API Library

General Framework for the data augmented Gaussian Processes

Class for Gaussian Processes models

GP(X::AbstractArray{T}, y::AbstractArray, kernel::Kernel;
    noise::Real=1e-5, opt_noise::Bool=true, verbose::Int=0,
    optimiser=ADAM(0.01),atfrequency::Int=1,
    mean::Union{<:Real,AbstractVector{<:Real},PriorMean}=ZeroMean(),
    IndependentPriors::Bool=true,ArrayType::UnionAll=Vector)

Argument list :

Mandatory arguments

• X : input features, should be a matrix N×D where N is the number of observation and D the number of dimension
• y : input labels, can be either a vector of labels for multiclass and single output or a matrix for multi-outputs (note that only one likelihood can be applied)
• kernel : covariance function, can be either a single kernel or a collection of kernels for multiclass and multi-outputs models

Keyword arguments

• noise : Initial noise of the model
• opt_noise : Flag for optimizing the noise σ=Σ(y-f)^2/N
• mean : Option for putting a prior mean
• verbose : How much does the model print (0:nothing, 1:very basic, 2:medium, 3:everything)
• optimiser : Optimiser used for the kernel parameters. Should be an Optimiser object from the Flux.jl library, see list here Optimisers and on this list. Default is ADAM(0.001)
• IndependentPriors : Flag for setting independent or shared parameters among latent GPs
• atfrequency : Choose how many variational parameters iterations are between hyperparameters optimization
• mean : PriorMean object, check the documentation on it MeanPrior
• ArrayType : Option for using different type of array for storage (allow for GPU usage)

Class for variational Gaussian Processes models (non-sparse)

VGP(X::AbstractArray{T},y::AbstractVector,
kernel::Kernel,
    likelihood::LikelihoodType,inference::InferenceType;
    verbose::Int=0,optimiser=ADAM(0.01),atfrequency::Integer=1,
    mean::Union{<:Real,AbstractVector{<:Real},PriorMean}=ZeroMean(),
    IndependentPriors::Bool=true,ArrayType::UnionAll=Vector)

Argument list :

Mandatory arguments

• X : input features, should be a matrix N×D where N is the number of observation and D the number of dimension
• y : input labels, can be either a vector of labels for multiclass and single output or a matrix for multi-outputs (note that only one likelihood can be applied)
• kernel : covariance function, a single kernel from the KernelFunctions.jl package
• likelihood : likelihood of the model, currently implemented : Gaussian, Bernoulli (with logistic link), Multiclass (softmax or logistic-softmax) see Likelihood Types
• inference : inference for the model, can be analytic, numerical or by sampling, check the model documentation to know what is available for your likelihood see the Compatibility Table

Keyword arguments

• verbose : How much does the model print (0:nothing, 1:very basic, 2:medium, 3:everything)
• optimiser : Optimiser used for the kernel parameters. Should be an Optimiser object from the Flux.jl library, see list here Optimisers and on this list. Default is ADAM(0.001)
• atfrequency : Choose how many variational parameters iterations are between hyperparameters optimization
• mean : PriorMean object, check the documentation on it MeanPrior
• IndependentPriors : Flag for setting independent or shared parameters among latent GPs
• ArrayType : Option for using different type of array for storage (allow for GPU usage)

Class for sparse variational Gaussian Processes

SVGP(X::AbstractArray{T1},y::AbstractVector{T2},kernel::Kernel,
    likelihood::LikelihoodType,inference::InferenceType, nInducingPoints::Int;
    verbose::Int=0,optimiser=ADAM(0.001),atfrequency::Int=1,
    mean::Union{<:Real,AbstractVector{<:Real},PriorMean}=ZeroMean(),
    Zoptimiser=false,
    ArrayType::UnionAll=Vector)

Argument list :

