Binary Classification with Laplace approximation

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This example demonstrates how to carry out non-conjugate Gaussian process inference using the Laplace approximation.

For a basic introduction to the functionality of this library, please refer to the User Guide.

Setup

using ApproximateGPs
using LinearAlgebra
using Distributions
using LogExpFunctions: logistic, softplus, invsoftplus
using Zygote
using Optim

using Plots
default(; legend=:outertopright, size=(700, 400))

using Random
Random.seed!(1);

Generate training data

We create a binary-labelled toy dataset:

Xgrid = -4:0.1:29  # for visualization
X = range(0, 23.5; length=48)  # training inputs
f(x) = 3 * sin(10 + 0.6x) + sin(0.1x) - 1  # latent function
fs = f.(X)  # latent function values at training inputs

lik = BernoulliLikelihood()  # has logistic invlink by default
# could use other invlink, e.g. normcdf(f) = cdf(Normal(), f)

invlink = lik.invlink  # logistic function
ps = invlink.(fs)  # probabilities at the training inputs
Y = [rand(Bernoulli(p)) for p in ps]  # observations at the training inputs
# could do this in one call as `Y = rand(lik(fs))`

function plot_data()
    plot(; xlims=extrema(Xgrid), xticks=0:6:24)
    plot!(Xgrid, invlink ∘ f; label="true probabilities")
    return scatter!(X, Y; label="observations", color=3)
end

plot_data()

Create a latent GP

Here we write a function that creates our latent GP prior, given the hyperparameter vector theta. Compared to a "vanilla" GP, the LatentGP requires a function or functor that maps from the latent GP f to the distribution of observations y. This functor is commonly called "the likelihood".

function build_latent_gp(theta)
    # `theta` is unconstrained, but kernel variance and lengthscale must be positive:
    variance = softplus(theta[1])
    lengthscale = softplus(theta[2])

    kernel = variance * with_lengthscale(SqExponentialKernel(), lengthscale)

    dist_y_given_f = BernoulliLikelihood()  # has logistic invlink by default
    # We could also be explicit and define it as a function:
    # dist_y_given_f(f) = Bernoulli(invlink(f))

    jitter = 1e-8  # required for numeric stability
    return LatentGP(GP(kernel), dist_y_given_f, jitter)
end;

We define a latent GP at our initial hyperparameter values, here with variance 1.0 and lengthscale 5.0:

theta0 = [invsoftplus(1.0), invsoftplus(5.0)]

lf = build_latent_gp(theta0)

lf.f.kernel

Squared Exponential Kernel (metric = Distances.Euclidean(0.0))
	- Scale Transform (s = 0.2)
	- σ² = 1.0

We can now compute the Laplace approximation $q(f)$ to the true posterior $p(f | y)$:

f_post = posterior(LaplaceApproximation(), lf(X), Y)

AbstractGPs.ApproxPosteriorGP{ApproximateGPs.LaplaceApproximationModule.LaplaceApproximation{NamedTuple{(), Tuple{}}}, AbstractGPs.FiniteGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}}, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}}, LinearAlgebra.Diagonal{Float64, FillArrays.Fill{Float64, 1, Tuple{Base.OneTo{Int64}}}}}, ApproximateGPs.LaplaceApproximationModule.LaplaceCache{Matrix{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Float64, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}}}(ApproximateGPs.LaplaceApproximationModule.LaplaceApproximation{NamedTuple{(), Tuple{}}}(NamedTuple()), AbstractGPs.FiniteGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}}, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}}, LinearAlgebra.Diagonal{Float64, FillArrays.Fill{Float64, 1, Tuple{Base.OneTo{Int64}}}}}(
f: AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}}(AbstractGPs.ZeroMean{Float64}(), Squared Exponential Kernel (metric = Distances.Euclidean(0.0))
	- Scale Transform (s = 0.2)
	- σ² = 1.0)
x: 0.0:0.5:23.5
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This finds a mode of the posterior (for the given values of the hyperparameters) using iterated Newton's method (i.e. solving an optimisation problem) and then constructs a Gaussian approximation to the posterior by matching the curvature at the mode.

Let's plot samples from this approximate posterior:

function plot_samples!(Xgrid, fpost; samples=100, color=2)
    fx = fpost(Xgrid, 1e-8)
    fsamples = rand(fx, samples)
    plot!(Xgrid, invlink.(fsamples); color, alpha=0.2, label="")
    return plot!(Xgrid, invlink.(mean(fx)); color, lw=2, label="posterior fit")
end

p1 = plot_data()
plot_samples!(Xgrid, f_post)

We can improve this fit by optimising the hyperparameters. For exact Gaussian process regression, the maximization objective is the marginal likelihood. Here, we can only optimise an approximation to the marginal likelihood.

Optimise the hyperparameters

ApproximateGPs provides a convenience function build_laplace_objective that constructs an objective function for optimising the hyperparameters, based on the Laplace approximation to the log marginal likelihood.

objective = build_laplace_objective(build_latent_gp, X, Y);

We pass this objective to Optim.jl's LBFGS optimiser:

training_results = Optim.optimize(
    objective, θ -> only(Zygote.gradient(objective, θ)), theta0, LBFGS(); inplace=false
)

 * Status: success

 * Candidate solution
    Final objective value:     2.566186e+01

 * Found with
    Algorithm:     L-BFGS

 * Convergence measures
    |x - x'|               = 4.87e-06 ≰ 0.0e+00
    |x - x'|/|x'|          = 6.32e-07 ≰ 0.0e+00
    |f(x) - f(x')|         = 3.96e-12 ≰ 0.0e+00
    |f(x) - f(x')|/|f(x')| = 1.54e-13 ≰ 0.0e+00
    |g(x)|                 = 1.64e-11 ≤ 1.0e-08

 * Work counters
    Seconds run:   1  (vs limit Inf)
    Iterations:    11
    f(x) calls:    33
    ∇f(x) calls:   33

Now that we have (hopefully) better hyperparameter values, we need to construct a LatentGP prior with these values:

lf2 = build_latent_gp(training_results.minimizer)

lf2.f.kernel

Squared Exponential Kernel (metric = Distances.Euclidean(0.0))
	- Scale Transform (s = 0.5826418867620505)
	- σ² = 7.709524987148875

Finally, we need to construct again the (approximate) posterior given the observations for the latent GP with optimised hyperparameters:

f_post2 = posterior(LaplaceApproximation(; f_init=objective.cache.f), lf2(X), Y)

