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Hierarchical Models with eXMC

notebooks/02_hierarchical_model.livemd

Igor O'sten

@borodark

exmc

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Hierarchical Models with eXMC

Section

# 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"}
])

==> exmc
    warning: default values for the optional arguments in do_compile/2 are never used
    │
 88 │   defp do_compile(%IR{} = ir, opts \\ []) do
    │        ~
    │
    └─ lib/exmc/compiler.ex:88:8: Exmc.Compiler (module)

     warning: default values for the optional arguments in build_step_fn/3 are never used
     │
 113 │   defp build_step_fn(logp_fn, _vag_fn, jit_opts \\ []) do
     │        ~
     │
     └─ lib/exmc/compiler.ex:113:8: Exmc.Compiler (module)

      warning: this clause of defp stream_from_compiled/4 is never used
      │
 1189 │   defp stream_from_compiled({vag_fn, step_fn, pm, ncp_info}, receiver_pid, init_values, opts) do
      │        ~
      │
      └─ (exmc 0.1.0) lib/exmc/nuts/sampler.ex:1189:8: Exmc.NUTS.Sampler.stream_from_compiled/4

:ok

Building a Hierarchical Model

Hierarchical models have parameters that depend on other parameters. eXMC uses string references to express these dependencies.

alias Exmc.{Builder, Sampler}

ir = Builder.new_ir()

# Hyperpriors
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)

# Group-level parameter depends on hyperpriors via string refs
ir = Builder.rv(ir, "alpha", Exmc.Dist.Normal, %{mu: "mu_group", sigma: "sigma_group"})

# Observation
data = Nx.tensor([4.1, 3.8, 4.5, 4.2, 3.9])
ir = Builder.rv(ir, "y", Exmc.Dist.Normal, %{mu: "alpha", sigma: Nx.tensor(0.5)})
ir = Builder.obs(ir, "y_obs", "y", data)

ir

%Exmc.IR{
  nodes: %{
    "alpha" => %Exmc.Node{
      id: "alpha",
      op: {:rv, Exmc.Dist.Normal, %{mu: "mu_group", sigma: "sigma_group"}},
      deps: ["mu_group", "sigma_group"],
      shape: nil,
      dtype: nil
    },
    "mu_group" => %Exmc.Node{
      id: "mu_group",
      op: {:rv, Exmc.Dist.Normal,
       %{
         mu: #Nx.Tensor<
           f32
           0.0
         >,
         sigma: #Nx.Tensor<
           f32
           10.0
         >
       }},
      deps: [],
      shape: nil,
      dtype: nil
    },
    "sigma_group" => %Exmc.Node{
      id: "sigma_group",
      op: {:rv, Exmc.Dist.HalfNormal,
       %{
         sigma: #Nx.Tensor<
           f32
           5.0
         >
       }, :log},
      deps: [],
      shape: nil,
      dtype: nil
    },
    "y" => %Exmc.Node{
      id: "y",
      op: {:rv, Exmc.Dist.Normal,
       %{
         mu: "alpha",
         sigma: #Nx.Tensor<
           f32
           0.5
         >
       }},
      deps: ["alpha"],
      shape: nil,
      dtype: nil
    },
    "y_obs" => %Exmc.Node{
      id: "y_obs",
      op: {:obs, "y",
       #Nx.Tensor<
         f32[5]
         [4.099999904632568, 3.799999952316284, 4.5, 4.199999809265137, 3.9000000953674316]
       >, %{reduce: :sum}},
      deps: ["y"],
      shape: nil,
      dtype: nil
    }
  },
  outputs: [],
  ncp_info: %{}
}

NCP Rewrite

eXMC automatically applies Non-Centered Parameterization (NCP) to hierarchical models. This improves sampling efficiency by reparameterizing alpha ~ Normal(mu, sigma) as alpha_raw ~ Normal(0, 1) with alpha = mu + sigma * alpha_raw.