Mandatory arguments

• X : input features, should be a matrix N×D where N is the number of observation and D the number of dimension
• y : input labels, can be either a vector of labels for multiclass and single output or a matrix for multi-outputs (note that only one likelihood can be applied)
• kernel : covariance function, can be either a single kernel or a collection of kernels for multiclass and multi-outputs models
• likelihood : likelihood of the model, currently implemented : Gaussian, Student-T, Laplace, Bernoulli (with logistic link), Bayesian SVM, Multiclass (softmax or logistic-softmax) see Likelihood
• inference : inference for the model, can be analytic, numerical or by sampling, check the model documentation to know what is available for your likelihood see the Compatibility table
• nInducingPoints : number of inducing points

Optional arguments

• verbose : How much does the model print (0:nothing, 1:very basic, 2:medium, 3:everything)
• optimiser : Optimiser used for the kernel parameters. Should be an Optimiser object from the Flux.jl library, see list here Optimisers and on this list. Default is ADAM(0.001)
• atfrequency : Choose how many variational parameters iterations are between hyperparameters optimization
• mean : PriorMean object, check the documentation on it MeanPrior
• IndependentPriors : Flag for setting independent or shared parameters among latent GPs
• Zoptimiser : Optimiser used for the inducing points locations. Should be an Optimiser object from the Flux.jl library, see list here Optimisers and on this list. Default is ADAM(0.001)
• ArrayType : Option for using different type of array for storage (allow for GPU usage)

GaussianLikelihood(σ²::T=1e-3) #σ² is the variance

Gaussian noise :

There is no augmentation needed for this likelihood which is already conjugate to a Gaussian prior

StudentTLikelihood(ν::T,σ::Real=one(T))

Student-t likelihood for regression:

ν is the number of degrees of freedom and σ is the variance for local scale of the data.

For the analytical solution, it is augmented via:

Where ω ~ IG(0.5ν,,0.5ν) where IG is the inverse gamma distribution See paper Robust Gaussian Process Regression with a Student-t Likelihood

LaplaceLikelihood(β::T=1.0)  #  Laplace likelihood with scale β

Laplace likelihood for regression:

see wiki page

For the analytical solution, it is augmented via:

where $ω ~ Exp(ω | 1/(2 β^2))$, and Exp is the Exponential distribution We use the variational distribution $q(ω) = GIG(ω | a,b,p)$

LogisticLikelihood()

Bernoulli likelihood with a logistic link for the Bernoulli likelihood

(for more info see : wiki page)

For the analytic version the likelihood, it is augmented via:

where $ω ~ PG(ω | 1, 0)$, and PG is the Polya-Gamma distribution See paper : Efficient Gaussian Process Classification Using Polya-Gamma Data Augmentation

HeteroscedasticLikelihood(λ::T=1.0)

Gaussian with heteroscedastic noise given by another gp:

Where σ is the logistic function

Augmentation will be described in a future paper

BayesianSVM()

The Bayesian SVM is a Bayesian interpretation of the classical SVM.

math p(y|f,ω) = 1/(sqrt(2πω) exp(-0.5((1+ω-yf)^2/ω)) `$where$ω ∼ 𝟙[0,∞)`` has an improper prior (his posterior is however has a valid distribution, a Generalized Inverse Gaussian). For reference see this paper

    SoftMaxLikelihood()

Multiclass likelihood with Softmax transformation:

There is no possible augmentation for this likelihood

    LogisticSoftMaxLikelihood()

The multiclass likelihood with a logistic-softmax mapping: :

where σ is the logistic function. This likelihood has the same properties as softmax. –-

For the analytical version, the likelihood is augmented multiple times. More details can be found in the paper Multi-Class Gaussian Process Classification Made Conjugate: Efficient Inference via Data Augmentation

    Poisson Likelihood(λ::T=1.0)

Poisson Likelihood where a Poisson distribution is defined at every point in space (careful, it's different from continous Poisson processes)

Where σ is the logistic function Augmentation details will be released at some point (open an issue if you want to see them)

    NegBinomialLikelihood(r::Int=10)