AbstractGPs.ApproxPosteriorGP{ApproximateGPs.LaplaceApproximationModule.LaplaceApproximation{NamedTuple{(:f_init,), Tuple{Vector{Float64}}}}, AbstractGPs.FiniteGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}}, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}}, LinearAlgebra.Diagonal{Float64, FillArrays.Fill{Float64, 1, Tuple{Base.OneTo{Int64}}}}}, ApproximateGPs.LaplaceApproximationModule.LaplaceCache{Matrix{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Float64, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}}}(ApproximateGPs.LaplaceApproximationModule.LaplaceApproximation{NamedTuple{(:f_init,), Tuple{Vector{Float64}}}}((f_init = [-1.4948149503896233, -1.379917437143532, -1.3087408466992618, -1.3793046700302378, -1.6120322035012802, -1.915195932267416, -2.109782356239393, -2.007015138095168, -1.4988960866946233, -0.6132330938792461, 0.4944937036063641, 1.603869904420273, 2.5103556904668647, 3.0839984599440182, 3.285259042106893, 3.14378076203779, 2.7224235538375665, 2.0884168956977565, 1.3018730942128485, 0.419751848903356, -0.4930223606308328, -1.354501180227789, -2.066086065388864, -2.526242302338458, -2.6560689662689634, -2.4287395530450238, -1.890002547059782, -1.1569713818410292, -0.3889558458728249, 0.2639426275401154, 0.715577061168969, 0.9757286730173219, 1.1389672183630433, 1.3298501442737394, 1.6286554545375735, 2.01603285268741, 2.367672925811998, 2.504240621654916, 2.272697042117624, 1.619472623748461, 0.6212866053834422, -0.5387217947808043, -1.6356539897839626, -2.47706710021416, -2.955093332936643, -3.0599317994573236, -2.8586959992356684, -2.4573726226072923],)), AbstractGPs.FiniteGP{AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}}, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}}, LinearAlgebra.Diagonal{Float64, FillArrays.Fill{Float64, 1, Tuple{Base.OneTo{Int64}}}}}(
f: AbstractGPs.GP{AbstractGPs.ZeroMean{Float64}, KernelFunctions.ScaledKernel{KernelFunctions.TransformedKernel{KernelFunctions.SqExponentialKernel{Distances.Euclidean}, KernelFunctions.ScaleTransform{Float64}}, Float64}}(AbstractGPs.ZeroMean{Float64}(), Squared Exponential Kernel (metric = Distances.Euclidean(0.0))
	- Scale Transform (s = 0.5826418867620505)
	- σ² = 7.709524987148875)
x: 0.0:0.5:23.5
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, ApproximateGPs.LaplaceApproximationModule.LaplaceCache{Matrix{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Float64, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}}([7.709524997148875 7.389223312822344 6.505973738932493 5.262209846722348 3.909906416415531 2.6687439340972445 1.6733615689474703 0.9638623552521226 0.510014577689387 0.24790911979591565 0.11069927075027368 0.0454087303728121 0.017111035853884404 0.005923188940601928 0.0018835502626318784 0.0005502260962126537 0.00014765477548800472 3.63995616541034e-5 8.243035163494889e-6 1.7148277156310213e-6 3.2771489314633513e-7 5.753262712280263e-8 9.278431712071924e-9 1.3746030036252505e-9 1.8707783880561366e-10 2.3388897062689236e-11 2.6862070443820357e-12 2.834076491429122e-13 2.7467933948243166e-14 2.4455850109320436e-15 2.000238933210323e-16 1.5028766307887804e-17 1.0373065716405028e-18 6.577082204685343e-20 3.830908273828148e-21 2.0498051492306868e-22 1.007548008288949e-23 4.549474767956423e-25 1.8871182531789343e-26 7.1908348214937e-28 2.5171080180016327e-29 8.09406590178001e-31 2.3909690913046246e-32 6.488189570713545e-34 1.6173925633545106e-35 3.703818496000439e-37 7.791593616396778e-39 1.5057235150574925e-40; 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0.00014765477548800472 0.0005502260962126528 0.0018835502626318784 0.005923188940601928 0.017111035853884373 0.04540873037281206 0.1106992707502736 0.24790911979591576 0.5100145776893873 0.9638623552521217 1.673361568947467 2.668743934097245 3.909906416415528 5.262209846722353 6.5059737389324805 7.389223312822354 7.709524997148875 7.3892233128223275 6.5059737389324805 5.262209846722334 3.9099064164155144 2.668743934097245 1.673361568947476 0.9638623552521252 0.5100145776893854 0.24790911979591576 0.11069927075027379 0.04540873037281186 0.017111035853884387 0.00592318894060197 0.001883550262631885 0.0005502260962126576 0.000147654775488005 3.63995616541036e-5 8.243035163494845e-6 1.7148277156310151e-6 3.2771489314633397e-7 5.753262712280243e-8 9.278431712071924e-9 1.3746030036252505e-9 1.8707783880561167e-10 2.3388897062689068e-11 2.6862070443820737e-12 2.8340764914291523e-13 2.7467933948243557e-14 2.4455850109320436e-15 2.000238933210323e-16 1.50287663078877e-17; 3.63995616541034e-5 0.00014765477548800448 0.0005502260962126528 0.0018835502626318784 0.005923188940601917 0.017111035853884387 0.04540873037281202 0.11069927075027379 0.24790911979591534 0.5100145776893873 0.9638623552521217 1.67336156894747 2.668743934097245 3.909906416415528 5.262209846722353 6.5059737389324805 7.3892233128223275 7.709524997148875 7.389223312822354 6.505973738932504 5.262209846722353 3.909906416415542 2.668743934097245 1.673361568947464 0.9638623552521252 0.5100145776893854 0.24790911979591576 0.11069927075027379 0.04540873037281186 0.017111035853884387 0.005923188940601928 0.001883550262631885 0.0005502260962126498 0.000147654775488005 3.639956165410334e-5 8.243035163494904e-6 1.7148277156310395e-6 3.277148931463363e-7 5.753262712280243e-8 9.278431712071924e-9 1.3746030036252407e-9 1.8707783880561167e-10 2.33888970626894e-11 2.6862070443820547e-12 2.8340764914291523e-13 2.7467933948243557e-14 