View Source Iris: Linear Regression

Mix.install([
  {:ex_stan, path: "~/shubham/ex_stan"},
  {:explorer, "~> 0.8.0"},
  {:kino_explorer, "~> 0.1.18"},
  {:vega_lite, "~> 0.1.6"},
  {:kino_vega_lite, "~> 0.1.10"},
  {:scidata, "~> 0.1.8"}
])

Introduction

Here, we will aim to perform simple linear regression using Stan. We'll also look at performing simple checks to evaluate if our model is correct.

alias ExStan.{Model, Fit}
require Explorer.DataFrame, as: DF
alias VegaLite, as: Vl
VegaLite

The dataset consists of 150 records, with 50 records per class of flowers. It describes the length and width of sepals and petals for each class.

Suppose we want to model the relationship between the sepal length and the petal length for the Setosa class of flowers i.e we want to measure the changes to petal length as the sepal length changes.

For the sake of simplicity, we will assume that this is a linear relationship.

Linear Regression has the following form: $$ $ y = \alpha + \beta x + \epsilon $$$

  • x referes to the input variable
  • y refers to the output variable
  • $\beta$ refers to the slope of the line
  • $\alpha$ represents the y-intercept for the line i.e the value of y when x = 0
  • $\epsilon$ is the irreducible random noise associated with the output y. This is drawn from a gaussian distribution with mean 0 and variance $\sigma$. This has the form: $$ \epsilon \sim \mathcal{N}(0, \sigma) $$

Key Assumption

We assume that $y$ scatters around the linear line $\alpha + \beta x$ i.e the distribution of the $y$ given the parameters and input is a gaussian distribution with mean equal to the line and variance(scale) equal to $\sigma$

$$ P(y|x,\alpha,\beta,\sigma) = \mathcal{N}(y|\alpha + \beta x, \sigma) $$

We will use bayesian inference to find all the lines and scatters consistent with the observed data from the posterior, which can be obtained from Bayes Rule as

$$ P(\alpha, \beta, \sigma |y, x) \propto P(y|x,\alpha,\beta,\sigma) P(\alpha, \beta, \sigma, x) $$

Dataset

{train_features, train_labels} = Scidata.Iris.download()
{[
   [5.1, 3.5, 1.4, 0.2],
   [4.9, 3.0, 1.4, 0.2],
   [4.7, 3.2, 1.3, 0.2],
   [4.6, 3.1, 1.5, 0.2],
   [5.0, 3.6, 1.4, 0.2],
   [5.4, 3.9, 1.7, 0.4],
   [4.6, 3.4, 1.4, 0.3],
   [5.0, 3.4, 1.5, 0.2],
   [4.4, 2.9, 1.4, 0.2],
   [4.9, 3.1, 1.5, 0.1],
   [5.4, 3.7, 1.5, 0.2],
   [4.8, 3.4, 1.6, 0.2],
   [4.8, 3.0, 1.4, 0.1],
   [4.3, 3.0, 1.1, 0.1],
   [5.8, 4.0, 1.2, 0.2],
   [5.7, 4.4, 1.5, 0.4],
   [5.4, 3.9, 1.3, 0.4],
   [5.1, 3.5, 1.4, 0.3],
   [5.7, 3.8, 1.7, 0.3],
   [5.1, 3.8, 1.5, 0.3],
   [5.4, 3.4, 1.7, 0.2],
   [5.1, 3.7, 1.5, 0.4],
   [4.6, 3.6, 1.0, 0.2],
   [5.1, 3.3, 1.7, 0.5],
   [4.8, 3.4, 1.9, 0.2],
   [5.0, 3.0, 1.6, 0.2],
   [5.0, 3.4, 1.6, 0.4],
   [5.2, 3.5, 1.5, 0.2],
   [5.2, 3.4, 1.4, 0.2],
   [4.7, 3.2, 1.6, 0.2],
   [4.8, 3.1, 1.6, 0.2],
   [5.4, 3.4, 1.5, 0.4],
   [5.2, 4.1, 1.5, 0.1],
   [5.5, 4.2, 1.4, 0.2],
   [4.9, 3.1, 1.5, 0.1],
   [5.0, 3.2, 1.2, 0.2],
   [5.5, 3.5, 1.3, 0.2],
   [4.9, 3.1, 1.5, 0.1],
   [4.4, 3.0, 1.3, 0.2],
   [5.1, 3.4, 1.5, 0.2],
   [5.0, 3.5, 1.3, 0.3],
   [4.5, 2.3, 1.3, 0.3],
   [4.4, 3.2, 1.3, 0.2],
   [5.0, 3.5, 1.6, 0.6],
   [5.1, 3.8, 1.9, 0.4],
   [4.8, 3.0, 1.4, ...],
   [5.1, 3.8, ...],
   [4.6, ...],
   [...],
   ...
 ],
 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]}
feature_names = ["sepal_length", "sepal_width", "petal_length", "petal_width"]

df =
  train_features
  |> Enum.with_index(fn x, idx ->
    Enum.zip(feature_names, x) ++ [{"class", Enum.at(train_labels, idx)}]
  end)
  |> DF.new()
#Explorer.DataFrame<
  Polars[150 x 5]
  sepal_length f64 [5.1, 4.9, 4.7, 4.6, 5.0, ...]
  sepal_width f64 [3.5, 3.0, 3.2, 3.1, 3.6, ...]
  petal_length f64 [1.4, 1.4, 1.3, 1.5, 1.4, ...]
  petal_width f64 [0.2, 0.2, 0.2, 0.2, 0.2, ...]
  class s64 [0, 0, 0, 0, 0, ...]
>
values =
  df
  |> DF.filter(class == 1)
  |> DF.select(["sepal_length", "petal_length"])
  |> DF.to_rows()

Vl.new(width: 600, height: 300, title: "Versicolor Sepal Length vs Petal Length")
|> Vl.data_from_values(values)
|> Vl.encode_field(:x, "sepal_length", type: :quantitative, scale: [domain: [0.0, 7.5]])
|> Vl.encode_field(:y, "petal_length", type: :quantitative, scale: [domain: [0.0, 6.5]])
|> Vl.layers([
  Vl.new()
  |> Vl.mark(:point)
  |> Vl.encode_field(:y, "petal_length", type: :quantitative)
])
{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"petal_length":4.7,"sepal_length":7.0},{"petal_length":4.5,"sepal_length":6.4},{"petal_length":4.9,"sepal_length":6.9},{"petal_length":4.0,"sepal_length":5.5},{"petal_length":4.6,"sepal_length":6.5},{"petal_length":4.5,"sepal_length":5.7},{"petal_length":4.7,"sepal_length":6.3},{"petal_length":3.3,"sepal_length":4.9},{"petal_length":4.6,"sepal_length":6.6},{"petal_length":3.9,"sepal_length":5.2},{"petal_length":3.5,"sepal_length":5.0},{"petal_length":4.2,"sepal_length":5.9},{"petal_length":4.0,"sepal_length":6.0},{"petal_length":4.7,"sepal_length":6.1},{"petal_length":3.6,"sepal_length":5.6},{"petal_length":4.4,"sepal_length":6.7},{"petal_length":4.5,"sepal_length":5.6},{"petal_length":4.1,"sepal_length":5.8},{"petal_length":4.5,"sepal_length":6.2},{"petal_length":3.9,"sepal_length":5.6},{"petal_length":4.8,"sepal_length":5.9},{"petal_length":4.0,"sepal_length":6.1},{"petal_length":4.9,"sepal_length":6.3},{"petal_length":4.7,"sepal_length":6.1},{"petal_length":4.3,"sepal_length":6.4},{"petal_length":4.4,"sepal_length":6.6},{"petal_length":4.8,"sepal_length":6.8},{"petal_length":5.0,"sepal_length":6.7},{"petal_length":4.5,"sepal_length":6.0},{"petal_length":3.5,"sepal_length":5.7},{"petal_length":3.8,"sepal_length":5.5},{"petal_length":3.7,"sepal_length":5.5},{"petal_length":3.9,"sepal_length":5.8},{"petal_length":5.1,"sepal_length":6.0},{"petal_length":4.5,"sepal_length":5.4},{"petal_length":4.5,"sepal_length":6.0},{"petal_length":4.7,"sepal_length":6.7},{"petal_length":4.4,"sepal_length":6.3},{"petal_length":4.1,"sepal_length":5.6},{"petal_length":4.0,"sepal_length":5.5},{"petal_length":4.4,"sepal_length":5.5},{"petal_length":4.6,"sepal_length":6.1},{"petal_length":4.0,"sepal_length":5.8},{"petal_length":3.3,"sepal_length":5.0},{"petal_length":4.2,"sepal_length":5.6},{"petal_length":4.2,"sepal_length":5.7},{"petal_length":4.2,"sepal_length":5.7},{"petal_length":4.3,"sepal_length":6.2},{"petal_length":3.0,"sepal_length":5.1},{"petal_length":4.1,"sepal_length":5.7}]},"encoding":{"x":{"field":"sepal_length","scale":{"domain":[0.0,7.5]},"type":"quantitative"},"y":{"field":"petal_length","scale":{"domain":[0.0,6.5]},"type":"quantitative"}},"height":300,"layer":[{"encoding":{"y":{"field":"petal_length","type":"quantitative"}},"mark":"point"}],"title":"Versicolor Sepal Length vs Petal Length","width":600}
x =
  df
  |> DF.filter(class == 1)
  |> DF.select(["sepal_length"])
  |> DF.to_columns()
  |> Map.get("sepal_length")

y =
  df
  |> DF.filter(class == 1)
  |> DF.select(["petal_length"])
  |> DF.to_columns()
  |> Map.get("petal_length")
[4.7, 4.5, 4.9, 4.0, 4.6, 4.5, 4.7, 3.3, 4.6, 3.9, 3.5, 4.2, 4.0, 4.7, 3.6, 4.4, 4.5, 4.1, 4.5, 3.9,
 4.8, 4.0, 4.9, 4.7, 4.3, 4.4, 4.8, 5.0, 4.5, 3.5, 3.8, 3.7, 3.9, 5.1, 4.5, 4.5, 4.7, 4.4, 4.1, 4.0,
 4.4, 4.6, 4.0, 3.3, 4.2, 4.2, 4.2, 4.3, 3.0, 4.1]

Model

The key part to encode as in the model is the expert knowledge into the prior distributions of the variable.

