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Bayesian Multinomial Logistic Regression

Multinomial logistic regression is an extension of logistic regression. Logistic regression is used to model problems in which there are exactly two possible discrete outcomes. Multinomial logistic regression is used to model problems in which there are two or more possible discrete outcomes.

In our example, we'll be using the iris dataset. The goal of the iris multiclass problem is to predict the species of a flower given measurements (in centimeters) of sepal length and width and petal length and width. There are three possible species: Iris setosa, Iris versicolor, and Iris virginica.

To start, let's import all the libraries we'll need.

# Load Turing.
using Turing

# Load RDatasets.
using RDatasets

# Load StatsPlots for visualizations and diagnostics.
using StatsPlots

# Functionality for splitting and normalizing the data.
using MLDataUtils: shuffleobs, splitobs, rescale!

# We need a softmax function which is provided by NNlib.
using NNlib: softmax

# Set a seed for reproducibility.
using Random
Random.seed!(0)

# Hide the progress prompt while sampling.
Turing.setprogress!(false);

Data Cleaning & Set Up

Now we're going to import our dataset. Twenty rows of the dataset are shown below so you can get a good feel for what kind of data we have.

# Import the "iris" dataset.
data = RDatasets.dataset("datasets", "iris");

# Show twenty random rows.
data[rand(1:size(data, 1), 20), :]
20×5 DataFrame
 Row │ SepalLength  SepalWidth  PetalLength  PetalWidth  Species
     │ Float64      Float64     Float64      Float64     Cat…
─────┼──────────────────────────────────────────────────────────────
   1 │         5.9         3.2          4.8         1.8  versicolor
   2 │         7.7         3.8          6.7         2.2  virginica
   3 │         5.9         3.0          4.2         1.5  versicolor
   4 │         6.3         2.9          5.6         1.8  virginica
   5 │         6.3         2.7          4.9         1.8  virginica
   6 │         4.8         3.4          1.6         0.2  setosa
   7 │         6.5         3.0          5.5         1.8  virginica
   8 │         5.0         3.4          1.5         0.2  setosa
  ⋮  │      ⋮           ⋮            ⋮           ⋮           ⋮
  14 │         5.6         2.8          4.9         2.0  virginica
  15 │         6.4         2.8          5.6         2.1  virginica
  16 │         7.2         3.6          6.1         2.5  virginica
  17 │         5.6         2.5          3.9         1.1  versicolor
  18 │         5.8         2.8          5.1         2.4  virginica
  19 │         5.8         2.7          4.1         1.0  versicolor
  20 │         5.0         3.4          1.5         0.2  setosa
                                                      5 rows omitted

In this data set, the outcome Species is currently coded as a string. We convert it to a numerical value by using indices 1, 2, and 3 to indicate species setosa, versicolor, and virginica, respectively.

# Recode the `Species` column.
species = ["setosa", "versicolor", "virginica"]
data[!, :Species_index] = indexin(data[!, :Species], species)

# Show twenty random rows of the new species columns
data[rand(1:size(data, 1), 20), [:Species, :Species_index]]
20×2 DataFrame
 Row │ Species     Species_index
     │ Cat…        Union…
─────┼───────────────────────────
   1 │ setosa      1
   2 │ virginica   3
   3 │ setosa      1
   4 │ versicolor  2
   5 │ setosa      1
   6 │ versicolor  2
   7 │ versicolor  2
   8 │ setosa      1
  ⋮  │     ⋮             ⋮
  14 │ setosa      1
  15 │ virginica   3
  16 │ virginica   3
  17 │ setosa      1
  18 │ versicolor  2
  19 │ virginica   3
  20 │ versicolor  2
                   5 rows omitted

After we've done that tidying, it's time to split our dataset into training and testing sets, and separate the features and target from the data. Additionally, we must rescale our feature variables so that they are centered around zero by subtracting each column by the mean and dividing it by the standard deviation. Without this step, Turing's sampler will have a hard time finding a place to start searching for parameter estimates.