2.445585010932009e-15 2.000238933210323e-16; 8.243035163494889e-6 3.639956165410347e-5 0.000147654775488005 0.0005502260962126537 0.0018835502626318784 0.005923188940601933 0.017111035853884387 0.04540873037281218 0.1106992707502736 0.24790911979591534 0.5100145776893863 0.9638623552521217 1.67336156894747 2.668743934097245 3.909906416415528 5.262209846722353 6.5059737389324805 7.389223312822354 7.709524997148875 7.3892233128223275 6.5059737389324575 5.262209846722353 3.909906416415528 2.668743934097245 1.673361568947464 0.9638623552521252 0.5100145776893854 0.24790911979591398 0.11069927075027379 0.04540873037281218 0.017111035853884387 0.00592318894060197 0.0018835502626318719 0.0005502260962126537 0.000147654775488005 3.639956165410308e-5 8.243035163494845e-6 1.7148277156310151e-6 3.277148931463363e-7 5.753262712280243e-8 9.278431712071792e-9 1.3746030036252407e-9 1.8707783880561565e-10 2.3388897062689236e-11 2.6862070443820357e-12 2.8340764914291523e-13 2.7467933948243166e-14 2.445585010932009e-15; 1.7148277156310213e-6 8.243035163494873e-6 3.639956165410347e-5 0.000147654775488005 0.0005502260962126537 0.0018835502626318818 0.005923188940601928 0.017111035853884387 0.0454087303728121 0.11069927075027379 0.24790911979591576 0.5100145776893873 0.9638623552521217 1.67336156894747 2.668743934097245 3.909906416415528 5.262209846722334 6.505973738932504 7.3892233128223275 7.709524997148875 7.3892233128223275 6.5059737389324805 5.262209846722353 3.909906416415528 2.668743934097245 1.673361568947476 0.9638623552521252 0.5100145776893854 0.24790911979591576 0.11069927075027457 0.04540873037281218 0.017111035853884512 0.005923188940601886 0.001883550262631885 0.0005502260962126498 0.00014765477548800396 3.639956165410334e-5 8.243035163494904e-6 1.7148277156310272e-6 3.277148931463363e-7 5.753262712280202e-8 9.278431712071858e-9 1.3746030036252604e-9 1.8707783880561565e-10 2.3388897062689565e-11 2.6862070443820737e-12 2.8340764914291523e-13 2.7467933948243557e-14; 3.2771489314633513e-7 1.7148277156310183e-6 8.243035163494873e-6 3.639956165410334e-5 0.00014765477548800448 0.0005502260962126528 0.0018835502626318751 0.005923188940601917 0.01711103585388436 0.04540873037281194 0.11069927075027379 0.24790911979591485 0.5100145776893854 0.9638623552521217 1.67336156894747 2.668743934097235 3.9099064164155144 5.262209846722353 6.5059737389324575 7.3892233128223275 7.709524997148875 7.3892233128223275 6.5059737389324805 5.262209846722316 3.909906416415528 2.668743934097245 1.673361568947476 0.9638623552521183 0.510014577689389 0.24790911979591754 0.11069927075027379 0.0454087303728125 0.01711103585388427 0.005923188940601928 0.0018835502626318719 0.0005502260962126498 0.00014765477548800396 3.63995616541036e-5 8.243035163494904e-6 1.7148277156310272e-6 3.277148931463363e-7 5.753262712280202e-8 9.27843171207199e-9 1.3746030036252604e-9 1.8707783880561565e-10 2.3388897062689236e-11 2.6862070443820357e-12 2.8340764914291523e-13; 5.753262712280263e-8 3.277148931463375e-7 1.7148277156310272e-6 8.243035163494904e-6 3.639956165410354e-5 0.00014765477548800527 0.0005502260962126546 0.001883550262631885 0.005923188940601949 0.01711103585388445 0.04540873037281218 0.11069927075027379 0.24790911979591576 0.510014577689389 0.9638623552521252 1.67336156894747 2.668743934097245 3.909906416415542 5.262209846722353 6.5059737389324805 7.3892233128223275 7.709524997148875 7.3892233128223275 6.5059737389324805 5.262209846722353 3.9099064164155557 2.6687439340972636 1.673361568947464 0.9638623552521252 0.510014577689389 0.24790911979591576 0.11069927075027457 0.04540873037281186 0.017111035853884512 0.005923188940601886 0.0018835502626318719 0.0005502260962126537 0.000147654775488005 3.639956165410334e-5 8.243035163494904e-6 1.7148277156310028e-6 3.277148931463363e-7 5.7532627122802837e-8 9.27843171207199e-9 1.3746030036252604e-9 1.87077838805613e-10 2.3388897062689236e-11 2.6862070443820357e-12; 9.278431712071924e-9 5.753262712280263e-8 3.277148931463375e-7 1.714827715631024e-6 8.243035163494889e-6 3.639956165410354e-5 0.000147654775488005 0.0005502260962126546 0.001883550262631885 0.005923188940601928 0.01711103585388445 0.04540873037281218 0.11069927075027379 0.24790911979591576 0.510014577689389 0.9638623552521217 1.673361568947476 2.668743934097245 3.909906416415528 5.262209846722353 6.5059737389324805 7.3892233128223275 7.709524997148875 7.3892233128223275 6.505973738932527 5.262209846722353 3.909906416415528 2.668743934097245 1.673361568947476 0.9638623552521252 0.5100145776893854 0.24790911979591576 0.11069927075027379 0.04540873037281186 0.017111035853884387 0.005923188940601928 0.0018835502626318719 0.0005502260962126537 0.000147654775488005 3.6399561654103856e-5 8.243035163494785e-6 1.7148277156310028e-6 3.277148931463363e-7 5.753262712280202e-8 9.27843171207199e-9 1.3746030036252407e-9 1.87077838805613e-10 2.3388897062689565e-11; 1.3746030036252505e-9 9.278431712071924e-9 5.753262712280243e-8 3.277148931463363e-7 1.7148277156310183e-6 8.243035163494873e-6 3.63995616541034e-5 0.00014765477548800472 0.0005502260962126537 0.0018835502626318784 0.005923188940601928 0.017111035853884387 0.04540873037281202 0.11069927075027379 0.24790911979591576 0.5100145776893854 0.9638623552521252 1.673361568947464 2.668743934097245 3.909906416415528 5.262209846722316 6.5059737389324805 7.3892233128223275 7.709524997148875 7.38922331282238 6.5059737389324805 5.262209846722353 3.909906416415528 2.668743934097245 