If we are confident that the value for $\alpha$ is normally distributed with mean 3 and variance 1, we can represent the prior for $alpha$ as $\mathcal{N}(3,1)$.

Let's assume that we don't have an expert knowledge about the parameters. In this case, we will assign "weakly informative" priors to the variables to increase the changes of finding the right parameter. Eg: Use a super-wide normal prior density.

model_code = ~S"""
data {
  int<lower=0> N; // number of points
  vector[N] x; // sepal length i.e covariate
  vector[N] y; // petal length i.e variate
}

parameters {
  real alpha; // intercept
  real beta; // slope
  real<lower=0> sigma; //scatter
}

model {
  // priors
  alpha ~ normal(0, 10);
  beta ~ normal(0, 10);
  sigma ~ normal(0, 1);

  // likelihood
  y ~ normal(alpha + beta * x, sigma); 
}
"""
"data {\n  int<lower=0> N; // number of points\n  vector[N] x; // sepal length i.e covariate\n  vector[N] y; // petal length i.e variate\n}\n\nparameters {\n  real alpha; // intercept\n  real beta; // slope\n  real<lower=0> sigma; //scatter\n}\n\nmodel {\n  // priors\n  alpha ~ normal(0, 10);\n  beta ~ normal(0, 10);\n  sigma ~ normal(0, 1);\n\n  // likelihood\n  y ~ normal(alpha + beta * x, sigma); \n}\n"
data = %{
  "x" => x,
  "y" => y,
  "N" => length(x)
}

model = ExStan.build(model_code, data)

20:08:07.064 [info] Building model..

20:08:07.064 [debug] POST http://localhost:8080/v1/models

20:08:07.080 [info] Model created

20:08:07.080 [info] Messages from stanc: 

20:08:07.080 [debug] POST http://localhost:8080/v1/models/d2ncp7rn/params
%ExStan.Model{
  model_name: "models/d2ncp7rn",
  program_code: "data {\n  int<lower=0> N; // number of points\n  vector[N] x; // sepal length i.e covariate\n  vector[N] y; // petal length i.e variate\n}\n\nparameters {\n  real alpha; // intercept\n  real beta; // slope\n  real<lower=0> sigma; //scatter\n}\n\nmodel {\n  // priors\n  alpha ~ normal(0, 10);\n  beta ~ normal(0, 10);\n  sigma ~ normal(0, 1);\n\n  // likelihood\n  y ~ normal(alpha + beta * x, sigma); \n}\n",
  data: %{
    "N" => 50,
    "x" => [7.0, 6.4, 6.9, 5.5, 6.5, 5.7, 6.3, 4.9, 6.6, 5.2, 5.0, 5.9, 6.0, 6.1, 5.6, 6.7, 5.6,
     5.8, 6.2, 5.6, 5.9, 6.1, 6.3, 6.1, 6.4, 6.6, 6.8, 6.7, 6.0, 5.7, 5.5, 5.5, 5.8, 6.0, 5.4, 6.0,
     6.7, 6.3, 5.6, 5.5, 5.5, 6.1, 5.8, 5.0, 5.6, ...],
    "y" => [4.7, 4.5, 4.9, 4.0, 4.6, 4.5, 4.7, 3.3, 4.6, 3.9, 3.5, 4.2, 4.0, 4.7, 3.6, 4.4, 4.5,
     4.1, 4.5, 3.9, 4.8, 4.0, 4.9, 4.7, 4.3, 4.4, 4.8, 5.0, 4.5, 3.5, 3.8, 3.7, 3.9, 5.1, 4.5, 4.5,
     4.7, 4.4, 4.1, 4.0, 4.4, 4.6, 4.0, 3.3, ...]
  },
  param_names: ["alpha", "beta", "sigma"],
  constrained_param_names: [["alpha"], ["beta"], ["sigma"]],
  dims: [[], [], []],
  random_seed: nil
}

Inference

fit = Model.sample(model, num_chains: 4, num_warmup: 1000, num_samples: 2000, stepsize: 0.1)

13:08:44.398 [info] `chain` id is set automatically.

13:08:44.398 [debug] POST http://localhost:8080/v1/models/d2ncp7rn/fits

13:08:44.407 [debug] POST http://localhost:8080/v1/models/d2ncp7rn/fits

13:08:44.433 [debug] POST http://localhost:8080/v1/models/d2ncp7rn/fits

13:08:44.808 [debug] POST http://localhost:8080/v1/models/d2ncp7rn/fits

13:08:45.039 [debug] GET http://localhost:8080/v1/operations/iwyj7bq6

13:08:45.271 [debug] GET http://localhost:8080/v1/operations/kadv2z6x

13:08:45.272 [debug] GET http://localhost:8080/v1/operations/wkda3ztn

13:08:45.273 [debug] GET http://localhost:8080/v1/operations/xmvjrcyr

13:08:45.274 [info] Sampling: 100%, done.

13:08:45.274 [debug] GET http://localhost:8080/v1/models/d2ncp7rn/fits/iwyj7bq6

13:08:45.288 [debug] DELETE http://localhost:8080/v1/models/d2ncp7rn/fits/iwyj7bq6

13:08:45.289 [debug] GET http://localhost:8080/v1/models/d2ncp7rn/fits/kadv2z6x

13:08:45.302 [debug] DELETE http://localhost:8080/v1/models/d2ncp7rn/fits/kadv2z6x

13:08:45.304 [debug] GET http://localhost:8080/v1/models/d2ncp7rn/fits/wkda3ztn

13:08:45.330 [debug] DELETE http://localhost:8080/v1/models/d2ncp7rn/fits/wkda3ztn

13:08:45.333 [debug] GET http://localhost:8080/v1/models/d2ncp7rn/fits/xmvjrcyr

13:08:45.343 [debug] DELETE http://localhost:8080/v1/models/d2ncp7rn/fits/xmvjrcyr

13:08:45.362 [info] Messages received during sampling:

13:08:45.363 [info]   Gradient evaluation took 1.4e-05 seconds

13:08:45.363 [info]   1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds.

13:08:45.363 [info]   Adjust your expectations accordingly!

13:08:45.363 [info]   Gradient evaluation took 1.4e-05 seconds

13:08:45.363 [info]   1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds.

13:08:45.363 [info]   Adjust your expectations accordingly!

13:08:45.363 [info]   Gradient evaluation took 3e-06 seconds

13:08:45.363 [info]   1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.

13:08:45.363 [info]   Adjust your expectations accordingly!

13:08:45.363 [info]   Gradient evaluation took 2e-06 seconds

13:08:45.363 [info]   1000 transitions using 10 leapfrog steps per transition would take 0.02 seconds.