# Split our dataset 50%/50% into training/test sets.
trainset, testset = splitobs(shuffleobs(data), 0.5)

# Define features and target.
features = [:SepalLength, :SepalWidth, :PetalLength, :PetalWidth]
target = :Species_index

# Turing requires data in matrix and vector form.
train_features = Matrix(trainset[!, features])
test_features = Matrix(testset[!, features])
train_target = trainset[!, target]
test_target = testset[!, target]

# Standardize the features.
μ, σ = rescale!(train_features; obsdim=1)
rescale!(test_features, μ, σ; obsdim=1);

Model Declaration

Finally, we can define our model logistic_regression. It is a function that takes three arguments where

  • x is our set of independent variables;
  • y is the element we want to predict;
  • σ is the standard deviation we want to assume for our priors.

We select the setosa species as the baseline class (the choice does not matter). Then we create the intercepts and vectors of coefficients for the other classes against that baseline. More concretely, we create scalar intercepts intercept_versicolor and intersept_virginica and coefficient vectors coefficients_versicolor and coefficients_virginica with four coefficients each for the features SepalLength, SepalWidth, PetalLength and PetalWidth. We assume a normal distribution with mean zero and standard deviation σ as prior for each scalar parameter. We want to find the posterior distribution of these, in total ten, parameters to be able to predict the species for any given set of features.

# Bayesian multinomial logistic regression
@model function logistic_regression(x, y, σ)
    n = size(x, 1)
    length(y) == n ||
        throw(DimensionMismatch("number of observations in `x` and `y` is not equal"))

    # Priors of intercepts and coefficients.
    intercept_versicolor ~ Normal(0, σ)
    intercept_virginica ~ Normal(0, σ)
    coefficients_versicolor ~ MvNormal(4, σ)
    coefficients_virginica ~ MvNormal(4, σ)

    # Compute the likelihood of the observations.
    values_versicolor = intercept_versicolor .+ x * coefficients_versicolor
    values_virginica = intercept_virginica .+ x * coefficients_virginica
    for i in 1:n
        # the 0 corresponds to the base category `setosa`
        v = softmax([0, values_versicolor[i], values_virginica[i]])
        y[i] ~ Categorical(v)
    end
end;

Sampling

Now we can run our sampler. This time we'll use HMC to sample from our posterior.

m = logistic_regression(train_features, train_target, 1)
chain = sample(m, HMC(0.05, 10), MCMCThreads(), 1_500, 3)
Chains MCMC chain (1500×19×3 Array{Float64, 3}):

Iterations        = 1:1:1500
Number of chains  = 3
Samples per chain = 1500
Wall duration     = 10.29 seconds
Compute duration  = 9.65 seconds
parameters        = intercept_versicolor, intercept_virginica, coefficients
_versicolor[1], coefficients_versicolor[2], coefficients_versicolor[3], coe
fficients_versicolor[4], coefficients_virginica[1], coefficients_virginica[
2], coefficients_virginica[3], coefficients_virginica[4]
internals         = lp, n_steps, is_accept, acceptance_rate, log_density, h
amiltonian_energy, hamiltonian_energy_error, step_size, nom_step_size

Summary Statistics
                  parameters      mean       std   naive_se      mcse      
  e ⋯
                      Symbol   Float64   Float64    Float64   Float64    Fl
oat ⋯

        intercept_versicolor    0.9354    0.5238     0.0078    0.0227   542
.78 ⋯
         intercept_virginica   -0.6837    0.6685     0.0100    0.0317   481
.34 ⋯
  coefficients_versicolor[1]    1.0630    0.6406     0.0096    0.0293   497
.97 ⋯
  coefficients_versicolor[2]   -1.4800    0.5730     0.0085    0.0248   632
.34 ⋯
  coefficients_versicolor[3]    1.0131    0.7155     0.0107    0.0330   449
.34 ⋯
  coefficients_versicolor[4]    0.3243    0.7044     0.0105    0.0364   357
.08 ⋯
   coefficients_virginica[1]    0.9780    0.6766     0.0101    0.0284   512
.07 ⋯
   coefficients_virginica[2]   -0.7076    0.6770     0.0101    0.0289   520
.45 ⋯
   coefficients_virginica[3]    2.1067    0.8158     0.0122    0.0379   388
.54 ⋯
   coefficients_virginica[4]    2.6082    0.7995     0.0119    0.0462   255
.90 ⋯
                                                               3 columns om
itted