{trace, stats} =
  Sampler.sample(ir, %{"mu_group" => 4.0, "sigma_group" => 1.0, "alpha" => 4.0},
    num_samples: 500,
    seed: 42,
    num_warmup: 300
  )

IO.inspect(Map.keys(trace), label: "Variables sampled")


02:25:26.913 [info] XLA service 0x73212d8a6960 initialized for platform CUDA (this does not guarantee that XLA will be used). Devices:

02:25:26.918 [info]   StreamExecutor device (0): NVIDIA GeForce RTX 3060 Ti, Compute Capability 8.6
Variables sampled: ["alpha", "mu_group", "sigma_group"]

["alpha", "mu_group", "sigma_group"]

Posterior Summary

summary = Exmc.Diagnostics.summary(trace)

for {name, s} <- Enum.sort(summary) do
  IO.puts("#{name}: mean=#{Float.round(s.mean, 3)} std=#{Float.round(s.std, 3)}")
end

alpha: mean=4.104 std=0.209
mu_group: mean=5.767 std=6.153
sigma_group: mean=5.209 std=3.264

[:ok, :ok, :ok]

Trace Plots

alias VegaLite, as: Vl

for name <- ["mu_group", "sigma_group", "alpha"] do
  samples = Nx.to_flat_list(trace[name]) |> Enum.filter(&amp;is_number/1)
  data = Enum.with_index(samples, fn val, i -> %{"iteration" => i, "value" => val} end)

  Vl.new(width: 600, height: 150, title: "Trace: #{name}")
  |> Vl.data_from_values(data)
  |> Vl.mark(:line, opacity: 0.7)
  |> Vl.encode_field(:x, "iteration", type: :quantitative)
  |> Vl.encode_field(:y, "value", type: :quantitative)
  |> Kino.render()
end

