Increasing model accuracy by using foreknowledge

Typically, when making predictions via a linear model, we fit the model on our data and make predictions from the fitted model. However, this doesn't take much foreknowledge into account. For example, when predicting a person's length given only the weight and gender, we already have an intuition about the effect size and direction. Bayesian analysis should be able to incorporate this prior information.

In this blog post, I aim to figure out whether foreknowledge can, in theory, increase model accuracy. To do this, I generate data and fit a linear model and a Bayesian binary regression. Next, I compare the accuracy of the model parameters from the linear and Bayesian model.

using AlgebraOfGraphics
using CairoMakie
using CategoricalArrays
using DataFrames
using GLM
using MLDataUtils: rescale!
using Random: seed!
using Statistics
using StatsFuns
using Turing

We define the model as g_i = a_e * a_i + r_e * r_i + \epsilon_i = 1.1 * a_i + 1.05 * r_i + \epsilon_i where a_e is the coefficient for the age, r_e is a coefficient for the nationality and \epsilon_i is some random noise for individual i.

We generate data for n individuals via:

function generate_data(i::Int)
  seed!(i)

  n = 120
  I = 1:n
  P = [i % 2 == 0 for i in I]
  r_2(x) = round(x; digits=2)

  A = r_2.([p ? rand(Normal(aₑ * 18, 1)) : rand(Normal(18, 1)) for p in P])
  R = r_2.([p ? rand(Normal(rₑ * 6, 3)) : rand(Normal(6, 3)) for p in P])
  E = r_2.(rand(Normal(0, 1), n))
  G = aₑ .* A + rₑ .* R .+ E
  G = r_2.(G)

  df = DataFrame(age=A, recent=R, error=E, grade=G, pass=P)
end;
df = generate_data(1)
age recent error grade pass
17.93 7.49 0.48 28.07 false
20.33 1.19 -0.8 22.81 true
17.19 10.25 -1.16 28.51 false
22.26 5.51 -0.47 29.8 true
19.16 0.07 -2.05 19.1 false
20.07 10.87 -0.42 33.07 true
19.75 6.37 -1.43 26.98 false
18.97 4.25 -0.46 24.87 true
16.96 5.22 -0.51 23.63 false
19.47 -0.88 -0.61 19.88 true
... ... ... ... ...
120 21.3 3.17 -0.4 26.36

data

Linear regression

First, we fit a linear model and verify that the coefficients are estimated reasonably well. Here, the only prior information that we give the model is the structure of the data, that is, a formula.

linear_model = lm(@formula(grade ~ age + recent), df)
StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}, Matrix{Float64}}

grade ~ 1 + age + recent

Coefficients:
────────────────────────────────────────────────────────────────────────
                Coef.  Std. Error      t  Pr(>|t|)  Lower 95%  Upper 95%
────────────────────────────────────────────────────────────────────────
(Intercept)  0.693364   1.23564     0.56    0.5758  -1.75376     3.14049
age          1.05597    0.0638938  16.53    <1e-31   0.929434    1.18251
recent       1.05645    0.0267033  39.56    <1e-68   1.00357     1.10934
────────────────────────────────────────────────────────────────────────
r5(x) = round(x; digits=5);
coefa = coef(linear_model)[2] |> r5

1.05597

coefr = coef(linear_model)[3] |> r5

1.05645

Notice how these estimated coefficients are close to the coefficients that we set for age and recent, namely a_e = aₑ ≈ \text{coefa} and r_e = rₑ ≈ \text{coefr}, as expected.

Bayesian regression

For the Bayesian regression we fit a model via Turing.jl. Now, we give the model information about the structure of the data as well as priors for the size of the coefficients. For demonstration purposes, I've set the priors to the correct values. This is reasonable because I was wondering whether finding a good prior could have a positive effect on the model accuracy.

function rescale_data(df)
    out = DataFrame(df)
    rescale!(out, [:age, :recent, :grade])
    out
end;
rescaled = let
    rescaled = rescale_data(df)
    rescaled[!, :pass_num] = [p ? 1.0 : 0.0 for p in rescaled.pass]
end;
@model function bayesian_model(ages, recents, grades, n)
    intercept ~ Normal(0, 5)
    βₐ ~ Normal(aₑ, 1)
    βᵣ ~ Normal(rₑ, 3)
    σ ~ truncated(Cauchy(0, 2), 0, Inf)

    μ = intercept .+ βₐ * ages .+ βᵣ * recents
    grades ~ MvNormal(μ, σ)
end;
chns = let
    n = nrow(df)
    bm = bayesian_model(df.age, df.recent, df.grade, n)
    chns = Turing.sample(bm, NUTS(), MCMCThreads(), 10_000, 3)
end;

Let's plot the density for the coefficient estimates \beta_a and \beta_r:

density

and compare the outputs from both models:

coefficient true value linear estimate linear error bayesian estimate bayesian error
aₑ 1.1 1.05597 4.0 % 1.057836887599545 3.8 %
rₑ 1.05 1.05645 0.6 % 1.0569010822285407 0.7 %

Conclusion

After giving the true coefficients to the Bayesian model in the form of priors, it does score better than the linear model. However, the differences aren't very big. This could be due to the particular random noise in this sample E or due to the relatively big sample size. The more samples, the more likely it is that the data will overrule the prior. In any way, there are real-world situations where gathering extra data is more expensive than gathering priors via reading papers. In those cases, the increased accuracy introduced by using priors could have serious benefits.

Built with Julia 1.11.5 and

AlgebraOfGraphics 0.8.13
CairoMakie 0.12.16
CategoricalArrays 0.10.8
DataFrames 1.7.0
GLM 1.9.0
MLDataUtils 0.5.4
Random 1.11.0
Statistics 1.11.1
StatsFuns 1.3.2
Turing 0.35.2