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Collinear Bayes

2021-11-17

In my post on Shapley values and multicollinearity, I looked into what happens when you fit a complex uninterpretable model on collinear or near-collinear data and try to figure out which features (variables) are important. The results were reasonable but not great. Luckily, there are still more things to try. Gelman et al. (2020) say that Bayesian models can do reasonably well on collinear data because they show high uncertainty in the estimated coefficients. Also, Bayesian models have a chance of fitting the data better as is beautifully shown in the Stan documentation. It can be quite tricky to implement though because a good parameterization is necessary (https://statmodeling.stat.columbia.edu/2019/07/07/collinearity-in-bayesian-models/).

Simulating data

Let's simulate some data with various columns are increasingly correlated with the outcome (and thus each other). Here, we assume that the data is centered around zero. This is easier for the Bayesian model to work with, but can often also make interpretation of the coefficients easier. There are various methods to rescale data, one is using MLDataUtils: rescale!. Note that rescale! bases the rescaling on the sample which is not recommended for small samples (Gelman, 2020). Instead, you can use knowledge that you have about the data such as the range of questionnaire scores or the weight of cars. Specifically, for example, it could be known for the data that the weight of a car is never below zero and unlikely to be above 3600 kg (8000 lbs); the weight of a Hummer H1.

begin
    using CairoMakie
    using CategoricalArrays: categorical
    using DataFrames: Not, DataFrame, select, stack, transform
    using GLM
    using Turing
    using Random: seed!
    using Statistics: rand, mean, cor
end
indexes = 1.0:150.0;
y_true(x) = x / last(indexes);
y_noise(x, corr_coefficient) = (corr_coefficient * y_true(x) - 0.5) + rand(Normal(0, 0.15));
df = let
    seed!(0)
    X = indexes
    A = y_noise.(indexes, 0)
    B = y_noise.(indexes, 0.05)
    C = y_noise.(indexes, 0.7)
    D = y_noise.(indexes, 1)
    E = y_noise.(indexes, 1)
    Y = y_noise.(indexes, 1)
    
    DataFrame(; X, A, B, C, D, E, Y)
end
XABCDEY
11.0-0.358554-0.522957-0.481998-0.492793-0.466424-0.552454
22.0-0.479912-0.748431-0.445867-0.437615-0.636698-0.931691
33.0-0.27124-0.5285-0.524026-0.45503-0.810672-0.369753
44.0-0.481415-0.50173-0.783662-0.437662-0.131537-0.0454284
55.0-0.680866-0.488108-0.338793-0.420014-0.646305-0.390959
66.0-0.453227-0.469888-0.300942-0.541693-0.352914-0.672081
77.0-0.535196-0.498834-0.543361-0.405727-0.385467-0.324522
88.0-0.663103-0.429192-0.428918-0.22254-0.328334-0.524071
99.0-0.430653-0.819506-0.398711-0.344799-0.506967-0.501721
1010.0-0.512089-0.614823-0.574396-0.678601-0.541517-0.357891
...
150150.0-0.567046-0.4210280.2886920.503390.193940.328947

The data and the correlations look as follows:

Defining the model

This is a basic linear regression model similar to the one mentioned in the tutorials on https://turing.ml. The priors of this model are visualized below.

@model function linear_regression(X::Matrix, y)
    σ₂ ~ truncated(Normal(0, 100), 0, Inf)
    
    intercept ~ Normal(0, 0.4)

    n_features = size(X, 2)
    coef ~ MvNormal(n_features, 0.4)

    mu = intercept .+ X * coef
    y ~ MvNormal(mu, sqrt(σ₂))
end;
X = select(df, Not([:X, :Y]));	
model = let
    y = df.Y
    model = linear_regression(Matrix(X), y)
end;

Inspecting the prior

To verify that the priors are correctly set, we can use sample(model, Prior(), n_samples) from Turing.jl. This is shown below with the raw sample values on the left and the density plot for these values on the right.

n_samples = 1_000;

In this plot, everything looks good. On average, we expect our data to be zero (centered) and the variance looks reasonable. We expect the coefficients for the linear model to be between -0.5 and 0.5. Thanks to these priors, the sampler should have useful samples right from the start.

function mysample(model, sampler)
    n_chains = 3
    chns = sample(model, sampler, MCMCThreads(), n_samples, n_chains)
    return fix_names(chns)
end;

Estimating the parameters

When we fit the model, we have to decide on a sampler for this complex collinear case. NUTS is normally the best bet in Turing.jl, but let's first try HMC.

In the plots below, the different colors indicate different chains. All plots show good mixing and stationarity on the leftmost plots; the chains properly converged to the same outcome:

let
    chns = mysample(model, HMC(0.005, 10))
    plot_chain(chns)
end

Obtaining this outcome required setting the leapfrog size to a very low number. Normally, it is 0.05 or 0.1 which both did not work. What I mean by did not work is that the different chains did not converge, that is, gave different outcomes. Note that, thanks to the low leapfrog size, it took quite a few iterations for the chains to converge.

Let's try a more modern sampler, namely NUTS:

let
    chns = mysample(model, NUTS())
    plot_chain(chns)
end

Compared to the Shapley values, this result is very promising. Both samplers correctly identified the most important coefficient and give reasonable estimates for all the coefficients. In the Shapley post, the most informative feature got too much credit compared to a slightly less informative feature.

GLM

As a sanity check, let's see what a Frequentists linear model concludes.

fitted_lm = lm(@formula(Y ~ A + B + C + D + E), df);

These outcomes are almost the same as the Bayesian output.

Discussion

Hmm. That's not what I expected. The outcomes for both models are nearly identical!

After me retreating to the Julia forum, Michael Creel kindly offered an answer to this. It turned out that Bayesian priors can help with things like multicollinearity, but only if the prior provides enough information to counter the imprecision.

This seems to go back again to my post on Frequentist and Bayesian coin flipping, see the last two figures. When there is a strong prior or weak data, then the Bayesian outcome will differ from the Frequentist one (also mentioned by Gelman, 2020). Apparently, here the data was pretty strong or the priors pretty weak.

So, the Frequentist and Bayesian world aren't so different. In many cases, they yield the same conclusions.

Built with Julia 1.10.5 and

CairoMakie 0.11.11
CategoricalArrays 0.10.8
DataFrames 1.6.1
GLM 1.9.0
Statistics 1.10.0
Turing 0.30.9

To run this blog post locally, open this notebook with Pluto.jl.
Rik Huijzer. The text is licensed under CC BY-SA 4.0. The code is licensed under the MIT License. Website built with Franklin.jl and the Julia programming language. Last update: 2024-10-26.