Quantiles
                  parameters      2.5%     25.0%     50.0%     75.0%     97
.5% ⋯
                      Symbol   Float64   Float64   Float64   Float64   Floa
t64 ⋯

        intercept_versicolor   -0.0782    0.5827    0.9308    1.2831    1.9
832 ⋯
         intercept_virginica   -2.0133   -1.1248   -0.6926   -0.2305    0.6
289 ⋯
  coefficients_versicolor[1]   -0.1748    0.6246    1.0593    1.4983    2.3
599 ⋯
  coefficients_versicolor[2]   -2.6648   -1.8507   -1.4605   -1.0870   -0.4
017 ⋯
  coefficients_versicolor[3]   -0.3586    0.5081    1.0048    1.4892    2.4
542 ⋯
  coefficients_versicolor[4]   -1.0370   -0.1608    0.3164    0.8088    1.7
027 ⋯
   coefficients_virginica[1]   -0.3566    0.5259    0.9830    1.4316    2.2
932 ⋯
   coefficients_virginica[2]   -2.0652   -1.1573   -0.6868   -0.2362    0.5
693 ⋯
   coefficients_virginica[3]    0.4721    1.5736    2.1018    2.6520    3.6
780 ⋯
   coefficients_virginica[4]    1.1115    2.0559    2.6021    3.1349    4.2
297 ⋯

Since we ran multiple chains, we may as well do a spot check to make sure each chain converges around similar points.

plot(chain)

Looks good!

We can also use the corner function from MCMCChains to show the distributions of the various parameters of our multinomial logistic regression. The corner function requires MCMCChains and StatsPlots.

corner(
    chain,
    MCMCChains.namesingroup(chain, :coefficients_versicolor);
    label=[string(i) for i in 1:4],
)

corner(
    chain,
    MCMCChains.namesingroup(chain, :coefficients_virginica);
    label=[string(i) for i in 1:4],
)

Fortunately the corner plots appear to demonstrate unimodal distributions for each of our parameters, so it should be straightforward to take the means of each parameter's sampled values to estimate our model to make predictions.

Making Predictions

How do we test how well the model actually predicts whether someone is likely to default? We need to build a prediction function that takes the test dataset and runs it through the average parameter calculated during sampling.

The prediction function below takes a Matrix and a Chains object. It computes the mean of the sampled parameters and calculates the species with the highest probability for each observation. Note that we do not have to evaluate the softmax function since it does not affect the order of its inputs.

function prediction(x::Matrix, chain)
    # Pull the means from each parameter's sampled values in the chain.
    intercept_versicolor = mean(chain, :intercept_versicolor)
    intercept_virginica = mean(chain, :intercept_virginica)
    coefficients_versicolor = [
        mean(chain, k) for k in MCMCChains.namesingroup(chain, :coefficients_versicolor)
    ]
    coefficients_virginica = [
        mean(chain, k) for k in MCMCChains.namesingroup(chain, :coefficients_virginica)
    ]

    # Compute the index of the species with the highest probability for each observation.
    values_versicolor = intercept_versicolor .+ x * coefficients_versicolor
    values_virginica = intercept_virginica .+ x * coefficients_virginica
    species_indices = [
        argmax((0, x, y)) for (x, y) in zip(values_versicolor, values_virginica)
    ]

    return species_indices
end;

Let's see how we did! We run the test matrix through the prediction function, and compute the accuracy for our prediction.

# Make the predictions.
predictions = prediction(test_features, chain)

# Calculate accuracy for our test set.
mean(predictions .== testset[!, :Species_index])
0.92

Perhaps more important is to see the accuracy per class.

for s in 1:3
    rows = testset[!, :Species_index] .== s
    println("Number of `", species[s], "`: ", count(rows))
    println(
        "Percentage of `",
        species[s],
        "` predicted correctly: ",
        mean(predictions[rows] .== testset[rows, :Species_index]),
    )
end
Number of `setosa`: 24
Percentage of `setosa` predicted correctly: 0.9583333333333334
Number of `versicolor`: 25
Percentage of `versicolor` predicted correctly: 0.88
Number of `virginica`: 26
Percentage of `virginica` predicted correctly: 0.9230769230769231

This tutorial has demonstrated how to use Turing to perform Bayesian multinomial logistic regression.

Appendix

These tutorials are a part of the TuringTutorials repository, found at: https://github.com/TuringLang/TuringTutorials.