{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"iteration":0,"value":4.034599217179492},{"iteration":1,"value":3.9149210729914086},{"iteration":2,"value":4.046017672272188},{"iteration":3,"value":3.634322988401384},{"iteration":4,"value":-1.7245225313473027},{"iteration":5,"value":7.989137530282781},{"iteration":6,"value":4.831960672883955},{"iteration":7,"value":4.883193520149542},{"iteration":8,"value":6.934495210963989},{"iteration":9,"value":5.833767907160437},{"iteration":10,"value":5.849676929845238},{"iteration":11,"value":8.764205263563516},{"iteration":12,"value":11.570782366462831},{"iteration":13,"value":14.843556113792463},{"iteration":14,"value":15.611754936167255},{"iteration":15,"value":15.863215685890287},{"iteration":16,"value":16.612534723979344},{"iteration":17,"value":16.76596794792567},{"iteration":18,"value":16.767013221194034},{"iteration":19,"value":16.767013221194034},{"iteration":20,"value":16.767013221194034},{"iteration":21,"value":16.685171098569505},{"iteration":22,"value":16.76164016418651},{"iteration":23,"value":16.894639432827667},{"iteration":24,"value":16.894639432827667},{"iteration":25,"value":16.894639432827667},{"iteration":26,"value":16.894639432827667},{"iteration":27,"value":16.894639432827667},{"iteration":28,"value":16.894639432827667},{"iteration":29,"value":16.810396323736466},{"iteration":30,"value":16.810396323736466},{"iteration":31,"value":16.615682171215017},{"iteration":32,"value":16.615682171215017},{"iteration":33,"value":16.615682171215017},{"iteration":34,"value":16.615682171215017},{"iteration":35,"value":16.615682171215017},{"iteration":36,"value":16.654976872859663},{"iteration":37,"value":16.654976872859663},{"iteration":38,"value":16.654976872859663},{"iteration":39,"value":16.59658947403292},{"iteration":40,"value":16.59658947403292},{"iteration":41,"value":16.59658947403292},{"iteration":42,"value":16.36789853744245},{"iteration":43,"value":16.54353589854072},{"iteration":44,"value":16.496892634919174},{"iteration":45,"value":16.73596749896757},{"iteration":46,"value":16.600683733190614},{"iteration":47,"value":16.19245928984826},{"iteration":48,"value":16.21480799364176},{"iteration":49,"value":16.222386909023733},{"iteration":50,"value":16.12781328822561},{"iteration":51,"value":16.12781328822561},{"iteration":52,"value":16.077209768534786},{"iteration":53,"value":15.934399848421315},{"iteration":54,"value":14.12490740898879},{"iteration":55,"value":14.12490740898879},{"iteration":56,"value":15.115775111701204},{"iteration":57,"value":15.115775111701204},{"iteration":58,"value":15.115775111701204},{"iteration":59,"value":15.115775111701204},{"iteration":60,"value":1.915484236861233},{"iteration":61,"value":2.1584118519154942},{"iteration":62,"value":1.9447521070700282},{"iteration":63,"value":0.09043706848882654},{"iteration":64,"value":-0.0735330662420178},{"iteration":65,"value":-0.019847967492684214},{"iteration":66,"value":-2.696390921906434},{"iteration":67,"value":1.5678604382165555},{"iteration":68,"value":5.511700618635155},{"iteration":69,"value":3.990977990073797},{"iteration":70,"value":3.8711678088174732},{"iteration":71,"value":4.650507943315033},{"iteration":72,"value":5.093869656865986},{"iteration":73,"value":3.8713526396129976},{"iteration":74,"value":4.049902621496914},{"iteration":75,"value":5.179376042516673},{"iteration":76,"value":5.042038887144308},{"iteration":77,"value":5.258744905102827},{"iteration":78,"value":5.066778899881508},{"iteration":79,"value":5.174984085596457},{"iteration":80,"value":3.0048029090341184},{"iteration":81,"value":3.2097901420921966},{"iteration":82,"value":5.287325309590299},{"iteration":83,"value":-1.9241273809247406},{"iteration":84,"value":11.88081434894182},{"iteration":85,"value":9.403004004392214},{"iteration":86,"value":9.403004004392214},{"iteration":87,"value":9.403004004392214},{"iteration":88,"value":9.472665333398302},{"iteration":89,"value":9.258865732762883},{"iteration":90,"value":9.385113565597702},{"iteration":91,"value":10.008250828444218},{"iteration":92,"value":10.368551610743609},{"iteration":93,"value":12.30376604377168},{"iteration":94,"value":12.078790188043833},{"iteration":95,"value":11.237421049499256},{"iteration":96,"value":11.19342502590734},{"iteration":97,"value":11.19342502590734},{"iteration":98,"value":11.19342502590734},{"iteration":99,"value":11.19342502590734},{"iteration":100,"value":11.12769186049664},{"iteration":101,"value":11.12769186049664},{"iteration":102,"value":11.12769186049664},{"iteration":103,"value":10.96949803734454},{"iteration":104,"value":10.902050039540185},{"iteration":105,"value":10.58031632837147},{"iteration":106,"value":10.531148829767389},{"iteration":107,"value":10.531148829767389},{"iteration":108,"value":10.533005284144524},{"iteration":109,"value":10.592946306495136},{"iteration":110,"value":10.58172880093962},{"iteration":111,"value":10.42905925990191},{"iteration":112,"value":10.42905925990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          %{"iteration" => 2, "value" => 0.7935484897980556},
          %{"iteration" => 3, "value" => 0.7395783375144994},
          %{"iteration" => 4, "value" => 5.663982887466837},
          %{"iteration" => 5, "value" => 1.8790198244511431},
          %{"iteration" => 6, "value" => 0.6052015549121406},
          %{"iteration" => 7, "value" => 0.3450430376863075},
          %{"iteration" => 8, "value" => 4.643060817908812},