13:08:45.363 [info]   Adjust your expectations accordingly!
%ExStan.Fit{
  stan_outputs: ["{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Gradient evaluation took 1.4e-05 seconds\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds.\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Adjust your expectations accordingly!\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:    1 / 3000 [  0%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  100 / 3000 [  3%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  200 / 3000 [  6%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  300 / 3000 [ 10%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  400 / 3000 [ 13%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  500 / 3000 [ 16%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  600 / 3000 [ 20%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  700 / 3000 [ 23%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  800 / 3000 [ 26%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  900 / 3000 [ 30%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1000 / 3000 [ 33%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1001 / 3000 [ 33%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1100 / 3000 [ 36%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1200 / 3000 [ 40%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1300 / 3000 [ 43%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1400 / 3000 [ 46%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1500 / 3000 [ 50%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1600 / 3000 [ 53%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1700 / 3000 [ 56%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1800 / 3000 [ 60%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1900 / 3000 [ 63%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2000 / 3000 [ 66%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2100 / 3000 [ 70%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2200 / 3000 [ 73%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2300 / 3000 [ 76%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2400 / 3000 [ 80%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2500 / 3000 [ 83%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2600 / 3000 [ 86%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2700 / 3000 [ 90%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2800 / 3000 [ 93%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2900 / 3000 [ 96%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 3000 / 3000 [100%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info: Elapsed Time: 0.092 seconds (Warm-up)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.286 seconds (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.378 seconds (Total)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"initialization\",\"values\":[-0.9783659596645611,-0.5573192648261118,1.4981911745129347]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Adaptation terminated\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Step size = 0.0576725\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Diagonal elements of inverse mass matrix:\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"0.236142, 0.006762" <> ...,
   "{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Gradient evaluation took 1.4e-05 seconds\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds.\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Adjust your expectations accordingly!\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:    1 / 3000 [  0%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  100 / 3000 [  3%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  200 / 3000 [  6%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  300 / 3000 [ 10%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  400 / 3000 [ 13%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  500 / 3000 [ 16%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  600 / 3000 [ 20%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  700 / 3000 [ 23%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  800 / 3000 [ 26%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  900 / 3000 [ 30%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1000 / 3000 [ 33%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1001 / 3000 [ 33%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1100 / 3000 [ 36%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1200 / 3000 [ 40%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1300 / 3000 [ 43%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1400 / 3000 [ 46%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1500 / 3000 [ 50%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1600 / 3000 [ 53%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1700 / 3000 [ 56%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1800 / 3000 [ 60%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1900 / 3000 [ 63%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2000 / 3000 [ 66%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2100 / 3000 [ 70%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2200 / 3000 [ 73%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2300 / 3000 [ 76%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2400 / 3000 [ 80%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2500 / 3000 [ 83%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2600 / 3000 [ 86%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2700 / 3000 [ 90%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2800 / 3000 [ 93%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2900 / 3000 [ 96%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 3000 / 3000 [100%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info: Elapsed Time: 0.076 seconds (Warm-up)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.13 seconds (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.206 seconds (Total)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"initialization\",\"values\":[0.05442034671090079,-1.4899873790046734,0.10489923554534775]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Adaptation terminated\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Step size = 0.063929\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Diagonal elements of inverse mass matrix:\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"0.304065, 0.0084158" <> ...,
   "{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Gradient evaluation took 3e-06 seconds\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Adjust your expectations accordingly!\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:    1 / 3000 [  0%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  100 / 3000 [  3%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  200 / 3000 [  6%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  300 / 3000 [ 10%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  400 / 3000 [ 13%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  500 / 3000 [ 16%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  600 / 3000 [ 20%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  700 / 3000 [ 23%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  800 / 3000 [ 26%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  900 / 3000 [ 30%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1000 / 3000 [ 33%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1001 / 3000 [ 33%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1100 / 3000 [ 36%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1200 / 3000 [ 40%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1300 / 3000 [ 43%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1400 / 3000 [ 46%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1500 / 3000 [ 50%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1600 / 3000 [ 53%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1700 / 3000 [ 56%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1800 / 3000 [ 60%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1900 / 3000 [ 63%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2000 / 3000 [ 66%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2100 / 3000 [ 70%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2200 / 3000 [ 73%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2300 / 3000 [ 76%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2400 / 3000 [ 80%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2500 / 3000 [ 83%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2600 / 3000 [ 86%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2700 / 3000 [ 90%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2800 / 3000 [ 93%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2900 / 3000 [ 96%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 3000 / 3000 [100%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info: Elapsed Time: 0.057 seconds (Warm-up)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.128 seconds (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.185 seconds (Total)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"initialization\",\"values\":[-0.2951205304732387,-0.9470579910310857,0.20042005425138632]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Adaptation terminated\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Step size = 0.0590723\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Diagonal elements of inverse mass matrix:\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"0.357472, 0.0101886" <> ...,
   "{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Gradient evaluation took 2e-06 seconds\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:1000 transitions using 10 leapfrog steps per transition would take 0.02 seconds.\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Adjust your expectations accordingly!\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:    1 / 3000 [  0%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  100 / 3000 [  3%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  200 / 3000 [  6%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  300 / 3000 [ 10%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  400 / 3000 [ 13%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  500 / 3000 [ 16%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  600 / 3000 [ 20%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  700 / 3000 [ 23%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  800 / 3000 [ 26%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  900 / 3000 [ 30%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1000 / 3000 [ 33%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1001 / 3000 [ 33%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1100 / 3000 [ 36%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1200 / 3000 [ 40%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1300 / 3000 [ 43%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1400 / 3000 [ 46%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1500 / 3000 [ 50%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1600 / 3000 [ 53%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1700 / 3000 [ 56%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1800 / 3000 [ 60%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1900 / 3000 [ 63%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2000 / 3000 [ 66%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2100 / 3000 [ 70%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2200 / 3000 [ 73%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2300 / 3000 [ 76%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2400 / 3000 [ 80%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2500 / 3000 [ 83%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2600 / 3000 [ 86%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2700 / 3000 [ 90%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2800 / 3000 [ 93%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2900 / 3000 [ 96%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 3000 / 3000 [100%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info: Elapsed Time: 0.062 seconds (Warm-up)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.126 seconds (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.188 seconds (Total)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"initialization\",\"values\":[-0.15819898881256257,-1.5319420582368118,-0.28985475046909806]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Adaptation terminated\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Step size = 0.0677414\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Diagonal elements of inverse mass matrix:\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"0.26831, 0.007586" <> ...],
  num_chains: 4,
  param_names: ["alpha", "beta", "sigma"],
  feature_names: ["accept_stat__", "alpha", "beta", "divergent__", "energy__", "lp__",
   "n_leapfrog__", "sigma", "stepsize__", "treedepth__"],
  constrained_param_names: ["alpha", "beta", "sigma"],
  dims: [[], [], []],
  num_warmup: 1000,
  num_samples: 2000,
  num_thin: 1,
  num_flat: 1,
  save_warmup: false,
  sample_and_sampler_param_names: ["accept_stat__", "divergent__", "energy__", "lp__",
   "n_leapfrog__", "stepsize__", "treedepth__"],
  draws: #Nx.Tensor<
    f32[10][2000][4]
    [
      [
        [0.996964156627655, 0.8868873119354248, 1.0, 0.9234753847122192],
        [0.9997256398200989, 1.0, 0.9932836890220642, 0.914984941482544],
        [0.9758890867233276, 0.9477329850196838, 0.9936863780021667, 0.8674825429916382],
        [0.9952867031097412, 0.9932151436805725, 0.9546643495559692, 0.7800149321556091],
        [0.8280965089797974, 0.9776605367660522, 0.8031880259513855, 0.684487521648407],
        [0.9999793767929077, 0.9282898306846619, 0.7713868021965027, 0.8920788168907166],
        [0.9396244287490845, 0.9382675290107727, 1.0, 0.9779354333877563],
        [0.9384660124778748, 0.9944924116134644, 0.9990277886390686, 0.8236435055732727],
        [1.0, 0.9970009922981262, 0.9454789757728577, 0.8700716495513916],
        [0.9888136386871338, ...],
        ...
      ],
      ...
    ]
  >
}

Evaluation

Traceplot

fit_df = Fit.to_frame(fit)
#Explorer.DataFrame<
  Polars[8000 x 12]
  accept_stat__ f64 [0.996964156627655, 0.9997256398200989, 0.9758890867233276, 0.9952867031097412, 0.8280965089797974, ...]
  alpha f64 [0.0051194410771131516, 0.08864973485469818, 0.5399718284606934, 0.24493956565856934, 0.2502175569534302, ...]
  beta f64 [0.7153146862983704, 0.7021792531013489, 0.6310850381851196, 0.6797246932983398, 0.6745905876159668, ...]
  chain_number s64 [1, 1, 1, 1, 1, ...]
  divergent__ f64 [0.0, 0.0, 0.0, 0.0, 0.0, ...]
  energy__ f64 [-29.677431106567383, -32.132598876953125, -32.335594177246094, -29.575349807739258, -29.73563003540039, ...]
  lp__ f64 [32.973697662353516, 32.965911865234375, 32.574947357177734, 31.358783721923828, 31.926576614379883, ...]
  n_leapfrog__ f64 [27.0, 43.0, 47.0, 51.0, 11.0, ...]
  sample_number s64 [1, 2, 3, 4, 5, ...]
  sigma f64 [0.3104642331600189, 0.320194810628891, 0.29795271158218384, 0.2608219385147095, 0.2673617899417877, ...]
  stepsize__ f64 [0.05767250806093216, 0.05767250806093216, 0.05767250806093216, 0.05767250806093216, 0.05767250806093216, ...]
  treedepth__ f64 [4.0, 5.0, 5.0, 5.0, 3.0, ...]
>
param_list = fit_df |> DF.select(["alpha", "sample_number", "chain_number"]) |> DF.to_rows()

Vl.new(width: 700, height: 400)
|> Vl.data_from_values(param_list)
|> Vl.mark(:line)
|> Vl.encode_field(:x, "sample_number", type: :quantitative)
|> Vl.encode_field(:y, "alpha", type: :quantitative)
|> Vl.encode_field(:color, "chain_number", type: :nominal, scale: [scheme: "blues"])
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The above diagram shows that all chains have mixed well with each other i.e they've all arrived at a similar stationary distribution, showing that the model has converged.