To locally run this tutorial, do the following commands:

using TuringTutorials
TuringTutorials.weave("08-multinomial-logistic-regression", "08_multinomial-logistic-regression.jmd")

Computer Information:

Julia Version 1.6.7
Commit 3b76b25b64 (2022-07-19 15:11 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-11.0.1 (ORCJIT, znver2)
Environment:
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  BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/7aa0085e-79a4-45f3-a5bd-9743c91cf3da

Package Information:

      Status `/cache/build/default-amdci4-3/julialang/turingtutorials/tutorials/08-multinomial-logistic-regression/Project.toml`
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  [9a3f8284] Random

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  [d7e528f0] FreeType2_jll v2.10.4+0
  [559328eb] FriBidi_jll v1.0.10+0
  [0656b61e] GLFW_jll v3.3.8+0
  [d2c73de3] GR_jll v0.66.0+0
  [78b55507] Gettext_jll v0.21.0+0
  [7746bdde] Glib_jll v2.68.3+2
  [3b182d85] Graphite2_jll v1.3.14+0
  [2e76f6c2] HarfBuzz_jll v2.8.1+1
  [1d5cc7b8] IntelOpenMP_jll v2018.0.3+2
  [aacddb02] JpegTurbo_jll v2.1.2+0
  [c1c5ebd0] LAME_jll v3.100.1+0
  [88015f11] LERC_jll v3.0.0+1
  [dd4b983a] LZO_jll v2.10.1+0
  [e9f186c6] Libffi_jll v3.2.2+1
  [d4300ac3] Libgcrypt_jll v1.8.7+0
  [7e76a0d4] Libglvnd_jll v1.3.0+3
  [7add5ba3] Libgpg_error_jll v1.42.0+0
  [94ce4f54] Libiconv_jll v1.16.1+1
  [4b2f31a3] Libmount_jll v2.35.0+0
  [89763e89] Libtiff_jll v4.4.0+0
  [38a345b3] Libuuid_jll v2.36.0+0
  [856f044c] MKL_jll v2022.0.0+0
  [e7412a2a] Ogg_jll v1.3.5+1
  [458c3c95] OpenSSL_jll v1.1.17+0
  [efe28fd5] OpenSpecFun_jll v0.5.5+0
  [91d4177d] Opus_jll v1.3.2+0
  [2f80f16e] PCRE_jll v8.44.0+0
  [30392449] Pixman_jll v0.40.1+0
  [ea2cea3b] Qt5Base_jll v5.15.3+1
  [f50d1b31] Rmath_jll v0.3.0+0
  [a2964d1f] Wayland_jll v1.19.0+0
  [2381bf8a] Wayland_protocols_jll v1.25.0+0
  [02c8fc9c] XML2_jll v2.9.14+0
  [aed1982a] XSLT_jll v1.1.34+0
  [4f6342f7] Xorg_libX11_jll v1.6.9+4
  [0c0b7dd1] Xorg_libXau_jll v1.0.9+4
  [935fb764] Xorg_libXcursor_jll v1.2.0+4
  [a3789734] Xorg_libXdmcp_jll v1.1.3+4
  [1082639a] Xorg_libXext_jll v1.3.4+4
  [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
  [a51aa0fd] Xorg_libXi_jll v1.7.10+4
  [d1454406] Xorg_libXinerama_jll v1.1.4+4
  [ec84b674] Xorg_libXrandr_jll v1.5.2+4
  [ea2f1a96] Xorg_libXrender_jll v0.9.10+4
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.0+3
  [c7cfdc94] Xorg_libxcb_jll v1.13.0+3
  [cc61e674] Xorg_libxkbfile_jll v1.1.0+4
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.2+4
  [33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
  [c5fb5394] Xorg_xtrans_jll v1.4.0+3
  [3161d3a3] Zstd_jll v1.5.2+0
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [b53b4c65] libpng_jll v1.6.38+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+0
  [0dad84c5] ArgTools
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions
  [44cfe95a] Pkg
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML
  [a4e569a6] Tar
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll
  [deac9b47] LibCURL_jll
  [29816b5a] LibSSH2_jll
  [c8ffd9c3] MbedTLS_jll
  [14a3606d] MozillaCACerts_jll
  [4536629a] OpenBLAS_jll
  [05823500] OpenLibm_jll
  [83775a58] Zlib_jll
  [8e850ede] nghttp2_jll
  [3f19e933] p7zip_jll