          %{"iteration" => 9, "value" => 3.3560273529816325},
          %{"iteration" => 10, "value" => 2.946707842153107},
          %{"iteration" => 11, "value" => 6.383857341948512},
          %{"iteration" => 12, "value" => 6.27171192375952},
          %{"iteration" => 13, "value" => 6.107696649691666},
          %{"iteration" => 14, "value" => 5.9423948456308535},
          %{"iteration" => 15, "value" => 5.908828077446881},
          %{"iteration" => 16, "value" => 6.769439930316577},
          %{"iteration" => 17, "value" => 7.135893052633832},
          %{"iteration" => 18, "value" => 7.361799738984826},
          %{"iteration" => 19, "value" => 7.361799738984826},
          %{"iteration" => 20, "value" => 7.361799738984826},
          %{"iteration" => 21, "value" => 7.451124421301343},
          %{"iteration" => 22, "value" => 7.362649707998059},
          %{"iteration" => 23, "value" => 7.499546661298129},
          %{"iteration" => 24, "value" => 7.499546661298129},
          %{"iteration" => 25, "value" => 7.499546661298129},
          %{"iteration" => 26, "value" => 7.499546661298129},
          %{"iteration" => 27, "value" => 7.499546661298129},
          %{"iteration" => 28, "value" => 7.499546661298129},
          %{"iteration" => 29, "value" => 7.680857162754908},
          %{"iteration" => 30, "value" => 7.680857162754908},
          %{"iteration" => 31, "value" => 7.566252213095387},
          %{"iteration" => 32, "value" => 7.566252213095387},
          %{"iteration" => 33, "value" => 7.566252213095387},
          %{"iteration" => 34, "value" => 7.566252213095387},
          %{"iteration" => 35, "value" => 7.566252213095387},
          %{"iteration" => 36, "value" => 7.548994425531527},
          %{"iteration" => 37, "value" => 7.548994425531527},
          %{"iteration" => 38, "value" => 7.548994425531527},
          %{"iteration" => 39, "value" => 7.596543675124188},
          %{"iteration" => 40, "value" => 7.596543675124188},
          %{"iteration" => 41, "value" => 7.596543675124188},
          %{"iteration" => 42, ...},
          %{...},
          ...
        ]
      },
      "encoding" => %{
        "x" => %{"field" => "iteration", "type" => "quantitative"},
        "y" => %{"field" => "value", "type" => "quantitative"}
      },
      "height" => 150,
      "mark" => %{"opacity" => 0.7, "type" => "line"},
      "title" => "Trace: sigma_group",
      "width" => 600
    }
  },
  %VegaLite{
    spec: %{
      "$schema" => "https://vega.github.io/schema/vega-lite/v5.json",
      "data" => %{
        "values" => [
          %{"iteration" => 0, "value" => 4.405998339931365},
          %{"iteration" => 1, "value" => 4.241480597405575},
          %{"iteration" => 2, "value" => 4.425855273937629},
          %{"iteration" => 3, "value" => 3.992279771764086},
          %{"iteration" => 4, "value" => 4.127185844351227},
          %{"iteration" => 5, "value" => 4.104568157578194},
          %{"iteration" => 6, "value" => 4.03912726980086},
          %{"iteration" => 7, "value" => 4.016898861325222},
          %{"iteration" => 8, "value" => 4.009225757509914},
          %{"iteration" => 9, "value" => 4.251500172243409},
          %{"iteration" => 10, "value" => 4.136665680840586},
          %{"iteration" => 11, "value" => 4.2746971711628},
          %{"iteration" => 12, "value" => 3.8157533671753434},
          %{"iteration" => 13, "value" => 3.8297354724909702},
          %{"iteration" => 14, "value" => 4.28941131591006},
          %{"iteration" => 15, "value" => 4.578131649642621},
          %{"iteration" => 16, "value" => 3.8323624117922765},
          %{"iteration" => 17, "value" => 4.340642093570125},
          %{"iteration" => 18, "value" => 4.090926826772375},
          %{"iteration" => 19, "value" => 4.090926826772375},
          %{"iteration" => 20, "value" => 4.090926826772375},
          %{"iteration" => 21, "value" => 3.922952429879466},
          %{"iteration" => 22, "value" => 4.310046373001104},
          %{"iteration" => 23, "value" => 4.1624145212575225},
          %{"iteration" => 24, "value" => 4.1624145212575225},
          %{"iteration" => 25, "value" => 4.1624145212575225},
          %{"iteration" => 26, "value" => 4.1624145212575225},
          %{"iteration" => 27, "value" => 4.1624145212575225},
          %{"iteration" => 28, "value" => 4.1624145212575225},
          %{"iteration" => 29, "value" => 4.141093212751871},
          %{"iteration" => 30, "value" => 4.141093212751871},
          %{"iteration" => 31, "value" => 4.002772854578888},
          %{"iteration" => 32, "value" => 4.002772854578888},
          %{"iteration" => 33, "value" => 4.002772854578888},
          %{"iteration" => 34, "value" => 4.002772854578888},
          %{"iteration" => 35, "value" => 4.002772854578888},
          %{"iteration" => 36, "value" => 4.0179683427047355},
          %{"iteration" => 37, "value" => 4.0179683427047355},
          %{"iteration" => 38, "value" => 4.0179683427047355},
          %{"iteration" => 39, "value" => 4.103858559171282},
          %{"iteration" => 40, "value" => 4.103858559171282},
          %{"iteration" => 41, ...},
          %{...},
          ...
        ]
      },
      "encoding" => %{
        "x" => %{"field" => "iteration", "type" => "quantitative"},
        "y" => %{"field" => "value", "type" => "quantitative"}
      },
      "height" => 150,
      "mark" => %{"opacity" => 0.7, "type" => "line"},
      "title" => "Trace: alpha",
      "width" => 600
    }
  }
]

