```
# Import Turing.
using Turing
# Package for loading the data set.
using RDatasets
# Package for visualization.
using StatsPlots
# Functionality for splitting the data.
using MLUtils: splitobs
# Functionality for constructing arrays with identical elements efficiently.
using FillArrays
# Functionality for normalizing the data and evaluating the model predictions.
using StatsBase
# Functionality for working with scaled identity matrices.
using LinearAlgebra
# Set a seed for reproducibility.
using Random
Random.seed!(0);
```

# Linear Regression

Turing is powerful when applied to complex hierarchical models, but it can also be put to task at common statistical procedures, like linear regression. This tutorial covers how to implement a linear regression model in Turing.

## Set Up

We begin by importing all the necessary libraries.

`setprogress!(false)`

We will use the `mtcars`

dataset from the RDatasets package. `mtcars`

contains a variety of statistics on different car models, including their miles per gallon, number of cylinders, and horsepower, among others.

We want to know if we can construct a Bayesian linear regression model to predict the miles per gallon of a car, given the other statistics it has. Let us take a look at the data we have.

```
# Load the dataset.
= RDatasets.dataset("datasets", "mtcars")
data
# Show the first six rows of the dataset.
first(data, 6)
```

Row | Model | MPG | Cyl | Disp | HP | DRat | WT | QSec | VS | AM | Gear | Carb |
---|---|---|---|---|---|---|---|---|---|---|---|---|

String31 | Float64 | Int64 | Float64 | Int64 | Float64 | Float64 | Float64 | Int64 | Int64 | Int64 | Int64 | |

1 | Mazda RX4 | 21.0 | 6 | 160.0 | 110 | 3.9 | 2.62 | 16.46 | 0 | 1 | 4 | 4 |

2 | Mazda RX4 Wag | 21.0 | 6 | 160.0 | 110 | 3.9 | 2.875 | 17.02 | 0 | 1 | 4 | 4 |

3 | Datsun 710 | 22.8 | 4 | 108.0 | 93 | 3.85 | 2.32 | 18.61 | 1 | 1 | 4 | 1 |

4 | Hornet 4 Drive | 21.4 | 6 | 258.0 | 110 | 3.08 | 3.215 | 19.44 | 1 | 0 | 3 | 1 |

5 | Hornet Sportabout | 18.7 | 8 | 360.0 | 175 | 3.15 | 3.44 | 17.02 | 0 | 0 | 3 | 2 |

6 | Valiant | 18.1 | 6 | 225.0 | 105 | 2.76 | 3.46 | 20.22 | 1 | 0 | 3 | 1 |

`size(data)`

`(32, 12)`

The next step is to get our data ready for testing. We’ll split the `mtcars`

dataset into two subsets, one for training our model and one for evaluating our model. Then, we separate the targets we want to learn (`MPG`

, in this case) and standardize the datasets by subtracting each column’s means and dividing by the standard deviation of that column. The resulting data is not very familiar looking, but this standardization process helps the sampler converge far easier.

```
# Remove the model column.
select!(data, Not(:Model))
# Split our dataset 70%/30% into training/test sets.
= map(DataFrame, splitobs(data; at=0.7, shuffle=true))
trainset, testset
# Turing requires data in matrix form.
= :MPG
target = Matrix(select(trainset, Not(target)))
train = Matrix(select(testset, Not(target)))
test = trainset[:, target]
train_target = testset[:, target]
test_target
# Standardize the features.
= fit(ZScoreTransform, train; dims=1)
dt_features transform!(dt_features, train)
StatsBase.transform!(dt_features, test)
StatsBase.
# Standardize the targets.
= fit(ZScoreTransform, train_target)
dt_targets transform!(dt_targets, train_target)
StatsBase.transform!(dt_targets, test_target); StatsBase.
```

## Model Specification

In a traditional frequentist model using OLS, our model might look like:

\[ \mathrm{MPG}_i = \alpha + \boldsymbol{\beta}^\mathsf{T}\boldsymbol{X_i} \]

where \(\boldsymbol{\beta}\) is a vector of coefficients and \(\boldsymbol{X}\) is a vector of inputs for observation \(i\). The Bayesian model we are more concerned with is the following:

\[ \mathrm{MPG}_i \sim \mathcal{N}(\alpha + \boldsymbol{\beta}^\mathsf{T}\boldsymbol{X_i}, \sigma^2) \]

where \(\alpha\) is an intercept term common to all observations, \(\boldsymbol{\beta}\) is a coefficient vector, \(\boldsymbol{X_i}\) is the observed data for car \(i\), and \(\sigma^2\) is a common variance term.