1.673361568947476 0.9638623552521183 0.510014577689389 0.24790911979591576 0.11069927075027379 0.04540873037281186 0.017111035853884387 0.005923188940601928 0.0018835502626318985 0.0005502260962126537 0.000147654775488005 3.6399561654102826e-5 8.243035163494785e-6 1.7148277156310272e-6 3.277148931463363e-7 5.7532627122802837e-8 9.278431712071858e-9 1.3746030036252604e-9 1.8707783880561565e-10; 1.8707783880561366e-10 1.3746030036252556e-9 9.278431712071924e-9 5.753262712280263e-8 3.277148931463363e-7 1.714827715631024e-6 8.243035163494889e-6 3.639956165410347e-5 0.000147654775488005 0.0005502260962126537 0.001883550262631885 0.005923188940601928 0.017111035853884387 0.04540873037281218 0.11069927075027379 0.24790911979591398 0.5100145776893854 0.9638623552521252 1.673361568947464 2.668743934097245 3.909906416415528 5.262209846722353 6.505973738932527 7.38922331282238 7.709524997148875 7.3892233128223275 6.5059737389324805 5.262209846722316 3.9099064164155 2.668743934097245 1.6733615689474521 0.9638623552521183 0.5100145776893854 0.24790911979591576 0.11069927075027379 0.04540873037281186 0.017111035853884387 0.005923188940601928 0.0018835502626318719 0.0005502260962126458 0.0001476547754880029 3.6399561654102826e-5 8.243035163494904e-6 1.7148277156310028e-6 3.277148931463363e-7 5.753262712280202e-8 9.278431712071858e-9 1.3746030036252407e-9; 2.3388897062689236e-11 1.8707783880561366e-10 1.3746030036252556e-9 9.278431712071924e-9 5.753262712280243e-8 3.277148931463363e-7 1.7148277156310213e-6 8.243035163494889e-6 3.639956165410347e-5 0.000147654775488005 0.0005502260962126537 0.001883550262631885 0.005923188940601928 0.017111035853884387 0.04540873037281202 0.11069927075027379 0.24790911979591576 0.5100145776893854 0.9638623552521252 1.673361568947476 2.668743934097245 3.9099064164155557 5.262209846722353 6.5059737389324805 7.3892233128223275 7.709524997148875 7.3892233128223275 6.5059737389324805 5.262209846722353 3.909906416415528 2.668743934097245 1.673361568947476 0.9638623552521114 0.5100145776893854 0.24790911979591576 0.11069927075027536 0.04540873037281186 0.017111035853884387 0.005923188940601928 0.0018835502626318719 0.0005502260962126458 0.0001476547754880029 3.6399561654103856e-5 8.243035163494904e-6 1.7148277156310272e-6 3.277148931463363e-7 5.753262712280202e-8 9.278431712071858e-9; 2.6862070443820357e-12 2.3388897062689236e-11 1.87077838805613e-10 1.3746030036252505e-9 9.278431712071891e-9 5.753262712280243e-8 3.277148931463363e-7 1.7148277156310183e-6 8.243035163494873e-6 3.639956165410334e-5 0.000147654775488005 0.0005502260962126537 0.0018835502626318784 0.005923188940601928 0.017111035853884328 0.04540873037281186 0.11069927075027379 0.24790911979591576 0.5100145776893854 0.9638623552521252 1.673361568947476 2.6687439340972636 3.909906416415528 5.262209846722353 6.5059737389324805 7.3892233128223275 7.709524997148875 7.3892233128223275 6.5059737389324805 5.262209846722353 3.9099064164155 2.668743934097245 1.673361568947476 0.9638623552521252 0.5100145776893854 0.24790911979591576 0.11069927075027379 0.04540873037281186 0.017111035853884387 0.005923188940601928 0.0018835502626318719 0.0005502260962126537 0.0001476547754880071 3.639956165410334e-5 8.243035163494904e-6 1.7148277156310272e-6 3.277148931463363e-7 5.7532627122802837e-8; 2.834076491429122e-13 2.6862070443820357e-12 2.3388897062689152e-11 1.8707783880561234e-10 1.3746030036252457e-9 9.278431712071858e-9 5.753262712280202e-8 3.2771489314633513e-7 1.7148277156310151e-6 8.243035163494816e-6 3.639956165410321e-5 0.00014765477548800396 0.0005502260962126517 0.0018835502626318719 0.005923188940601928 0.01711103585388427 0.04540873037281186 0.11069927075027379 0.24790911979591398 0.5100145776893854 0.9638623552521183 1.673361568947464 2.668743934097245 3.909906416415528 5.262209846722316 6.5059737389324805 7.3892233128223275 7.709524997148875 7.3892233128223275 6.505973738932434 5.262209846722316 3.9099064164155 2.668743934097245 1.673361568947476 0.9638623552521252 0.5100145776893926 0.24790911979591576 0.11069927075027379 0.04540873037281186 0.017111035853884387 0.005923188940601844 0.0018835502626318719 0.0005502260962126537 0.000147654775488005 3.639956165410334e-5 8.243035163494904e-6 1.7148277156310028e-6 3.277148931463363e-7; 2.7467933948243166e-14 2.834076491429112e-13 2.6862070443820167e-12 2.3388897062689068e-11 1.8707783880561234e-10 1.3746030036252505e-9 9.278431712071891e-9 5.753262712280243e-8 3.2771489314633397e-7 1.7148277156310151e-6 8.243035163494873e-6 3.639956165410347e-5 0.00014765477548800396 0.0005502260962126517 0.0018835502626318719 0.005923188940601886 0.017111035853884387 0.04540873037281186 0.11069927075027379 0.24790911979591576 0.510014577689389 0.9638623552521252 1.673361568947476 2.668743934097245 3.9099064164155 5.262209846722353 6.5059737389324805 7.3892233128223275 7.709524997148875 7.3892233128223275 6.505973738932434 5.26220984672239 3.9099064164155 2.668743934097245 1.673361568947476 0.9638623552521388 0.5100145776893854 0.24790911979591576 0.11069927075027379 0.04540873037281186 0.017111035853884387 0.005923188940601928 0.0018835502626318985 0.0005502260962126537 0.0001476547754880071 3.6399561654103856e-5 8.243035163494904e-6 1.7148277156310272e-6; 2.4455850109320436e-15 2.746793394824336e-14 2.8340764914291523e-13 2.6862070443820547e-12 2.33888970626894e-11 1.8707783880561433e-10 1.3746030036252654e-9 9.27843171207199e-9 5.753262712280263e-8 3.277148931463375e-7 