Summary Statistics

DF.describe(fit_df, percentiles: [0.05, 0.5, 0.95])
#Explorer.DataFrame<
  Polars[9 x 13]
  describe string ["count", "nil_count", "mean", "std", "min", ...]
  accept_stat__ f64 [8.0e3, 0.0, 0.9250108583872206, 0.11017736305494416, 0.061433132737874985, ...]
  alpha f64 [8.0e3, 0.0, 0.1960959957432842, 0.5198474697061236, -1.6676594018936157, ...]
  beta f64 [8.0e3, 0.0, 0.684586221717298, 0.08708407236657734, 0.3472535014152527, ...]
  chain_number f64 [8.0e3, 0.0, 2.5, 1.1181038724258545, 1.0, ...]
  divergent__ f64 [8.0e3, 0.0, 0.0, 0.0, 0.0, ...]
  energy__ f64 [8.0e3, 0.0, -30.015236579418183, 1.735527576239903, -32.958709716796875, ...]
  lp__ f64 [8.0e3, 0.0, 31.507167670965195, 1.2479477885346828, 23.338132858276367, ...]
  n_leapfrog__ f64 [8.0e3, 0.0, 39.50925, 24.756428953593385, 1.0, ...]
  sample_number f64 [8.0e3, 0.0, 1000.5, 577.3862847914152, 1.0, ...]
  sigma f64 [8.0e3, 0.0, 0.32030238229595126, 0.0340225085795164, 0.23222212493419647, ...]
  stepsize__ f64 [8.0e3, 0.0, 0.06210380233824253, 0.003998393212977617, 0.05767250806093216, ...]
  treedepth__ f64 [8.0e3, 0.0, 4.486875, 1.413002939324019, 1.0, ...]
>

The above parameters can be explained as follows:

  • lp__: Log probability representing the log of the posterior density; used for debugging and model comparison.
  • acceptstat_: Acceptance probability of proposed samples in HMC/NUTS; indicates sampling efficiency.
  • stepsize__: Step size used in numerical integration of Hamiltonian dynamics; controls resolution of exploration.
  • treedepth__: Depth of tree built during each NUTS iteration; indicates exploration depth.
  • nleapfrog_: Number of leapfrog steps taken during HMC/NUTS integration; indicates exploration efficiency.
  • divergent__: Binary indicator showing if an iteration experienced divergence; indicates numerical integration issues.
  • energy__: Hamiltonian energy of the system used by NUTS; indicates exploration behavior.

Parameter Plot

tmp = DF.select(fit_df, fit.param_names) |> DF.to_rows()

new_list =
  Enum.flat_map(tmp, fn map ->
    Enum.map(map, fn {key, value} ->
      %{
        "param_name" => key,
        "param_value" => value
      }
    end)
  end)

Vl.new(width: 800, height: 300)
|> Vl.data_from_values(new_list)
|> Vl.encode_field(:y, "param_name", type: :ordinal, title: "Parameter Name")
|> Vl.layers([
  Vl.new()
  |> Vl.mark(:point, filled: true)
  |> Vl.encode_field(:x, "param_value",
    aggregate: :mean,
    type: :quantitative,
    scale: [zero: false]
  )
  |> Vl.encode(:color, value: :black),
  Vl.new()
  |> Vl.mark(:errorbar, extent: :stdev)
  |> Vl.encode_field(:x, "param_value", type: :quantitative, title: "Parameter Value")
])
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Name","type":"ordinal"}},"height":300,"layer":[{"encoding":{"color":{"value":"black"},"x":{"aggregate":"mean","field":"param_value","scale":{"zero":false},"type":"quantitative"}},"mark":{"filled":true,"type":"point"}},{"encoding":{"x":{"field":"param_value","title":"Parameter Value","type":"quantitative"}},"mark":{"extent":"stdev","type":"errorbar"}}],"width":800}

Posterior Plot

key = Nx.Random.key(42)

alpha_mean =
  fit_df
  |> DF.select(["alpha"])
  |> DF.to_series()
  |> Map.get("alpha")
  |> Explorer.Series.mean()

beta_mean =
  fit_df
  |> DF.select(["beta"])
  |> DF.to_series()
  |> Map.get("beta")
  |> Explorer.Series.mean()

sigma_mean =
  fit_df
  |> DF.select(["sigma"])
  |> DF.to_series()
  |> Map.get("sigma")
  |> Explorer.Series.mean()

x_tensor = Nx.tensor(x)
y_tensor = Nx.tensor(y)
predicted = Nx.multiply(beta_mean, x_tensor) |> Nx.add(alpha_mean)

predicted_list = Nx.to_list(predicted)