Forest Plot (Credible Intervals)

alias VegaLite, as: Vl

forest_data =
  for {name, s} <- Enum.sort(summary) do
    %{"param" => name, "mean" => s.mean, "q5" => s.q5, "q95" => s.q95}
  end

Vl.new(width: 500, height: 150, title: "90% Credible Intervals")
|> Vl.data_from_values(forest_data)
|> Vl.layers([
  Vl.new()
  |> Vl.mark(:rule, stroke_width: 2)
  |> Vl.encode_field(:y, "param", type: :nominal)
  |> Vl.encode_field(:x, "q5", type: :quantitative, title: "value")
  |> Vl.encode_field(:x2, "q95"),
  Vl.new()
  |> Vl.mark(:point, filled: true, size: 60, color: "black")
  |> Vl.encode_field(:y, "param", type: :nominal)
  |> Vl.encode_field(:x, "mean", type: :quantitative)
])

{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"mean":4.103633638699562,"param":"alpha","q5":3.747892835345853,"q95":4.434515732324446},{"mean":5.767134129669193,"param":"mu_group","q5":-4.331011965260686,"q95":16.596794186990806},{"mean":5.208658201466352,"param":"sigma_group","q5":0.4682526704438327,"q95":11.85993943011782}]},"height":150,"layer":[{"encoding":{"x":{"field":"q5","title":"value","type":"quantitative"},"x2":{"field":"q95"},"y":{"field":"param","type":"nominal"}},"mark":{"strokeWidth":2,"type":"rule"}},{"encoding":{"x":{"field":"mean","type":"quantitative"},"y":{"field":"param","type":"nominal"}},"mark":{"color":"black","filled":true,"size":60,"type":"point"}}],"title":"90% Credible Intervals","width":500}

Multi-Chain Sampling

Run multiple chains for convergence diagnostics:

{traces, stats_list} =
  Sampler.sample_chains(ir, 4,
    init_values: %{"mu_group" => 4.0, "sigma_group" => 1.0, "alpha" => 4.0},
    num_samples: 300,
    seed: 42,
    num_warmup: 200
  )

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

Chains: 4

:ok

R-hat Convergence Diagnostic

R-hat measures between-chain vs within-chain variance. Values near 1.0 indicate convergence.