1.7148277156310333e-6 8.243035163494933e-6 3.63995616541036e-5 0.0001476547754880055 0.0005502260962126537 0.001883550262631885 0.00592318894060197 0.017111035853884387 0.04540873037281218 0.11069927075027457 0.24790911979591754 0.510014577689389 0.9638623552521252 1.673361568947476 2.668743934097245 3.909906416415528 5.262209846722353 6.505973738932434 7.3892233128223275 7.709524997148875 7.3892233128223275 6.505973738932527 5.26220984672239 3.9099064164155557 2.6687439340972072 1.6733615689474521 0.9638623552521252 0.5100145776893854 0.24790911979591576 0.11069927075027379 0.04540873037281186 0.017111035853884387 0.005923188940601928 0.0018835502626318719 0.0005502260962126537 0.000147654775488005 3.639956165410334e-5 8.243035163494904e-6; 2.000238933210323e-16 2.4455850109320262e-15 2.746793394824336e-14 2.8340764914291326e-13 2.6862070443820357e-12 2.3388897062689236e-11 1.8707783880561366e-10 1.3746030036252556e-9 9.278431712071891e-9 5.753262712280243e-8 3.277148931463363e-7 1.7148277156310213e-6 8.243035163494845e-6 3.639956165410334e-5 0.00014765477548800396 0.0005502260962126537 0.001883550262631885 0.005923188940601928 0.017111035853884387 0.04540873037281218 0.11069927075027379 0.24790911979591576 0.5100145776893854 0.9638623552521183 1.6733615689474521 2.668743934097245 3.9099064164155 5.262209846722316 6.505973738932434 7.3892233128223275 7.709524997148875 7.3892233128223275 6.505973738932527 5.262209846722316 3.9099064164155 2.6687439340972072 1.673361568947476 0.9638623552521114 0.5100145776893854 0.24790911979591576 0.11069927075027221 0.04540873037281186 0.017111035853884387 0.005923188940601928 0.0018835502626318719 0.0005502260962126537 0.0001476547754880029 3.639956165410334e-5; 1.5028766307887804e-17 2.000238933210323e-16 2.4455850109320262e-15 2.746793394824336e-14 2.8340764914291523e-13 2.6862070443820547e-12 2.33888970626894e-11 1.87077838805615e-10 1.3746030036252556e-9 9.278431712071956e-9 5.7532627122802837e-8 3.277148931463375e-7 1.7148277156310213e-6 8.243035163494904e-6 3.63995616541036e-5 0.000147654775488005 0.0005502260962126576 0.001883550262631885 0.00592318894060197 0.017111035853884512 0.0454087303728125 0.11069927075027457 0.24790911979591576 0.510014577689389 0.9638623552521183 1.673361568947476 2.668743934097245 3.9099064164155 5.26220984672239 6.505973738932527 7.3892233128223275 7.709524997148875 7.3892233128223275 6.505973738932527 5.26220984672239 3.9099064164155557 2.668743934097245 1.673361568947476 0.9638623552521252 0.5100145776893854 0.24790911979591576 0.11069927075027379 0.0454087303728125 0.017111035853884387 0.005923188940601928 0.0018835502626318985 0.0005502260962126458 0.0001476547754880029; 1.0373065716405028e-18 1.50287663078877e-17 2.0002389332103086e-16 2.445585010932009e-15 2.7467933948243166e-14 2.8340764914291326e-13 2.6862070443820357e-12 2.3388897062689236e-11 1.8707783880561234e-10 1.3746030036252457e-9 9.278431712071891e-9 5.753262712280243e-8 3.2771489314633397e-7 1.7148277156310151e-6 8.243035163494904e-6 3.639956165410334e-5 0.000147654775488005 0.0005502260962126498 0.0018835502626318719 0.005923188940601886 0.01711103585388427 0.04540873037281186 0.11069927075027379 0.24790911979591576 0.5100145776893854 0.9638623552521114 1.673361568947476 2.668743934097245 3.9099064164155 5.26220984672239 6.505973738932527 7.3892233128223275 7.709524997148875 7.3892233128223275 6.505973738932434 5.262209846722316 3.9099064164155 2.668743934097245 1.6733615689474521 0.9638623552521252 0.5100145776893782 0.24790911979591224 0.11069927075027379 0.04540873037281186 0.017111035853884633 0.005923188940601928 0.0018835502626318985 0.0005502260962126537; 6.577082204685343e-20 1.0373065716405028e-18 1.50287663078877e-17 2.000238933210323e-16 2.4455850109320262e-15 2.7467933948243166e-14 2.834076491429142e-13 2.686207044382045e-12 2.3388897062689152e-11 1.87077838805613e-10 1.3746030036252505e-9 9.278431712071924e-9 5.753262712280243e-8 3.277148931463363e-7 1.7148277156310151e-6 8.243035163494845e-6 3.63995616541036e-5 0.000147654775488005 0.0005502260962126537 0.001883550262631885 0.005923188940601928 0.017111035853884512 0.04540873037281186 0.11069927075027379 0.24790911979591576 0.5100145776893854 0.9638623552521252 1.673361568947476 2.668743934097245 3.9099064164155557 5.262209846722316 6.505973738932527 7.3892233128223275 7.709524997148875 7.389223312822433 6.505973738932527 5.262209846722316 3.9099064164155557 2.668743934097245 1.673361568947476 0.9638623552521114 0.5100145776893854 0.24790911979591929 0.11069927075027379 0.04540873037281186 0.017111035853884387 0.005923188940601928 0.001883550262631845; 3.830908273828148e-21 6.577082204685296e-20 1.0373065716404955e-18 1.50287663078877e-17 2.0002389332103086e-16 2.445585010932009e-15 2.7467933948243166e-14 2.8340764914291326e-13 2.6862070443820167e-12 2.3388897062689152e-11 1.87077838805613e-10 1.3746030036252505e-9 9.278431712071858e-9 5.753262712280243e-8 3.277148931463363e-7 1.7148277156310151e-6 8.243035163494845e-6 3.639956165410334e-5 0.000147654775488005 0.0005502260962126498 0.0018835502626318719 0.005923188940601886 0.017111035853884387 0.04540873037281186 0.11069927075027379 0.24790911979591576 0.5100145776893854 0.9638623552521252 1.673361568947476 2.6687439340972072 3.9099064164155 5.26220984672239 6.505973738932434 7.389223312822433 7.709524997148875 7.3892233128223275 6.505973738932527 5.262209846722316 3.9099064164155557 2.668743934097245 1.673361568947476 