For \(\sigma^2\), we assign a prior of `truncated(Normal(0, 100); lower=0)`

. This is consistent with Andrew Gelman’s recommendations on noninformative priors for variance. The intercept term (\(\alpha\)) is assumed to be normally distributed with a mean of zero and a variance of three. This represents our assumptions that miles per gallon can be explained mostly by our assorted variables, but a high variance term indicates our uncertainty about that. Each coefficient is assumed to be normally distributed with a mean of zero and a variance of 10. We do not know that our coefficients are different from zero, and we don’t know which ones are likely to be the most important, so the variance term is quite high. Lastly, each observation \(y_i\) is distributed according to the calculated `mu`

term given by \(\alpha + \boldsymbol{\beta}^\mathsf{T}\boldsymbol{X_i}\).

```
# Bayesian linear regression.
@model function linear_regression(x, y)
# Set variance prior.
~ truncated(Normal(0, 100); lower=0)
σ²
# Set intercept prior.
~ Normal(0, sqrt(3))
intercept
# Set the priors on our coefficients.
= size(x, 2)
nfeatures ~ MvNormal(Zeros(nfeatures), 10.0 * I)
coefficients
# Calculate all the mu terms.
= intercept .+ x * coefficients
mu return y ~ MvNormal(mu, σ² * I)
end
```

`linear_regression (generic function with 2 methods)`

With our model specified, we can call the sampler. We will use the No U-Turn Sampler (NUTS) here.

```
= linear_regression(train, train_target)
model = sample(model, NUTS(), 5_000) chain
```

```
┌ Info: Found initial step size
└ ϵ = 0.05
```

```
Chains MCMC chain (5000×24×1 Array{Float64, 3}):
Iterations = 1001:1:6000
Number of chains = 1
Samples per chain = 5000
Wall duration = 10.71 seconds
Compute duration = 10.71 seconds
parameters = σ², intercept, coefficients[1], coefficients[2], coefficients[3], coefficients[4], coefficients[5], coefficients[6], coefficients[7], coefficients[8], coefficients[9], coefficients[10]
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size
Summary Statistics
parameters mean std mcse ess_bulk ess_tail ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Flo ⋯
σ² 0.2854 0.1743 0.0052 935.4479 1168.3157 1. ⋯
intercept -0.0000 0.1112 0.0017 4118.9135 2875.0643 1. ⋯
coefficients[1] 0.0715 0.5017 0.0096 2727.0462 2492.6577 1. ⋯
coefficients[2] -0.0439 0.5773 0.0134 1849.2766 1819.3889 1. ⋯
coefficients[3] -0.2490 0.4071 0.0087 2196.9694 2250.1085 1. ⋯
coefficients[4] 0.0643 0.2304 0.0042 3021.8708 2590.5764 0. ⋯
coefficients[5] -0.4099 0.5357 0.0133 1631.0509 2120.7863 1. ⋯
coefficients[6] 0.1252 0.3425 0.0075 2112.2940 1784.5923 1. ⋯
coefficients[7] 0.0036 0.3830 0.0078 2383.9210 2431.6343 1. ⋯
coefficients[8] 0.1258 0.2816 0.0058 2318.1258 2583.1122 1. ⋯
coefficients[9] 0.1419 0.2820 0.0053 2758.8579 2750.6807 1. ⋯
coefficients[10] -0.1808 0.3652 0.0091 1608.6368 1937.4585 1. ⋯
2 columns omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
σ² 0.1166 0.1793 0.2394 0.3371 0.7218
intercept -0.2188 -0.0679 -0.0017 0.0690 0.2222
coefficients[1] -0.9277 -0.2443 0.0829 0.3898 1.0542
coefficients[2] -1.2258 -0.3968 -0.0434 0.3273 1.0851
coefficients[3] -1.0796 -0.4960 -0.2371 0.0080 0.5639
coefficients[4] -0.3818 -0.0805 0.0607 0.2049 0.5466
coefficients[5] -1.4732 -0.7628 -0.4226 -0.0560 0.6459
coefficients[6] -0.5725 -0.0857 0.1420 0.3474 0.7671
coefficients[7] -0.7846 -0.2450 0.0082 0.2497 0.7529
coefficients[8] -0.4322 -0.0508 0.1316 0.3063 0.6703
coefficients[9] -0.4153 -0.0294 0.1388 0.3131 0.7107
coefficients[10] -0.9318 -0.4079 -0.1691 0.0528 0.5508
```

We can also check the densities and traces of the parameters visually using the `plot`

functionality.

`plot(chain)`