result =
  DF.new(
    sepal_length: x,
    petal_length: y,
    predicted: predicted_list
  )
  |> DF.to_rows()
[
  %{"petal_length" => 4.7, "predicted" => 4.988199710845947, "sepal_length" => 7.0},
  %{"petal_length" => 4.5, "predicted" => 4.577447891235352, "sepal_length" => 6.4},
  %{"petal_length" => 4.9, "predicted" => 4.919741153717041, "sepal_length" => 6.9},
  %{"petal_length" => 4.0, "predicted" => 3.961320161819458, "sepal_length" => 5.5},
  %{"petal_length" => 4.6, "predicted" => 4.645906448364258, "sepal_length" => 6.5},
  %{"petal_length" => 4.5, "predicted" => 4.09823751449585, "sepal_length" => 5.7},
  %{"petal_length" => 4.7, "predicted" => 4.508989334106445, "sepal_length" => 6.3},
  %{"petal_length" => 3.3, "predicted" => 3.5505685806274414, "sepal_length" => 4.9},
  %{"petal_length" => 4.6, "predicted" => 4.714365005493164, "sepal_length" => 6.6},
  %{"petal_length" => 3.9, "predicted" => 3.75594425201416, "sepal_length" => 5.2},
  %{"petal_length" => 3.5, "predicted" => 3.6190271377563477, "sepal_length" => 5.0},
  %{"petal_length" => 4.2, "predicted" => 4.235154628753662, "sepal_length" => 5.9},
  %{"petal_length" => 4.0, "predicted" => 4.303613185882568, "sepal_length" => 6.0},
  %{"petal_length" => 4.7, "predicted" => 4.372071743011475, "sepal_length" => 6.1},
  %{"petal_length" => 3.6, "predicted" => 4.029778957366943, "sepal_length" => 5.6},
  %{"petal_length" => 4.4, "predicted" => 4.78282356262207, "sepal_length" => 6.7},
  %{"petal_length" => 4.5, "predicted" => 4.029778957366943, "sepal_length" => 5.6},
  %{"petal_length" => 4.1, "predicted" => 4.166696071624756, "sepal_length" => 5.8},
  %{"petal_length" => 4.5, "predicted" => 4.440530300140381, "sepal_length" => 6.2},
  %{"petal_length" => 3.9, "predicted" => 4.029778957366943, "sepal_length" => 5.6},
  %{"petal_length" => 4.8, "predicted" => 4.235154628753662, "sepal_length" => 5.9},
  %{"petal_length" => 4.0, "predicted" => 4.372071743011475, "sepal_length" => 6.1},
  %{"petal_length" => 4.9, "predicted" => 4.508989334106445, "sepal_length" => 6.3},
  %{"petal_length" => 4.7, "predicted" => 4.372071743011475, "sepal_length" => 6.1},
  %{"petal_length" => 4.3, "predicted" => 4.577447891235352, "sepal_length" => 6.4},
  %{"petal_length" => 4.4, "predicted" => 4.714365005493164, "sepal_length" => 6.6},
  %{"petal_length" => 4.8, "predicted" => 4.851282596588135, "sepal_length" => 6.8},
  %{"petal_length" => 5.0, "predicted" => 4.78282356262207, "sepal_length" => 6.7},
  %{"petal_length" => 4.5, "predicted" => 4.303613185882568, "sepal_length" => 6.0},
  %{"petal_length" => 3.5, "predicted" => 4.09823751449585, "sepal_length" => 5.7},
  %{"petal_length" => 3.8, "predicted" => 3.961320161819458, "sepal_length" => 5.5},
  %{"petal_length" => 3.7, "predicted" => 3.961320161819458, "sepal_length" => 5.5},
  %{"petal_length" => 3.9, "predicted" => 4.166696071624756, "sepal_length" => 5.8},
  %{"petal_length" => 5.1, "predicted" => 4.303613185882568, "sepal_length" => 6.0},
  %{"petal_length" => 4.5, "predicted" => 3.8928616046905518, "sepal_length" => 5.4},
  %{"petal_length" => 4.5, "predicted" => 4.303613185882568, "sepal_length" => 6.0},
  %{"petal_length" => 4.7, "predicted" => 4.78282356262207, "sepal_length" => 6.7},
  %{"petal_length" => 4.4, "predicted" => 4.508989334106445, "sepal_length" => 6.3},
  %{"petal_length" => 4.1, "predicted" => 4.029778957366943, "sepal_length" => 5.6},
  %{"petal_length" => 4.0, "predicted" => 3.961320161819458, "sepal_length" => 5.5},
  %{"petal_length" => 4.4, "predicted" => 3.961320161819458, "sepal_length" => 5.5},
  %{"petal_length" => 4.6, "predicted" => 4.372071743011475, "sepal_length" => 6.1},
  %{"petal_length" => 4.0, "predicted" => 4.166696071624756, "sepal_length" => 5.8},
  %{"petal_length" => 3.3, "predicted" => 3.6190271377563477, "sepal_length" => 5.0},
  %{"petal_length" => 4.2, "predicted" => 4.029778957366943, "sepal_length" => 5.6},
  %{"petal_length" => 4.2, "predicted" => 4.09823751449585, "sepal_length" => 5.7},
  %{"petal_length" => 4.2, "predicted" => 4.09823751449585, "sepal_length" => 5.7},
  %{"petal_length" => 4.3, "predicted" => 4.440530300140381, ...},
  %{"petal_length" => 3.0, ...},
  %{...}
]
Vl.new(width: 700, height: 300, title: "Linear Regression Plot")
|> Vl.data_from_values(result)
|> Vl.encode_field(:x, "sepal_length", type: :quantitative, scale: [domain: [4.5, 7.5]])
|> Vl.layers([
  Vl.new()
  |> Vl.mark(:point, shape: :square)
  |> Vl.encode_field(:y, "petal_length", type: :quantitative, scale: [domain: [0.5, 6.5]]),
  Vl.new()
  |> Vl.mark(:line)
  |> Vl.encode_field(:y, "predicted", type: :quantitative, aggregate: :mean)
])
{"$schema":"https://vega.github.io/schema/vega-lite/v5.json","data":{"values":[{"petal_length":4.7,"predicted":4.988199710845947,"sepal_length":7.0},{"petal_length":4.5,"predicted":4.577447891235352,"sepal_length":6.4},{"petal_length":4.9,"predicted":4.919741153717041,"sepal_length":6.9},{"petal_length":4.0,"predicted":3.961320161819458,"sepal_length":5.5},{"petal_length":4.6,"predicted":4.645906448364258,"sepal_length":6.5},{"petal_length":4.5,"predicted":4.09823751449585,"sepal_length":5.7},{"petal_length":4.7,"predicted":4.508989334106445,"sepal_length":6.3},{"petal_length":3.3,"predicted":3.5505685806274414,"sepal_length":4.9},{"petal_length":4.6,"predicted":4.714365005493164,"sepal_length":6.6},{"petal_length":3.9,"predicted":3.75594425201416,"sepal_length":5.2},{"petal_length":3.5,"predicted":3.6190271377563477,"sepal_length":5.0},{"petal_length":4.2,"predicted":4.235154628753662,"sepal_length":5.9},{"petal_length":4.0,"predicted":4.303613185882568,"sepal_length":6.0},{"petal_length":4.7,"predicted":4.372071743011475,"sepal_length":6.1},{"petal_length":3.6,"predicted":4.029778957366943,"sepal_length":5.6},{"petal_length":4.4,"predicted":4.78282356262207,"sepal_length":6.7},{"petal_length":4.5,"predicted":4.029778957366943,"sepal_length":5.6},{"petal_length":4.1,"predicted":4.166696071624756,"sepal_length":5.8},{"petal_length":4.5,"predicted":4.440530300140381,"sepal_length":6.2},{"petal_length":3.9,"predicted":4.029778957366943,"sepal_length":5.6},{"petal_length":4.8,"predicted":4.235154628753662,"sepal_length":5.9},{"petal_length":4.0,"predicted":4.372071743011475,"sepal_length":6.1},{"petal_length":4.9,"predicted":4.508989334106445,"sepal_length":6.3},{"petal_length":4.7,"predicted":4.372071743011475,"sepal_length":6.1},{"petal_length":4.3,"predicted":4.577447891235352,"sepal_length":6.4},{"petal_length":4.4,"predicted":4.714365005493164,"sepal_length":6.6},{"petal_length":4.8,"predicted":4.851282596588135,"sepal_length":6.8},{"petal_length":5.0,"predicted":4.78282356262207,"sepal_length":6.7},{"petal_length":4.5,"predicted":4.303613185882568,"sepal_length":6.0},{"petal_length":3.5,"predicted":4.09823751449585,"sepal_length":5.7},{"petal_length":3.8,"predicted":3.961320161819458,"sepal_length":5.5},{"petal_length":3.7,"predicted":3.961320161819458,"sepal_length":5.5},{"petal_length":3.9,"predicted":4.166696071624756,"sepal_length":5.8},{"petal_length":5.1,"predicted":4.303613185882568,"sepal_length":6.0},{"petal_length":4.5,"predicted":3.8928616046905518,"sepal_length":5.4},{"petal_length":4.5,"predicted":4.303613185882568,"sepal_length":6.0},{"petal_length":4.7,"predicted":4.78282356262207,"sepal_length":6.7},{"petal_length":4.4,"predicted":4.508989334106445,"sepal_length":6.3},{"petal_length":4.1,"predicted":4.029778957366943,"sepal_length":5.6},{"petal_length":4.0,"predicted":3.961320161819458,"sepal_length":5.5},{"petal_length":4.4,"predicted":3.961320161819458,"sepal_length":5.5},{"petal_length":4.6,"predicted":4.372071743011475,"sepal_length":6.1},{"petal_length":4.0,"predicted":4.166696071624756,"sepal_length":5.8},{"petal_length":3.3,"predicted":3.6190271377563477,"sepal_length":5.0},{"petal_length":4.2,"predicted":4.029778957366943,"sepal_length":5.6},{"petal_length":4.2,"predicted":4.09823751449585,"sepal_length":5.7},{"petal_length":4.2,"predicted":4.09823751449585,"sepal_length":5.7},{"petal_length":4.3,"predicted":4.440530300140381,"sepal_length":6.2},{"petal_length":3.0,"predicted":3.687485694885254,"sepal_length":5.1},{"petal_length":4.1,"predicted":4.09823751449585,"sepal_length":5.7}]},"encoding":{"x":{"field":"sepal_length","scale":{"domain":[4.5,7.5]},"type":"quantitative"}},"height":300,"layer":[{"encoding":{"y":{"field":"petal_length","scale":{"domain":[0.5,6.5]},"type":"quantitative"}},"mark":{"shape":"square","type":"point"}},{"encoding":{"y":{"aggregate":"mean","field":"predicted","type":"quantitative"}},"mark":"line"}],"title":"Linear Regression Plot","width":700}

Posterior Predictive Checks

Here, we will aim simulate data from our posterior distribution. From Stan's documentation:

Posterior predictive checks are a way of measuring whether a model does a good job of capturing relevant aspects of the data, such as means, standard deviations, and quantiles (Rubin 1984; Andrew Gelman, Meng, and Stern 1996). Posterior predictive checking works by simulating new replicated data sets based on the fitted model parameters and then comparing statistics applied to the replicated data set with the same statistic applied to the original data set.

We will aim to do this with Stan, using the generated_quantities block.

ppc_model_code = ~S"""
data {
  int<lower=0> N; // number of points
  vector[N] x; // sepal length i.e covariate
  vector[N] y; // petal length i.e variate
}

parameters {
  real alpha; // intercept
  real beta; // slope
  real<lower=0> sigma; //scatter
}

model {
  // priors
  alpha ~ normal(0, 10);
  beta ~ normal(0, 10);
  sigma ~ normal(0, 1);

  // likelihood
  y ~ normal(alpha + beta * x, sigma); 
}

generated quantities {
  vector[N] y_sim; // simulated data from posterior

  for (i in 1:N)
      y_sim[i] = normal_rng(alpha + beta * x[i], sigma);
}
"""
"data {\n  int<lower=0> N; // number of points\n  vector[N] x; // sepal length i.e covariate\n  vector[N] y; // petal length i.e variate\n}\n\nparameters {\n  real alpha; // intercept\n  real beta; // slope\n  real<lower=0> sigma; //scatter\n}\n\nmodel {\n  // priors\n  alpha ~ normal(0, 10);\n  beta ~ normal(0, 10);\n  sigma ~ normal(0, 1);\n\n  // likelihood\n  y ~ normal(alpha + beta * x, sigma); \n}\n\ngenerated quantities {\n  vector[N] y_sim; // simulated data from posterior\n\n  for (i in 1:N)\n      y_sim[i] = normal_rng(alpha + beta * x[i], sigma);\n}\n"
ppc_model = ExStan.build(ppc_model_code, data)
ppc_fit = Model.sample(ppc_model, num_samples: 1000, num_chains: 4, stepsize: 0.1)

13:09:17.904 [info] Building model..

13:09:17.904 [debug] POST http://localhost:8080/v1/models

13:09:17.910 [info] Model created

13:09:17.910 [info] Messages from stanc: 

13:09:17.910 [debug] POST http://localhost:8080/v1/models/4fz34g6a/params

13:09:17.913 [info] `chain` id is set automatically.