0.9638623552521252 0.5100145776893926 0.24790911979591929 0.11069927075027379 0.04540873037281315 0.017111035853884144 0.005923188940601928; 2.0498051492306868e-22 3.830908273828148e-21 6.577082204685296e-20 1.0373065716405028e-18 1.502876630788759e-17 2.0002389332103086e-16 2.445585010932009e-15 2.746793394824336e-14 2.834076491429112e-13 2.686207044382026e-12 2.3388897062689152e-11 1.87077838805613e-10 1.3746030036252407e-9 9.278431712071924e-9 5.7532627122802837e-8 3.2771489314633397e-7 1.7148277156310151e-6 8.243035163494904e-6 3.639956165410308e-5 0.00014765477548800396 0.0005502260962126498 0.0018835502626318719 0.005923188940601928 0.017111035853884387 0.04540873037281186 0.11069927075027536 0.24790911979591576 0.5100145776893926 0.9638623552521388 1.6733615689474521 2.6687439340972072 3.9099064164155557 5.262209846722316 6.505973738932527 7.3892233128223275 7.709524997148875 7.3892233128223275 6.505973738932434 5.26220984672239 3.9099064164155 2.668743934097245 1.673361568947476 0.9638623552521388 0.5100145776893926 0.24790911979591576 0.11069927075027693 0.04540873037281186 0.017111035853884633; 1.007548008288949e-23 2.0498051492307012e-22 3.830908273828176e-21 6.577082204685389e-20 1.0373065716405103e-18 1.5028766307887804e-17 2.000238933210323e-16 2.4455850109320436e-15 2.7467933948243166e-14 2.834076491429142e-13 2.686207044382045e-12 2.3388897062689236e-11 1.87077838805613e-10 1.3746030036252505e-9 9.27843171207199e-9 5.753262712280202e-8 3.2771489314633397e-7 1.7148277156310395e-6 8.243035163494845e-6 3.639956165410334e-5 0.00014765477548800396 0.0005502260962126537 0.0018835502626318719 0.005923188940601928 0.017111035853884387 0.04540873037281186 0.11069927075027379 0.24790911979591576 0.5100145776893854 0.9638623552521252 1.673361568947476 2.668743934097245 3.9099064164155 5.262209846722316 6.505973738932527 7.3892233128223275 7.709524997148875 7.3892233128223275 6.505973738932434 5.26220984672239 3.9099064164154442 2.6687439340972072 1.673361568947476 0.9638623552521114 0.5100145776893854 0.24790911979591576 0.11069927075027379 0.04540873037281186; 4.549474767956423e-25 1.0075480082889418e-23 2.0498051492307012e-22 3.830908273828176e-21 6.577082204685343e-20 1.0373065716405103e-18 1.5028766307887804e-17 2.000238933210323e-16 2.445585010932009e-15 2.7467933948243166e-14 2.834076491429142e-13 2.6862070443820357e-12 2.3388897062689236e-11 1.87077838805613e-10 1.3746030036252505e-9 9.278431712071858e-9 5.753262712280243e-8 3.277148931463363e-7 1.7148277156310151e-6 8.243035163494904e-6 3.63995616541036e-5 0.000147654775488005 0.0005502260962126537 0.0018835502626318985 0.005923188940601928 0.017111035853884387 0.04540873037281186 0.11069927075027379 0.24790911979591576 0.5100145776893854 0.9638623552521114 1.673361568947476 2.668743934097245 3.9099064164155557 5.262209846722316 6.505973738932434 7.3892233128223275 7.709524997148875 7.3892233128223275 6.505973738932434 5.262209846722316 3.9099064164155 2.6687439340972827 1.6733615689474284 0.9638623552521114 0.5100145776893854 0.24790911979591576 0.11069927075027379; 1.8871182531789343e-26 4.549474767956423e-25 1.0075480082889418e-23 2.0498051492307012e-22 3.830908273828176e-21 6.577082204685343e-20 1.0373065716405103e-18 1.5028766307887804e-17 2.000238933210323e-16 2.4455850109320262e-15 2.746793394824336e-14 2.8340764914291326e-13 2.6862070443820357e-12 2.3388897062689236e-11 1.87077838805613e-10 1.3746030036252505e-9 9.278431712071924e-9 5.753262712280243e-8 3.277148931463363e-7 1.7148277156310272e-6 8.243035163494904e-6 3.639956165410334e-5 0.000147654775488005 0.0005502260962126537 0.0018835502626318719 0.005923188940601928 0.017111035853884387 0.04540873037281186 0.11069927075027379 0.24790911979591576 0.5100145776893854 0.9638623552521252 1.6733615689474521 2.668743934097245 3.9099064164155557 5.26220984672239 6.505973738932434 7.3892233128223275 7.709524997148875 7.3892233128223275 6.505973738932434 5.262209846722316 3.9099064164155 2.6687439340972827 1.673361568947476 0.9638623552521388 0.5100145776893854 0.24790911979591576; 7.1908348214937e-28 1.8871182531789208e-26 4.549474767956423e-25 1.0075480082889348e-23 2.0498051492307012e-22 3.830908273828176e-21 6.577082204685343e-20 1.0373065716405103e-18 1.502876630788759e-17 2.000238933210323e-16 2.4455850109320262e-15 2.7467933948243166e-14 2.834076491429112e-13 2.6862070443820357e-12 2.3388897062689068e-11 1.87077838805613e-10 1.3746030036252505e-9 9.278431712071924e-9 5.753262712280243e-8 3.277148931463363e-7 1.7148277156310272e-6 8.243035163494904e-6 3.6399561654103856e-5 0.000147654775488005 0.0005502260962126458 0.0018835502626318719 0.005923188940601928 0.017111035853884387 0.04540873037281186 0.11069927075027379 0.24790911979591576 0.5100145776893854 0.9638623552521252 1.673361568947476 2.668743934097245 3.9099064164155 5.26220984672239 6.505973738932434 7.3892233128223275 7.709524997148875 7.389223312822433 6.505973738932434 5.262209846722465 3.9099064164155 2.6687439340972827 1.673361568947476 0.9638623552521114 0.5100145776893854; 2.5171080180016327e-29 7.190834821493598e-28 1.8871182531789208e-26 4.549474767956325e-25 1.0075480082889418e-23 2.0498051492306718e-22 3.830908273828122e-21 6.577082204685343e-20 1.0373065716404955e-18 1.502876630788759e-17 2.0002389332103086e-16 2.445585010931991e-15 2.7467933948242973e-14 2.834076491429112e-13 2.6862070443820167e-12 2.3388897062688903e-11 1.8707783880561167e-10 1.3746030036252407e-9 9.278431712071792e-9 