Gaussian Processes: Introduction

JuliaGPs packages integrate well with Turing.jl because they implement the Distributions.jl interface. You should be able to understand what is going on in this tutorial if you know what a GP is. For a more in-depth understanding of the JuliaGPs functionality used here, please consult the JuliaGPs docs.

In this tutorial, we will model the putting dataset discussed in Chapter 21 of Bayesian Data Analysis. The dataset comprises the result of measuring how often a golfer successfully gets the ball in the hole, depending on how far away from it they are. The goal of inference is to estimate the probability of any given shot being successful at a given distance.

Let’s download the data and take a look at it:

using CSV, DataFrames

df = CSV.read("golf.dat", DataFrame; delim=' ', ignorerepeated=true)
df[1:5, :]
5×3 DataFrame
Row distance n y
Int64 Int64 Int64
1 2 1443 1346
2 3 694 577
3 4 455 337
4 5 353 208
5 6 272 149

We’ve printed the first 5 rows of the dataset (which comprises only 19 rows in total). Observe it has three columns:

  1. distance – how far away from the hole. I’ll refer to distance as d throughout the rest of this tutorial
  2. n – how many shots were taken from a given distance
  3. y – how many shots were successful from a given distance

We will use a Binomial model for the data, whose success probability is parametrised by a transformation of a GP. Something along the lines of: \[ \begin{aligned} f & \sim \operatorname{GP}(0, k) \\ y_j \mid f(d_j) & \sim \operatorname{Binomial}(n_j, g(f(d_j))) \\ g(x) & := \frac{1}{1 + e^{-x}} \end{aligned} \]

To do this, let’s define our Turing.jl model:

using AbstractGPs, LogExpFunctions, Turing

@model function putting_model(d, n; jitter=1e-4)
    v ~ Gamma(2, 1)
    l ~ Gamma(4, 1)
    f = GP(v * with_lengthscale(SEKernel(), l))
    f_latent ~ f(d, jitter)
    y ~ product_distribution(Binomial.(n, logistic.(f_latent)))
    return (fx=f(d, jitter), f_latent=f_latent, y=y)
end
putting_model (generic function with 2 methods)

We first define an AbstractGPs.GP, which represents a distribution over functions, and is entirely separate from Turing.jl. We place a prior over its variance v and length-scale l. f(d, jitter) constructs the multivariate Gaussian comprising the random variables in f whose indices are in d (plus a bit of independent Gaussian noise with variance jitter – see the docs for more details). f(d, jitter) has the type AbstractMvNormal, and is the bit of AbstractGPs.jl that implements the Distributions.jl interface, so it’s legal to put it on the right-hand side of a ~. From this you should deduce that f_latent is distributed according to a multivariate Gaussian. The remaining lines comprise standard Turing.jl code that is encountered in other tutorials and Turing documentation.

Before performing inference, we might want to inspect the prior that our model places over the data, to see whether there is anything obviously wrong. These kinds of prior predictive checks are straightforward to perform using Turing.jl, since it is possible to sample from the prior easily by just calling the model:

m = putting_model(Float64.(df.distance), df.n)
m().y
19-element Vector{Int64}:
 343
 214
 166
 133
  99
  84
  62
  61
  70
 106
 110
  94
  84
  69
  80
  79
  73
  69
  65

We make use of this to see what kinds of datasets we simulate from the prior:

using Plots

function plot_data(d, n, y, xticks, yticks)
    ylims = (0, round(maximum(n), RoundUp; sigdigits=2))
    margin = -0.5 * Plots.mm
    plt = plot(; xticks=xticks, yticks=yticks, ylims=ylims, margin=margin, grid=false)
    bar!(plt, d, n; color=:red, label="", alpha=0.5)
    bar!(plt, d, y; label="", color=:blue, alpha=0.7)
    return plt
end

# Construct model and run some prior predictive checks.
m = putting_model(Float64.(df.distance), df.n)
hists = map(1:20) do j
    xticks = j > 15 ? :auto : nothing
    yticks = rem(j, 5) == 1 ? :auto : nothing
    return plot_data(df.distance, df.n, m().y, xticks, yticks)
end
plot(hists...; layout=(4, 5))

In this case, the only prior knowledge I have is that the proportion of successful shots ought to decrease monotonically as the distance from the hole increases, which should show up in the data as the blue lines generally go down as we move from left to right on each graph. Unfortunately, there is not a simple way to enforce monotonicity in the samples from a GP, and we can see this in some of the plots above, so we must hope that we have enough data to ensure that this relationship holds approximately under the posterior. In any case, you can judge for yourself whether you think this is the most useful visualisation that we can perform – if you think there is something better to look at, please let us know!