13:09:17.913 [debug] POST http://localhost:8080/v1/models/4fz34g6a/fits

13:09:17.916 [debug] POST http://localhost:8080/v1/models/4fz34g6a/fits

13:09:17.934 [debug] POST http://localhost:8080/v1/models/4fz34g6a/fits

13:09:18.188 [debug] POST http://localhost:8080/v1/models/4fz34g6a/fits

13:09:18.362 [debug] GET http://localhost:8080/v1/operations/z5ufub2z

13:09:18.538 [debug] GET http://localhost:8080/v1/operations/7oyoubhj

13:09:18.538 [debug] GET http://localhost:8080/v1/operations/fnpqspp4

13:09:18.539 [debug] GET http://localhost:8080/v1/operations/443nrna2

13:09:18.540 [info] Sampling: 100%, done.

13:09:18.540 [debug] GET http://localhost:8080/v1/models/4fz34g6a/fits/z5ufub2z

13:09:18.555 [debug] DELETE http://localhost:8080/v1/models/4fz34g6a/fits/z5ufub2z

13:09:18.556 [debug] GET http://localhost:8080/v1/models/4fz34g6a/fits/7oyoubhj

13:09:18.588 [debug] DELETE http://localhost:8080/v1/models/4fz34g6a/fits/7oyoubhj

13:09:18.589 [debug] GET http://localhost:8080/v1/models/4fz34g6a/fits/fnpqspp4

13:09:18.607 [debug] DELETE http://localhost:8080/v1/models/4fz34g6a/fits/fnpqspp4

13:09:18.609 [debug] GET http://localhost:8080/v1/models/4fz34g6a/fits/443nrna2

13:09:18.633 [debug] DELETE http://localhost:8080/v1/models/4fz34g6a/fits/443nrna2

13:09:18.679 [info] Messages received during sampling:

13:09:18.680 [info]   Gradient evaluation took 6e-06 seconds

13:09:18.680 [info]   1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.

13:09:18.680 [info]   Adjust your expectations accordingly!

13:09:18.680 [info]   Gradient evaluation took 5e-06 seconds

13:09:18.680 [info]   1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.

13:09:18.680 [info]   Adjust your expectations accordingly!

13:09:18.680 [info]   Gradient evaluation took 3e-06 seconds

13:09:18.680 [info]   1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.

13:09:18.680 [info]   Adjust your expectations accordingly!

13:09:18.680 [info]   Gradient evaluation took 4e-06 seconds

13:09:18.680 [info]   1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.