5.753262712280202e-8 3.277148931463363e-7 1.7148277156310028e-6 8.243035163494785e-6 3.6399561654102826e-5 0.0001476547754880029 0.0005502260962126458 0.0018835502626318719 0.005923188940601844 0.017111035853884387 0.04540873037281186 0.11069927075027221 0.24790911979591576 0.5100145776893782 0.9638623552521114 1.673361568947476 2.668743934097245 3.9099064164154442 5.262209846722316 6.505973738932434 7.389223312822433 7.709524997148875 7.389223312822223 6.505973738932434 5.262209846722316 3.9099064164155 2.6687439340972072 1.6733615689474284 0.9638623552521114; 8.09406590178001e-31 2.5171080180016327e-29 7.1908348214937e-28 1.887118253178894e-26 4.5494747679563895e-25 1.0075480082889348e-23 2.0498051492306718e-22 3.830908273828148e-21 6.57708220468525e-20 1.0373065716404955e-18 1.502876630788759e-17 2.0002389332102943e-16 2.445585010931991e-15 2.7467933948243166e-14 2.834076491429112e-13 2.6862070443820167e-12 2.3388897062689068e-11 1.8707783880561167e-10 1.3746030036252407e-9 9.278431712071858e-9 5.753262712280202e-8 3.277148931463363e-7 1.7148277156310028e-6 8.243035163494785e-6 3.6399561654102826e-5 0.0001476547754880029 0.0005502260962126537 0.0018835502626318719 0.005923188940601928 0.017111035853884387 0.04540873037281186 0.11069927075027379 0.24790911979591224 0.5100145776893854 0.9638623552521252 1.673361568947476 2.6687439340972072 3.9099064164155 5.262209846722316 6.505973738932434 7.389223312822223 7.709524997148875 7.389223312822433 6.505973738932434 5.262209846722465 3.9099064164155 2.6687439340972827 1.6733615689475234; 2.3909690913046246e-32 8.094065901780242e-31 2.5171080180017044e-29 7.190834821493801e-28 1.8871182531789613e-26 4.549474767956487e-25 1.0075480082889561e-23 2.0498051492307303e-22 3.830908273828204e-21 6.577082204685437e-20 1.0373065716405177e-18 1.5028766307887804e-17 2.0002389332103372e-16 2.445585010932061e-15 2.7467933948243557e-14 2.8340764914291523e-13 2.6862070443820737e-12 2.33888970626894e-11 1.8707783880561565e-10 1.3746030036252604e-9 9.27843171207199e-9 5.7532627122802837e-8 3.277148931463363e-7 1.7148277156310272e-6 8.243035163494904e-6 3.6399561654103856e-5 0.0001476547754880071 0.0005502260962126537 0.0018835502626318985 0.005923188940601928 0.017111035853884387 0.0454087303728125 0.11069927075027379 0.24790911979591929 0.5100145776893926 0.9638623552521388 1.673361568947476 2.6687439340972827 3.9099064164155 5.262209846722465 6.505973738932434 7.389223312822433 7.709524997148875 7.389223312822433 6.505973738932434 5.262209846722316 3.9099064164155 2.6687439340972072; 6.488189570713545e-34 2.3909690913045904e-32 8.094065901780126e-31 2.5171080180016689e-29 7.190834821493801e-28 1.8871182531789343e-26 4.549474767956423e-25 1.0075480082889561e-23 2.0498051492307012e-22 3.830908273828176e-21 6.577082204685389e-20 1.0373065716405028e-18 1.50287663078877e-17 2.000238933210323e-16 2.4455850109320262e-15 2.746793394824336e-14 2.8340764914291523e-13 2.6862070443820547e-12 2.3388897062689236e-11 1.8707783880561565e-10 1.3746030036252604e-9 9.27843171207199e-9 5.753262712280202e-8 3.277148931463363e-7 1.7148277156310028e-6 8.243035163494904e-6 3.639956165410334e-5 0.000147654775488005 0.0005502260962126537 0.0018835502626318719 0.005923188940601928 0.017111035853884387 0.04540873037281186 0.11069927075027379 0.24790911979591929 0.5100145776893926 0.9638623552521114 1.6733615689474284 2.6687439340972827 3.9099064164155 5.262209846722316 6.505973738932434 7.389223312822433 7.709524997148875 7.389223312822433 6.505973738932434 5.262209846722465 3.909906416415611; 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3.2450077675538476e-7 1.6160362429834667e-6 7.286563956177326e-6 2.8952828170082608e-5 0.0001002244541490421 0.00030824094536213215 0.000899888455718359 0.0027050228814244 0.008581684900978891 0.025851548188687255 0.05882957201395027 0.09315281543539929 0.12429420968250374 0.16747267254210804 0.24346544058749586 0.3807531658058674 0.6192053906147487 0.9967166809961703 1.482110615109364 1.327989819562123 0.6295768293945043 0.5149090512290838 0.355665333596151 0.23524013141152983 0.16108223050848267 0.11694807655053932 0.0869412123349731 0.06024685031892815 0.03474406862058482 0.01609924012583959 0.006405442173813243 0.0023394358510745936 0.0007927310086976271 0.00024310178336740368 6.558474987840219e-5 1.5503489695504796e-5 3.3594768017220145e-6 7.258008833199098e-7 1.6778205425068522e-7 4.1265012233839456e-8 9.36908611977588e-9 1.5389812351407258e-9 1.756675057673641e-10 1.7397844691841565e-11 1.77348238860382e-12 1.9512078758326055e-13 2.2633237164453017e-14 2.6209332787671784e-15; 6.15067623253349e-8 3.334386044840097e-7 1.6366100794180327e-6 7.078986603638099e-6 2.667543157076197e-5 8.93070491823968e-5 0.00028381918569248465 0.0009287133094839643 0.0032073100702219885 0.010517501870349139 0.026054298647582966 0.044909409089504616 0.06523038017995875 0.09567547356935045 0.15140898553036677 0.2577599905125014 0.4563146056451123 0.7995746656834952 1.2942720525822244 1.753796868368034 1.2762218052983292 0.4701177980550057 0.3710729182821518 0.2768828670929443 0.21208546337291573 0.17122511515857197 0.14094635047621723 0.10780315992413105 0.06845246173434165 0.03485786693724613 0.015218733936294089 0.0060918571241338594 0.002260198680952992 0.000758298256138088 0.0002236632036560689 5.77719505568706e-5 1.3672548237650943e-5 3.224860910879503e-6 8.135899432435745e-7 2.1831347411965264e-7 5.4065936636699025e-8 9.684827056211229e-9 