Moving on, we generate samples from the posterior using the default NUTS sampler. We’ll make use of ReverseDiff.jl, as it has better performance than ForwardDiff.jl on this example. See Turing.jl’s docs on Automatic Differentiation for more info.

using Random, ReverseDiff

m_post = m | (y=df.y,)
chn = sample(Xoshiro(123456), m_post, NUTS(; adtype=AutoReverseDiff()), 1_000, progress=false)
Info: Found initial step size
  ϵ = 0.2
Chains MCMC chain (1000×33×1 Array{Float64, 3}):

Iterations        = 501:1:1500
Number of chains  = 1
Samples per chain = 1000
Wall duration     = 148.17 seconds
Compute duration  = 148.17 seconds
parameters        = v, l, f_latent[1], f_latent[2], f_latent[3], f_latent[4], f_latent[5], f_latent[6], f_latent[7], f_latent[8], f_latent[9], f_latent[10], f_latent[11], f_latent[12], f_latent[13], f_latent[14], f_latent[15], f_latent[16], f_latent[17], f_latent[18], f_latent[19]
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      rhat  ⋯
        Symbol   Float64   Float64   Float64     Float64    Float64   Float64  ⋯

             v    2.8740    1.2900    0.0418   1067.5214   787.6270    1.0003  ⋯
             l    3.6385    0.8280    0.0519    225.8803   286.8892    1.0033  ⋯
   f_latent[1]    2.5453    0.0979    0.0030   1077.4782   490.0008    1.0005  ⋯
   f_latent[2]    1.7002    0.0683    0.0019   1257.5024   704.7677    1.0013  ⋯
   f_latent[3]    0.9733    0.0794    0.0033    582.2903   566.8537    1.0026  ⋯
   f_latent[4]    0.4780    0.0762    0.0026    876.7129   525.4170    1.0013  ⋯
   f_latent[5]    0.1911    0.0734    0.0024    933.8486   716.6565    0.9998  ⋯
   f_latent[6]   -0.0141    0.0875    0.0039    521.0220   721.1299    0.9992  ⋯
   f_latent[7]   -0.2437    0.0828    0.0029    835.3462   702.7869    1.0007  ⋯
   f_latent[8]   -0.5026    0.0878    0.0028   1025.6213   535.9936    1.0047  ⋯
   f_latent[9]   -0.7226    0.0997    0.0037    732.5870   627.0753    1.0012  ⋯
  f_latent[10]   -0.8661    0.0966    0.0033    865.2909   675.3767    1.0007  ⋯
  f_latent[11]   -0.9529    0.0980    0.0031   1003.0110   611.8971    1.0004  ⋯
  f_latent[12]   -1.0450    0.1074    0.0043    639.0534   682.9616    0.9999  ⋯
  f_latent[13]   -1.1966    0.1136    0.0045    629.2198   743.7114    1.0000  ⋯
  f_latent[14]   -1.4113    0.1118    0.0037    928.4357   656.2373    1.0011  ⋯
  f_latent[15]   -1.6110    0.1204    0.0047    700.0815   491.5819    1.0041  ⋯
       ⋮            ⋮         ⋮         ⋮          ⋮          ⋮          ⋮     ⋱
                                                     1 column and 4 rows omitted

Quantiles
    parameters      2.5%     25.0%     50.0%     75.0%     97.5%
        Symbol   Float64   Float64   Float64   Float64   Float64

             v    1.0905    1.9373    2.6189    3.5243    5.9021
             l    2.4117    3.1094    3.4718    3.9832    5.8400
   f_latent[1]    2.3648    2.4759    2.5432    2.6132    2.7529
   f_latent[2]    1.5706    1.6537    1.6976    1.7457    1.8365
   f_latent[3]    0.8334    0.9190    0.9704    1.0227    1.1449
   f_latent[4]    0.3325    0.4269    0.4772    0.5279    0.6299
   f_latent[5]    0.0512    0.1416    0.1881    0.2420    0.3425
   f_latent[6]   -0.1823   -0.0692   -0.0160    0.0442    0.1433
   f_latent[7]   -0.4100   -0.2984   -0.2433   -0.1898   -0.0828
   f_latent[8]   -0.6755   -0.5618   -0.4982   -0.4444   -0.3421
   f_latent[9]   -0.9189   -0.7881   -0.7216   -0.6587   -0.5317
  f_latent[10]   -1.0635   -0.9291   -0.8627   -0.8002   -0.6758
  f_latent[11]   -1.1413   -1.0151   -0.9532   -0.8866   -0.7573
  f_latent[12]   -1.2495   -1.1169   -1.0456   -0.9743   -0.8328
  f_latent[13]   -1.4153   -1.2762   -1.1978   -1.1191   -0.9755
  f_latent[14]   -1.6249   -1.4912   -1.4057   -1.3325   -1.1948
  f_latent[15]   -1.8592   -1.6882   -1.6062   -1.5254   -1.3887
       ⋮            ⋮         ⋮         ⋮         ⋮         ⋮
                                                    4 rows omitted

We can use these samples and the posterior function from AbstractGPs to sample from the posterior probability of success at any distance we choose:

d_pred = 1:0.2:21
samples = map(returned(m_post, chn)[1:10:end]) do x
    return logistic.(rand(posterior(x.fx, x.f_latent)(d_pred, 1e-4)))
end
p = plot()
plot!(d_pred, reduce(hcat, samples); label="", color=:blue, alpha=0.2)
scatter!(df.distance, df.y ./ df.n; label="", color=:red)

We can see that the general trend is indeed down as the distance from the hole increases, and that if we move away from the data, the posterior uncertainty quickly inflates. This suggests that the model is probably going to do a reasonable job of interpolating between observed data, but less good a job at extrapolating to larger distances.

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