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)
  IO.puts("#{name}: R-hat = #{Float.round(rhat, 3)}")
end

alpha: R-hat = 0.999
mu_group: R-hat = 1.028
sigma_group: R-hat = 1.011

[:ok, :ok, :ok]

Multi-Chain Trace Overlay

alias VegaLite, as: Vl

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

  Vl.new(width: 600, height: 200, title: "#{name} — 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)
  |> Kino.render()
end

{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"chain":"chain 0","iteration":0,"value":1.407314907777561},{"chain":"chain 0","iteration":1,"value":8.626870003993595},{"chain":"chain 0","iteration":2,"value":4.859773345618677},{"chain":"chain 0","iteration":3,"value":4.058459792121015},{"chain":"chain 0","iteration":4,"value":3.596002467788411},{"chain":"chain 0","iteration":5,"value":3.7877951719803766},{"chain":"chain 0","iteration":6,"value":3.0560186777000475},{"chain":"chain 0","iteration":7,"value":3.1800254510456507},{"chain":"chain 0","iteration":8,"value":3.0123179833662994},{"chain":"chain 0","iteration":9,"value":3.0753301479908806},{"chain":"chain 0","iteration":10,"value":3.8072580907230953},{"chain":"chain 0","iteration":11,"value":3.9480584321136325},{"chain":"chain 0","iteration":12,"value":3.568133409707207},{"chain":"chain 0","iteration":13,"value":3.5704356367830745},{"chain":"chain 0","iteration":14,"value":6.780742763518279},{"chain":"chain 0","iteration":15,"value":6.753528130944616},{"chain":"chain 0","iteration":16,"value":6.8693885319044075},{"chain":"chain 0","iteration":17,"value":2.4221313048015807},{"chain":"chain 0","iteration":18,"value":2.3135938758994095},{"chain":"chain 0","iteration":19,"value":9.693665470074176},{"chain":"chain 0","iteration":20,"value":6.471447069285834},{"chain":"chain 0","iteration":21,"value":6.354415758137976},{"chain":"chain 0","iteration":22,"value":6.521791359110807},{"chain":"chain 0","iteration":23,"value":7.618973183499924},{"chain":"chain 0","iteration":24,"value":7.660366825984794},{"chain":"chain 0","iteration":25,"value":7.45366602505528},{"chain":"chain 0","iteration":26,"value":7.43691031120514},{"chain":"chain 0","iteration":27,"value":7.171238893806681},{"chain":"chain 0","iteration":28,"value":7.555452008776931},{"chain":"chain 0","iteration":29,"value":3.1836356642288544},{"chain":"chain 0","iteration":30,"value":-2.4838547311230035},{"chain":"chain 0","iteration":31,"value":5.195947927178689},{"chain":"chain 0","iteration":32,"value":7.727692600871187},{"chain":"chain 0","iteration":33,"value":7.590526342180961},{"chain":"chain 0","iteration":34,"value":12.291562461963531},{"chain":"chain 0","iteration":35,"value":-1.6465499411631952},{"chain":"chain 0","iteration":36,"value":-0.3725197967172812},{"chain":"chain 0","iteration":37,"value":-1.2935603685540664},{"chain":"chain 0","iteration":38,"value":0.5448555887303219},{"chain":"chain 0","iteration":39,"value":5.0245277006288385},{"chain":"chain 0","iteration":40,"value":5.1864483729983375},{"chain":"chain 0","iteration":41,"value":6.469717115148696},{"chain":"chain 0","iteration":42,"value":4.944461460297289},{"chain":"chain 0","iteration":43,"value":4.9966489114184265},{"chain":"chain 0","iteration":44,"value":1.5203273022373973},{"chain":"chain 0","iteration":45,"value":0.6035169347637113},{"chain":"chain 0","iteration":46,"value":-0.24421696530064274},{"chain":"chain 0","iteration":47,"value":-1.3954212141606164},{"chain":"chain 0","iteration":48,"value":5.916730008251573},{"chain":"chain 0","iteration":49,"value":10.255662902951293},{"chain":"chain 0","iteration":50,"value":9.866283436505062},{"chain":"chain 