13:09:18.680 [info]   Adjust your expectations accordingly!
%ExStan.Fit{
  stan_outputs: ["{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Gradient evaluation took 6e-06 seconds\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Adjust your expectations accordingly!\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:    1 / 2000 [  0%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  100 / 2000 [  5%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  200 / 2000 [ 10%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  300 / 2000 [ 15%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  400 / 2000 [ 20%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  500 / 2000 [ 25%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  600 / 2000 [ 30%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  700 / 2000 [ 35%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  800 / 2000 [ 40%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  900 / 2000 [ 45%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1000 / 2000 [ 50%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1001 / 2000 [ 50%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1100 / 2000 [ 55%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1200 / 2000 [ 60%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1300 / 2000 [ 65%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1400 / 2000 [ 70%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1500 / 2000 [ 75%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1600 / 2000 [ 80%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1700 / 2000 [ 85%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1800 / 2000 [ 90%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1900 / 2000 [ 95%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2000 / 2000 [100%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info: Elapsed Time: 0.091 seconds (Warm-up)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.163 seconds (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.254 seconds (Total)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"initialization\",\"values\":[-0.9783659596645611,-0.5573192648261118,1.4981911745129347]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Adaptation terminated\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Step size = 0.0576725\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Diagonal elements of inverse mass matrix:\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"0.236142, 0.00676247, 0.0112447\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":{\"lp__\":32.97369668557888,\"accept_stat__\":0.9969641363127649,\"stepsize__\":0.057672507174732188,\"treedepth__\":4.0,\"n_leapfrog__\":27.0,\"divergent__\":0.0,\"energy__\":-29.677431615610318,\"alpha\":0.005119441075884412,\"beta\":0.7153146994410063,\"sigma\":0.31046424650747547,\"y_sim.1\":4.82349759944606,\"y_sim.2\":4.7389115294472889,\"y_sim.3\":4.620237197295994,\"y_sim.4\":3.1115716923192475,\"y_sim.5\":4.852241530178888,\"y_sim.6\":3.8406128903727887,\"y_sim.7\":4.078493293366337,\"y_sim.8\":3.1633729835238579,\"y_sim.9\":4.542629373638011,\"y_sim.10\":3.346443011570357,\"y_sim.11\":3.6862971302939329,\"y_sim.12\":3.659597155166908,\"y_sim.13\":4.302844716876097,\"y_sim.14\":4.263521303526952,\"y_sim.15\":3.869114558043055,\"y_sim.16\":4.609311522130971,\"y_sim.17\":4.182280288201898,\"y_sim.18\":4.598414093801347,\"y_sim.19\":4.210564131153034,\"y_sim.20\":3.8367378032274699,\"y_sim.21\":4.33" <> ...,
   "{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Gradient evaluation took 5e-06 seconds\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Adjust your expectations accordingly!\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:    1 / 2000 [  0%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  100 / 2000 [  5%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  200 / 2000 [ 10%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  300 / 2000 [ 15%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  400 / 2000 [ 20%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  500 / 2000 [ 25%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  600 / 2000 [ 30%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  700 / 2000 [ 35%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  800 / 2000 [ 40%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  900 / 2000 [ 45%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1000 / 2000 [ 50%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1001 / 2000 [ 50%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1100 / 2000 [ 55%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1200 / 2000 [ 60%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1300 / 2000 [ 65%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1400 / 2000 [ 70%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1500 / 2000 [ 75%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1600 / 2000 [ 80%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1700 / 2000 [ 85%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1800 / 2000 [ 90%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1900 / 2000 [ 95%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2000 / 2000 [100%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info: Elapsed Time: 0.076 seconds (Warm-up)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.07 seconds (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.146 seconds (Total)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"initialization\",\"values\":[0.05442034671090079,-1.4899873790046734,0.10489923554534775]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Adaptation terminated\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Step size = 0.063929\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Diagonal elements of inverse mass matrix:\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"0.304065, 0.00841584, 0.0102994\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":{\"lp__\":32.46995470330302,\"accept_stat__\":0.8868873265432437,\"stepsize__\":0.06392903555262691,\"treedepth__\":4.0,\"n_leapfrog__\":23.0,\"divergent__\":0.0,\"energy__\":-31.73574304675399,\"alpha\":0.20424749330827958,\"beta\":0.6910574313443162,\"sigma\":0.3186003043255849,\"y_sim.1\":5.122578650522407,\"y_sim.2\":4.562051644389223,\"y_sim.3\":4.810301266037838,\"y_sim.4\":4.7003993119269549,\"y_sim.5\":3.9059712573175,\"y_sim.6\":3.6856157258111748,\"y_sim.7\":4.945223356827866,\"y_sim.8\":4.070591941498224,\"y_sim.9\":5.241796515864197,\"y_sim.10\":3.421046060846399,\"y_sim.11\":3.800776131898477,\"y_sim.12\":3.9975353834629865,\"y_sim.13\":4.0370073083689859,\"y_sim.14\":4.550947317161923,\"y_sim.15\":3.8164819953047308,\"y_sim.16\":4.554120850219271,\"y_sim.17\":4.37459603470757,\"y_sim.18\":3.993306958736087,\"y_sim.19\":4.598908678847114,\"y_sim.20\":3.80328831588541,\"y_sim.21\":4.73079062685" <> ...,
   "{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Gradient evaluation took 3e-06 seconds\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Adjust your expectations accordingly!\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:    1 / 2000 [  0%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  100 / 2000 [  5%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  200 / 2000 [ 10%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  300 / 2000 [ 15%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  400 / 2000 [ 20%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  500 / 2000 [ 25%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  600 / 2000 [ 30%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  700 / 2000 [ 35%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  800 / 2000 [ 40%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  900 / 2000 [ 45%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1000 / 2000 [ 50%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1001 / 2000 [ 50%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1100 / 2000 [ 55%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1200 / 2000 [ 60%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1300 / 2000 [ 65%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1400 / 2000 [ 70%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1500 / 2000 [ 75%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1600 / 2000 [ 80%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1700 / 2000 [ 85%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1800 / 2000 [ 90%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1900 / 2000 [ 95%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2000 / 2000 [100%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info: Elapsed Time: 0.059 seconds (Warm-up)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.074 seconds (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.133 seconds (Total)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"initialization\",\"values\":[-0.2951205304732387,-0.9470579910310857,0.20042005425138632]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Adaptation terminated\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Step size = 0.0590723\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Diagonal elements of inverse mass matrix:\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"0.357472, 0.0101886, 0.0105952\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":{\"lp__\":31.588690693378685,\"accept_stat__\":1.0,\"stepsize__\":0.059072263089433637,\"treedepth__\":1.0,\"n_leapfrog__\":1.0,\"divergent__\":0.0,\"energy__\":-30.7418924218134,\"alpha\":0.06963091062628729,\"beta\":0.6937046784631527,\"sigma\":0.3328321554878919,\"y_sim.1\":4.6533026028622059,\"y_sim.2\":4.602971776826893,\"y_sim.3\":4.844395404327267,\"y_sim.4\":3.6324321682801368,\"y_sim.5\":4.460097409964932,\"y_sim.6\":4.187836119198266,\"y_sim.7\":4.222040339521549,\"y_sim.8\":3.5523426866639324,\"y_sim.9\":4.033565024419359,\"y_sim.10\":3.9136923756645567,\"y_sim.11\":3.4333822211416544,\"y_sim.12\":3.951062895791963,\"y_sim.13\":4.2857153004457209,\"y_sim.14\":4.17831241268849,\"y_sim.15\":4.482991167999929,\"y_sim.16\":4.274668566549412,\"y_sim.17\":4.194775080301997,\"y_sim.18\":3.7610101594058897,\"y_sim.19\":4.082780246524915,\"y_sim.20\":4.456824283742072,\"y_sim.21\":4.3272861790492789,\"y_" <> ...,
   "{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Gradient evaluation took 4e-06 seconds\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Adjust your expectations accordingly!\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:    1 / 2000 [  0%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  100 / 2000 [  5%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  200 / 2000 [ 10%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  300 / 2000 [ 15%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  400 / 2000 [ 20%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  500 / 2000 [ 25%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  600 / 2000 [ 30%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  700 / 2000 [ 35%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  800 / 2000 [ 40%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration:  900 / 2000 [ 45%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1000 / 2000 [ 50%]  (Warmup)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1001 / 2000 [ 50%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1100 / 2000 [ 55%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1200 / 2000 [ 60%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1300 / 2000 [ 65%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1400 / 2000 [ 70%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1500 / 2000 [ 75%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1600 / 2000 [ 80%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1700 / 2000 [ 85%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1800 / 2000 [ 90%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 1900 / 2000 [ 95%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:Iteration: 2000 / 2000 [100%]  (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info: Elapsed Time: 0.063 seconds (Warm-up)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.069 seconds (Sampling)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:               0.132 seconds (Total)\"]}\n{\"version\":1,\"topic\":\"logger\",\"values\":[\"info:\"]}\n{\"version\":1,\"topic\":\"initialization\",\"values\":[-0.15819898881256257,-1.5319420582368118,-0.28985475046909806]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Adaptation terminated\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Step size = 0.0677414\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"Diagonal elements of inverse mass matrix:\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":[\"0.26831, 0.00758626, 0.0111768\"]}\n{\"version\":1,\"topic\":\"sample\",\"values\":{\"lp__\":31.57164404707967,\"accept_stat__\":0.9234753652457923,\"stepsize__\":0.06774140344667262,\"treedepth__\":5.0,\"n_leapfrog__\":63.0,\"divergent__\":0.0,\"energy__\":-29.688534131874499,\"alpha\":-0.15156067984921826,\"beta\":0.7515841376160921,\"sigma\":0.28360737467769289,\"y_sim.1\":5.0310322096736999,\"y_sim.2\":4.376444469663497,\"y_sim.3\":5.264353406422264,\"y_sim.4\":4.184760035226238,\"y_sim.5\":4.785879601986433,\"y_sim.6\":4.378459342566148,\"y_sim.7\":4.932531879362846,\"y_sim.8\":4.074046066892218,\"y_sim.9\":5.335066468458714,\"y_sim.10\":3.400329199233303,\"y_sim.11\":3.4201969428254785,\"y_sim.12\":4.18596015778274,\"y_sim.13\":4.066756941312525,\"y_sim.14\":4.531530668254802,\"y_sim.15\":4.024982663474163,\"y_sim.16\":5.118482491140103,\"y_sim.17\":4.141143215483905,\"y_sim.18\":4.2079021808234489,\"y_sim.19\":4.5551355278315139,\"y_sim.20\":4.142215925554809,\"y_sim.21\":4.578" <> ...],
  num_chains: 4,
  param_names: ["alpha", "beta", "sigma", "y_sim"],
  feature_names: ["y_sim.36", "sigma", "y_sim.11", "y_sim.18", "energy__", "y_sim.26", "y_sim.1",
   "y_sim.2", "y_sim.39", "y_sim.7", "y_sim.24", "y_sim.23", "y_sim.28", "y_sim.37", "beta",
   "y_sim.31", "y_sim.40", "y_sim.3", "lp__", "y_sim.15", "y_sim.22", "y_sim.34", "y_sim.38",
   "y_sim.5", "accept_stat__", "y_sim.4", "y_sim.16", "y_sim.21", "y_sim.25", "y_sim.29",
   "y_sim.27", "y_sim.9", "y_sim.33", "y_sim.17", "y_sim.8", "y_sim.47", "treedepth__", "y_sim.44",
   "y_sim.13", "y_sim.46", "y_sim.19", "y_sim.45", "y_sim.35", "y_sim.43", "y_sim.6", "y_sim.48",
   ...],
  constrained_param_names: ["alpha", "beta", "sigma", "y_sim.1", "y_sim.2", "y_sim.3", "y_sim.4",
   "y_sim.5", "y_sim.6", "y_sim.7", "y_sim.8", "y_sim.9", "y_sim.10", "y_sim.11", "y_sim.12",
   "y_sim.13", "y_sim.14", "y_sim.15", "y_sim.16", "y_sim.17", "y_sim.18", "y_sim.19", "y_sim.20",
   "y_sim.21", "y_sim.22", "y_sim.23", "y_sim.24", "y_sim.25", "y_sim.26", "y_sim.27", "y_sim.28",
   "y_sim.29", "y_sim.30", "y_sim.31", "y_sim.32", "y_sim.33", "y_sim.34", "y_sim.35", "y_sim.36",
   "y_sim.37", "y_sim.38", "y_sim.39", "y_sim.40", "y_sim.41", "y_sim.42", ...],
  dims: [[], [], [], ~c"2"],
  num_warmup: 1000,
  num_samples: 1000,
  num_thin: 1,
  num_flat: 1,
  save_warmup: false,
  sample_and_sampler_param_names: ["energy__", "lp__", "accept_stat__", "treedepth__",
   "divergent__", "n_leapfrog__", "stepsize__"],
  draws: #Nx.Tensor<
    f32[60][1000][4]
    [
      [
        [4.252145767211914, 4.249105930328369, 3.934061288833618, 4.269288063049316],
        [4.157285690307617, 3.845665454864502, 4.777755260467529, 3.8428361415863037],
        [4.196473598480225, 4.521337032318115, 4.215762138366699, 4.276737689971924],
        [4.449012756347656, 4.311137676239014, 3.8536674976348877, 4.311818599700928],
        [4.066036224365234, 4.394589424133301, 4.086877346038818, 3.804987668991089],
        [3.797910451889038, 4.341098308563232, 4.177387237548828, 4.1655497550964355],
        [4.001308441162109, 4.372124671936035, 4.441122531890869, 3.6728451251983643],
        [4.650207042694092, 4.255675315856934, 5.043158531188965, 3.8664402961730957],
        [4.519861221313477, 4.500011444091797, 4.780529022216797, 4.075672149658203],
        [4.851667881011963, ...],
        ...
      ],
      ...
    ]
  >
}
ppc_fit_df = Fit.to_frame(ppc_fit)
#Explorer.DataFrame<
  Polars[4000 x 62]
  accept_stat__ f64 [0.996964156627655, 0.9037833213806152, 0.9991267323493958, 0.8821033239364624, 0.7828251719474792, ...]
  alpha f64 [0.0051194410771131516, -0.055669162422418594, 0.08807269483804703, 0.11085198074579239, 0.13069532811641693, ...]
  beta f64 [0.7153146862983704, 0.7339161038398743, 0.6984054446220398, 0.709664523601532, 0.7107914090156555, ...]
  chain_number s64 [1, 1, 1, 1, 1, ...]
  divergent__ f64 [0.0, 0.0, 0.0, 0.0, 0.0, ...]
  energy__ f64 [-29.677431106567383, -31.442651748657227, -31.85980796813965, -31.320493698120117, -30.65032958984375, ...]
  lp__ f64 [32.973697662353516, 32.46551513671875, 32.75627517700195, 31.882719039916992, 30.759618759155273, ...]
  n_leapfrog__ f64 [27.0, 27.0, 39.0, 11.0, 1.0, ...]
  sample_number s64 [1, 2, 3, 4, 5, ...]
  sigma f64 [0.3104642331600189, 0.3126705586910248, 0.29566335678100586, 0.30152904987335205, 0.30151671171188354, ...]
  stepsize__ f64 [0.05767250806093216, 0.05767250806093216, 0.05767250806093216, 0.05767250806093216, 0.05767250806093216, ...]
  treedepth__ f64 [4.0, 4.0, 5.0, 3.0, 1.0, ...]
  y_sim.1 f64 [4.823497772216797, 4.7544426918029785, 5.211258411407471, 5.448543071746826, 4.683837890625, ...]
  y_sim.10 f64 [3.346442937850952, 4.259782314300537, 3.794682741165161, 3.9531168937683105, 4.257172107696533, ...]
  y_sim.11 f64 [3.6862971782684326, 3.5009124279022217, 3.7921512126922607, 3.4635982513427734, 3.467628240585327, ...]
  y_sim.12 f64 [3.659597158432007, 4.578444004058838, 3.7494394779205322, 4.257068634033203, 4.675756931304932, ...]
  y_sim.13 f64 [4.302844524383545, 4.2765421867370605, 3.913215398788452, 4.4041948318481445, 4.183792591094971, ...]
  y_sim.14 f64 [4.263521194458008, 4.555456638336182, 4.649669170379639, 4.502903461456299, 4.232295513153076, ...]
  y_sim.15 f64 [3.869114637374878, 4.184559345245361, 4.037680149078369, 3.8788270950317383, 4.486353874206543, ...]
  y_sim.16 f64 [4.609311580657959, 5.287688255310059, 4.997692108154297, 4.80945348739624, 3.7722768783569336, ...]
  y_sim.17 f64 [4.18228006362915, 4.249373435974121, 4.570245265960693, 4.196048736572266, 4.191522598266602, ...]
  y_sim.18 f64 [4.598413944244385, 4.20850944519043, 4.379755020141602, 4.470010280609131, 4.328097343444824, ...]
  y_sim.19 f64 [4.210564136505127, 4.216538429260254, 4.136019706726074, 4.071269512176514, 5.057249069213867, ...]
  y_sim.2 f64 [4.7389116287231445, 4.533236026763916, 4.082890033721924, 4.425895690917969, 4.310957431793213, ...]
  y_sim.20 f64 [3.836737871170044, 4.085955619812012, 4.859762191772461, 4.083754062652588, 3.839002847671509, ...]
  y_sim.21 f64 [4.33606481552124, 4.829834461212158, 4.083838939666748, 4.885776996612549, 4.202927112579346, ...]
  y_sim.22 f64 [5.0340375900268555, 4.445311069488525, 4.880989074707031, 4.769937992095947, 4.365612506866455, ...]
  y_sim.23 f64 [4.4671950340271, 5.216767311096191, 5.000742435455322, 4.926114082336426, 4.620131015777588, ...]
  y_sim.24 f64 [4.764279365539551, 4.583964824676514, 4.7220282554626465, 4.946085453033447, 3.9678092002868652, ...]
  y_sim.25 f64 [4.3173980712890625, 4.686129093170166, 4.9100446701049805, 4.308849334716797, 4.6112260818481445, ...]
  y_sim.26 f64 [4.934433460235596, 4.999464511871338, 4.549232482910156, 4.509341239929199, 4.467954158782959, ...]
  y_sim.27 f64 [4.571176528930664, 5.58125114440918, 4.7039947509765625, 5.1712164878845215, 5.2149505615234375, ...]
  y_sim.28 f64 [4.336417198181152, 4.099513053894043, 4.037928104400635, 4.680697441101074, 4.9466872215271, ...]
  y_sim.29 f64 [4.2963032722473145, 4.502266883850098, 4.177097797393799, 4.127522945404053, 4.134654521942139, ...]
  y_sim.3 f64 [4.620237350463867, 4.534210681915283, 5.026480674743652, 5.120565414428711, 4.882562160491943, ...]
  y_sim.30 f64 [3.952723264694214, 4.022928237915039, 3.9047634601593018, 4.249373912811279, 4.16011381149292, ...]
  y_sim.31 f64 [4.07875919342041, 4.030707836151123, 3.9181840419769287, 4.009066581726074, 4.416684150695801, ...]
  y_sim.32 f64 [4.335236072540283, 3.618971586227417, 4.424781799316406, 3.725738525390625, 4.282963275909424, ...]
  y_sim.33 f64 [4.376975059509277, 3.9650280475616455, 3.9938864707946777, 4.208399295806885, 4.212228775024414, ...]
  y_sim.34 f64 [4.17669677734375, 4.110809803009033, 4.5173516273498535, 4.602537631988525, 4.8181843757629395, ...]
  y_sim.35 f64 [4.111159324645996, 4.034000396728516, 4.281088829040527, 3.9240896701812744, 3.5911705493927, ...]
  y_sim.36 f64 [4.252145767211914, 4.157285690307617, 4.196473598480225, 4.449012756347656, 4.066036224365234, ...]
  y_sim.37 f64 [4.070067882537842, 4.8128557205200195, 4.4437456130981445, 5.364345073699951, 4.358670234680176, ...]
  y_sim.38 f64 [4.860083103179932, 4.651970863342285, 4.560464859008789, 4.839892387390137, 4.261796474456787, ...]
  y_sim.39 f64 [4.786148548126221, 4.126878261566162, 4.275202751159668, 4.109866619110107, 4.045827388763428, ...]
  y_sim.4 f64 [3.111571788787842, 4.201290130615234, 4.071994304656982, 4.188502311706543, 3.98710298538208, ...]
  y_sim.40 f64 [3.4521305561065674, 4.566705703735352, 4.076889514923096, 4.573021411895752, 3.6686506271362305, ...]
  y_sim.41 f64 [3.675605058670044, 4.320471286773682, 3.913073778152466, 4.148974895477295, 3.4520411491394043, ...]
  y_sim.42 f64 [4.481626510620117, 4.111468315124512, 4.234259128570557, 4.485671520233154, 4.03279972076416, ...]
  y_sim.43 f64 [4.335338115692139, 3.756316661834717, 4.015466690063477, 4.348787307739258, 3.9999876022338867, ...]
  y_sim.44 f64 [3.630326747894287, 4.047121524810791, 3.689958333969116, 4.114184379577637, 3.5506844520568848, ...]
  y_sim.45 f64 [4.338044166564941, 4.89019250869751, 3.9310686588287354, 3.5241105556488037, 4.278080463409424, ...]
  y_sim.46 f64 [4.823855400085449, 4.472609043121338, 4.228863716125488, 3.9417896270751953, 3.934272527694702, ...]
  y_sim.47 f64 [4.569578647613525, 3.6864655017852783, 4.0571088790893555, 3.9496653079986572, 4.179078102111816, ...]
  y_sim.48 f64 [4.518703460693359, 4.264554500579834, 4.026238918304443, 4.605221748352051, 5.182692527770996, ...]
  y_sim.49 f64 [3.9334943294525146, 3.572613000869751, 3.291825294494629, 3.7053864002227783, 3.9992780685424805, ...]
  y_sim.5 f64 [4.852241516113281, 4.272439956665039, 4.378479480743408, 4.545758247375488, 4.931586265563965, ...]
  y_sim.50 f64 [4.250868797302246, 4.228835105895996, 4.2437849044799805, 3.770556926727295, 4.285182476043701, ...]
  y_sim.6 f64 [3.8406128883361816, 4.467121124267578, 4.050739765167236, 4.193012714385986, 4.236725330352783, ...]
  y_sim.7 f64 [4.078493118286133, 4.312259197235107, 4.5459723472595215, 4.644079208374023, 4.5999627113342285, ...]
  y_sim.8 f64 [3.1633729934692383, 3.4033663272857666, 3.2289087772369385, 4.317210674285889, 3.589508295059204, ...]
  y_sim.9 f64 [4.542629241943359, 4.636689186096191, 5.198631286621094, 5.444518089294434, 5.011138916015625, ...]
>
y_sim_col_names =
  ppc_fit.feature_names
  |> Enum.filter(fn x -> String.contains?(x, "y_sim") end)
  |> Enum.sort()
  |> Enum.take(5)

