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:
usingCSV, DataDeps, DataFramesENV["DATADEPS_ALWAYS_ACCEPT"] =trueregister(DataDep("putting","Putting data from BDA","http://www.stat.columbia.edu/~gelman/book/data/golf.dat","fc28d83896af7094d765789714524d5a389532279b64902866574079c1a977cc", ),)fname =joinpath(datadep"putting", "golf.dat")df = CSV.read(fname, 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:
distance – how far away from the hole. I’ll refer to distance as d throughout the rest of this tutorial
n – how many shots were taken from a given distance
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:
usingAbstractGPs, LogExpFunctions, Turing@modelfunctionputting_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
We make use of this to see what kinds of datasets we simulate from the prior:
usingPlotsfunctionplot_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 pltend# 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 :nothingreturnplot_data(df.distance, df.n, m().y, xticks, yticks)endplot(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.
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.