0","iteration":51,"value":9.866283436505062},{"chain":"chain 0","iteration":52,"value":10.055398236114206},{"chain":"chain 0","iteration":53,"value":10.055398236114206},{"chain":"chain 0","iteration":54,"value":10.073850851868707},{"chain":"chain 0","iteration":55,"value":10.124570273394653},{"chain":"chain 0","iteration":56,"value":0.9837377704309511},{"chain":"chain 0","iteration":57,"value":4.637499095131152},{"chain":"chain 0","iteration":58,"value":1.059177534070452},{"chain":"chain 0","iteration":59,"value":0.9185857207833673},{"chain":"chain 0","iteration":60,"value":1.513221598695428},{"chain":"chain 0","iteration":61,"value":0.0841541242604115},{"chain":"chain 0","iteration":62,"value":0.44391930640802163},{"chain":"chain 0","iteration":63,"value":-2.5722807171481334},{"chain":"chain 0","iteration":64,"value":-2.8197981260673752},{"chain":"chain 0","iteration":65,"value":-3.5049137594879167},{"chain":"chain 0","iteration":66,"value":4.568658275083029},{"chain":"chain 0","iteration":67,"value":6.15422177863558},{"chain":"chain 0","iteration":68,"value":6.977395315037785},{"chain":"chain 0","iteration":69,"value":0.42690682583370254},{"chain":"chain 0","iteration":70,"value":13.629450039115376},{"chain":"chain 0","iteration":71,"value":3.7563695571644136},{"chain":"chain 0","iteration":72,"value":4.051101062372528},{"chain":"chain 0","iteration":73,"value":3.770360651969327},{"chain":"chain 0","iteration":74,"value":3.4635808218061395},{"chain":"chain 0","iteration":75,"value":3.7122390749164604},{"chain":"chain 0","iteration":76,"value":4.279126159708021},{"chain":"chain 0","iteration":77,"value":-0.7510523426205247},{"chain":"chain 0","iteration":78,"value":-0.2778930054987432},{"chain":"chain 0","iteration":79,"value":3.5846306254804507},{"chain":"chain 0","iteration":80,"value":3.088943652175924},{"chain":"chain 0","iteration":81,"value":2.68523717729571},{"chain":"chain 0","iteration":82,"value":3.2848889046240317},{"chain":"chain 0","iteration":83,"value":5.800746383595646},{"chain":"chain 0","iteration":84,"value":5.37287177480597},{"chain":"chain 0","iteration":85,"value":0.4914127945330029},{"chain":"chain 0","iteration":86,"value":-2.721184131000247},{"chain":"chain 0","iteration":87,"value":-3.244081951431154},{"chain":"chain 0","iteration":88,"value":-2.104906729909725},{"chain":"chain 0","iteration":89,"value":5.011516896294952},{"chain":"chain 0","iteration":90,"value":1.174544497413484},{"chain":"chain 0","iteration":91,"value":0.9610624689922099},{"chain":"chain 0","iteration":92,"value":1.8066986580757038},{"chain":"chain 0","iteration":93,"value":1.5942533643526104},{"chain":"chain 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          %{"chain" => "chain 0", "iteration" => 34, "value" => 12.291562461963531},
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          %{"chain" => "chain 0", "iteration" => 41, "value" => 6.469717115148696},
          %{"chain" => "chain 0", "iteration" => 42, ...},
          %{"chain" => "chain 0", ...},
          %{...},
          ...
        ]
      },
      "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
    }
  },
  %VegaLite{
    spec: %{
      "$schema" => "https://vega.github.io/schema/vega-lite/v5.json",
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          %{"chain" => "chain 0", "iteration" => 41, ...},
          %{"chain" => "chain 0", ...},
          %{...},
          ...
        ]
      },
      "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" => "alpha — 4 chains",
      "width" => 600
    }
  }
]

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