Fixed variance","mu":1.9331924091328558},{"model":"Fixed variance","mu":2.5508163490063382},{"model":"Fixed variance","mu":2.449740437260235},{"model":"Fixed variance","mu":2.0594981807388417},{"model":"Fixed variance","mu":2.610407889022574},{"model":"Fixed variance","mu":2.588765943621714},{"model":"Fixed variance","mu":2.672000133085727},{"model":"Fixed variance","mu":2.756495531467049},{"model":"Fixed variance","mu":2.3628282662151134},{"model":"Fixed variance","mu":1.919151620813109},{"model":"Fixed variance","mu":1.7992037206504758},{"model":"Fixed variance","mu":1.6410974376421137},{"model":"Fixed variance","mu":1.7851451946269783},{"model":"Fixed variance","mu":1.6772562166508327},{"model":"Fixed variance","mu":1.4609123468976195},{"model":"Fixed variance","mu":1.8760920038483566},{"model":"Fixed variance","mu":2.374430988264271},{"model":"Fixed variance","mu":1.9081833943155955},{"model":"Fixed variance","mu":2.035725035829592},{"model":"Fixed variance","mu":2.0418838953612064},{"model":"Fixed variance","mu":1.9955516874146721},{"model":"Fixed variance","mu":2.654867305421889},{"model":"Fixed variance","mu":2.9979386119763523},{"model":"Fixed variance","mu":1.6403350290908922},{"model":"Fixed variance","mu":1.8468626173742229},{"model":"Fixed variance","mu":2.1092843944682973},{"model":"Fixed variance","mu":2.004881245796986},{"model":"Fixed 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Interpreting Results

  • 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

Other notebooks:

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