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.

Let's say that the data generation formula for the grade $g_i$ for some individual $i$, with age $a_i$ and recent grade $r_i$, is

$g_i = a_e * a_i + r_e * r_i + \epsilon_i = 1.1 * a_i + 1.05 * r_i + \epsilon_i$ 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

```
using DataFrames
using Distributions
using Random
function generate_data(i::Int)
Random.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)
first(df, 8)
```

age | recent | error | grade | pass |
---|---|---|---|---|

18.3 | 2.26 | -0.74 | 21.76 | false |

20.18 | 2.31 | 1.14 | 25.76 | true |

17.4 | 1.92 | -0.9 | 20.26 | false |

19.79 | 7.93 | 0.02 | 30.12 | true |

17.16 | -1.9 | -1.55 | 15.33 | false |

20.11 | 6.75 | 0.19 | 29.4 | true |

20.3 | 1.02 | 0.92 | 24.32 | false |

17.53 | -1.95 | 0.16 | 17.4 | true |

We can see the positive correlations of `age`

and `grade`

, and `recent`

and `grade`

.

as well as the differences in densities when splitting the individuals on pass or fail:

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.

```
using GLM
linear_model = lm(@formula(grade ~ age + recent), df)
```

```
StatsModels.TableRegressionModel{GLM.LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}}}}, Matrix{Float64}}
grade ~ 1 + age + recent
Coefficients:
───────────────────────────────────────────────────────────────────────
Coef. Std. Error t Pr(>|t|) Lower 95% Upper 95%
───────────────────────────────────────────────────────────────────────
(Intercept) 0.85718 1.44699 0.59 0.5547 -2.00851 3.72287
age 1.0436 0.0750508 13.91 <1e-25 0.894965 1.19223
recent 1.07894 0.0337224 31.99 <1e-58 1.01216 1.14573
───────────────────────────────────────────────────────────────────────
```

Notice how these estimated coefficients are close to the coefficients that we set for `age`

and `recent`

, namely $a_e = 1.1 \approx 1.0436$ and $r_e = 1.05 \approx 1.07894$, as expected.
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.

```
using Statistics
using StatsFuns: logistic
using Turing
@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
n = nrow(df)
bm = bayesian_model(df.age, df.recent, df.grade, n)
chns = Turing.sample(bm, NUTS(), MCMCThreads(), 10_000, 3)
```

Lets plot the density for the coefficient estimates $\beta_a$ and $\beta_r$.

coefficient | true value | linear estimate | linear error | bayesian estimate | bayesian error |
---|---|---|---|---|---|

aₑ | 1.1 | 1.044 | 5.1 % | 1.048 | 4.8 % |

rₑ | 1.05 | 1.079 | 2.8 % | 1.079 | 2.8 % |

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.

CC BY-SA 4.0 Rik Huijzer. Website built with Franklin.jl and the Julia programming language.
Last update: 2021-09-21.