1.2053092717277393e-9 1.301301653708345e-10 1.4458417651089788e-11 1.7336136255403378e-12 2.1912929578759775e-13 2.764843332831972e-14; 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1.3922724377357506e-10 9.73621622560978e-10 6.164424300849856e-9 3.439468721715997e-8 1.6718784502193902e-7 7.220245376666126e-7 2.9599265786031636e-6 1.2493779298809398e-5 5.5657805641056484e-5 0.00023543482415376595 0.0007523336068128064 0.001672789861534217 0.0031342016019689635 0.005929944254467602 0.012105273647690528 0.02658345596885801 0.06070623165994368 0.1372147825902839 0.28651042032527296 0.5008032172365164 0.6684983689019425 0.6878685519221045 0.6111942970156187 1.108185298057205 0.218742063916546 0.23914240718426946 0.2625120936851158 0.2650490331058483 0.2206307779582007 0.14656991539081227 0.08319194938602902 0.043184040966032386 0.0207395995600781 0.00899491878024157 0.003426370020891026 0.0011421760436128852 0.0003486747971472744 0.0001060423410878435 3.448741656973417e-5 1.19274459712774e-5 3.8067479486275343e-6 8.787336491145916e-7 1.4092407240674797e-7 1.96057376615074e-8 2.8070469221283445e-9 4.3372636211865933e-10 7.0650209701463e-11 1.1488164439112572e-11; 1.7918862519678088e-11 1.3640619017682607e-10 9.40143288328269e-10 5.710189849995491e-9 3.021492638789551e-8 1.420451647739149e-7 6.338888219453289e-7 2.9126187577720727e-6 1.4124515166529333e-5 6.503929547443048e-5 0.00022624200766875342 0.0005475979689749269 0.001116876174375459 0.0023003099934278304 0.005111731384321929 0.012219755331541066 0.030376798611311377 0.07474244252082071 0.1698886883377326 0.3232576481391436 0.4697207091398036 0.526141390095914 0.5089017725634825 0.47906273098760777 1.111030807330771 0.254841654335728 0.3071999851648308 0.33936545017657066 0.3083879128909619 0.22333118596620669 0.13806018139059573 0.07800938557707954 0.04076696193534793 0.019235324788211113 0.007970427470135313 0.0028900353996617503 0.0009596406848409388 0.0003174620317020071 0.00011230804049430521 4.2252524891773554e-5 1.4670122055780532e-5 3.6840998782416134e-6 6.427971041427609e-7 9.72984534697091e-8 1.5157407928970463e-8 2.5483611389016975e-9 4.516943561282224e-10 7.992503825654441e-11; 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5.213808811115846e-15 5.573260877882179e-14 5.39386375010738e-13 4.600310001388541e-12 3.4181311343778705e-11 2.2564424599709328e-10 1.4139743482284173e-9 9.123105734048915e-9 6.212454832872218e-8 4.0169474656727803e-7 1.962113828941647e-6 6.668734235805327e-6 1.9099290606287137e-5 5.523688831842201e-5 0.0001723619870085207 0.0005785842498993084 0.002019652854065885 0.006978026921015478 0.02227206301087396 0.059508009141346206 0.12142187888125991 0.19098107682075524 0.25938954194583846 0.34287949450850264 0.475053067806915 0.7045219744544576 1.0779772220043178 1.5468002922564585 1.3165262774520694 0.6275638798356131 0.5886935757311466 0.49627724986151256 0.3823178871582921 0.2635813203068526 0.15852044504921697 0.08299187385033929 0.03962588494862018 0.01878694250208678 0.00949921116120024 0.005096385997361043 0.0025185781719459827 0.0008988152741331276 0.0002225550035224107 4.775116232877553e-5 1.0533585693872273e-5 2.5055403730711857e-6 6.278242662886553e-7 1.5694006078312863e-7; 4.689240253203912e-16 5.456503175425033e-15 5.748607657255589e-14 5.337125963135835e-13 4.316847747976582e-12 3.102129560013032e-11 2.1160935171931107e-10 1.4862562414109185e-9 1.1017216519694156e-8 7.754654437778049e-8 4.123330566516206e-7 1.5255449976197339e-6 4.756160546646627e-6 1.4973596704046717e-5 5.0862309433142184e-5 0.00018585696804030314 0.0007062307539149666 0.0026561964741669895 0.00922880902547724 0.026842211297935025 0.0596207556027981 0.10208184759927247 0.15092747400859488 0.21717759123568547 0.32754674106604537 0.5287901496787646 0.8807569002400538 1.3757463612892902 1.7972404446550168 1.2833522677238502 0.5497752099122223 0.5282143905485358 0.45812498667456425 0.3527205277815523 0.23556618399750126 0.13639837318010373 0.07181006487442727 0.03745370773618714 0.02079607951550401 0.012234822731652344 0.006622784808847468 0.00258646706925663 0.0007003233984425523 0.0001642101268343363 3.9565962662110994e-5 1.0275127470062286e-5 2.809986150842096e-6 7.663796481223943e-7; 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By passing f_init=objective.cache.f we let the Laplace approximation "warm-start" at the last point of the inner-loop Newton optimisation; objective.cache is a field on the objective closure.

Let's plot samples from the approximate posterior for the optimised hyperparameters:

p2 = plot_data()
plot_samples!(Xgrid, f_post2)

Package and system information

      Status `~/work/ApproximateGPs.jl/ApproximateGPs.jl/examples/c-comparisons/Project.toml`
  [298c2ebc] ApproximateGPs v0.4.5 `/home/runner/work/ApproximateGPs.jl/ApproximateGPs.jl#master`
  [31c24e10] Distributions v0.25.89
  [98b081ad] Literate v2.14.0
  [2ab3a3ac] LogExpFunctions v0.3.23
  [429524aa] Optim v1.7.5
  [91a5bcdd] Plots v1.38.11
  [e88e6eb3] Zygote v0.6.60
  [37e2e46d] LinearAlgebra
  [9a3f8284] Random

Julia Version 1.6.7
Commit 3b76b25b64 (2022-07-19 15:11 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: Intel(R) Xeon(R) Platinum 8370C CPU @ 2.80GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-11.0.1 (ORCJIT, icelake-server)
Environment:
  JULIA_DEBUG = Documenter
  JULIA_LOAD_PATH = :/home/runner/.julia/packages/JuliaGPsDocs/7M86H/src

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