y_sim_rows = ppc_fit_df |> DF.select(y_sim_col_names) |> DF.to_rows()

flat_data =
  Enum.flat_map(y_sim_rows, fn map ->
    Enum.map(map, fn {mkey, mvalue} ->
      %{"group" => mkey, "value" => mvalue}
    end)
  end)

tdata = Enum.map(y, fn elem -> %{"group" => "y_original", "value" => elem} end)

Vl.new(width: 700, height: 300)
|> Vl.data_from_values(tdata ++ flat_data)
|> Vl.transform(density: "value", groupby: ["group"])
|> Vl.mark(:line)
|> Vl.encode_field(:x, "value", type: :quantitative, title: "Y Value")
|> Vl.encode_field(:y, "density", type: :quantitative)
|> Vl.encode(:color,
  field: "group",
  type: :nominal,
  scale: [
    domain: ["y_sim.1", "y_sim.10", "y_sim.11", "y_sim.12", "y_sim.13", "y_original"],
    range: ["#b3e2cd", "#fdcdac", "#cbd5e8", "#f4cae4", "#b6f5d9", "#000000"]
  ]
)
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As we can see, the posterior samples are a very close match to the original dataset, indicating that our model has acheived a good fit.

If posterior is not a close match to the original dataset, it indicates a problem with the model specification, and the priors chosen for the parameters.