Negative Binomial likelihood with number of failures r

Where σ is the logistic function

AnalyticVI

Variational Inference solver for conjugate or conditionally conjugate likelihoods (non-gaussian are made conjugate via augmentation) All data is used at each iteration (use AnalyticSVI for Stochastic updates)

AnalyticVI(;ϵ::T=1e-5)

Keywords arguments - ϵ::T : convergence criteria

AnalyticSVI Stochastic Variational Inference solver for conjugate or conditionally conjugate likelihoods (non-gaussian are made conjugate via augmentation)

AnalyticSVI(nMinibatch::Integer;ϵ::T=1e-5,optimiser=RobbinsMonro())

- `nMinibatch::Integer` : Number of samples per mini-batches

Keywords arguments

- `ϵ::T` : convergence criteria
- `optimiser` : Optimiser used for the variational updates. Should be an Optimiser object from the [Flux.jl](https://github.com/FluxML/Flux.jl) library, see list here [Optimisers](https://fluxml.ai/Flux.jl/stable/training/optimisers/) and on [this list](https://github.com/theogf/AugmentedGaussianProcesses.jl/tree/master/src/inference/optimisers.jl). Default is `RobbinsMonro()` (ρ=(τ+iter)^-κ)

GibbsSampling(;ϵ::T=1e-5,nBurnin::Int=100,samplefrequency::Int=1)

Draw samples from the true posterior via Gibbs Sampling.

Keywords arguments - ϵ::T : convergence criteria - nBurnin::Int : Number of samples discarded before starting to save samples - samplefrequency::Int : Frequency of sampling

QuadratureVI

Variational Inference solver by approximating gradients via numerical integration via Quadrature

QuadratureVI(ϵ::T=1e-5,nGaussHermite::Integer=20,optimiser=Momentum(0.0001))

Keyword arguments

- `ϵ::T` : convergence criteria
- `nGaussHermite::Int` : Number of points for the integral estimation
- `natural::Bool` : Use natural gradients
- `optimiser` : Optimiser used for the variational updates. Should be an Optimiser object from the [Flux.jl](https://github.com/FluxML/Flux.jl) library, see list here [Optimisers](https://fluxml.ai/Flux.jl/stable/training/optimisers/) and on [this list](https://github.com/theogf/AugmentedGaussianProcesses.jl/tree/master/src/inference/optimisers.jl). Default is `Momentum(0.0001)`

QuadratureSVI

Stochastic Variational Inference solver by approximating gradients via numerical integration via Quadrature

QuadratureSVI(nMinibatch::Integer;ϵ::T=1e-5,nGaussHermite::Integer=20,optimiser=Momentum(0.0001))

-`nMinibatch::Integer` : Number of samples per mini-batches

Keyword arguments

- `ϵ::T` : convergence criteria, which can be user defined
- `nGaussHermite::Int` : Number of points for the integral estimation (for the QuadratureVI)
- `natural::Bool` : Use natural gradients
- `optimiser` : Optimiser used for the variational updates. Should be an Optimiser object from the [Flux.jl](https://github.com/FluxML/Flux.jl) library, see list here [Optimisers](https://fluxml.ai/Flux.jl/stable/training/optimisers/) and on [this list](https://github.com/theogf/AugmentedGaussianProcesses.jl/tree/master/src/inference/optimisers.jl). Default is `Momentum(0.0001)`

MCIntegrationVI(;ϵ::T=1e-5,nMC::Integer=1000,optimiser=Momentum(0.001))

Variational Inference solver by approximating gradients via MC Integration.

Keyword arguments

- `ϵ::T` : convergence criteria, which can be user defined
- `nMC::Int` : Number of samples per data point for the integral evaluation
- `natural::Bool` : Use natural gradients
- `optimiser` : Optimiser used for the variational updates. Should be an Optimiser object from the [Flux.jl](https://github.com/FluxML/Flux.jl) library, see list here [Optimisers](https://fluxml.ai/Flux.jl/stable/training/optimisers/) and on [this list](https://github.com/theogf/AugmentedGaussianProcesses.jl/tree/master/src/inference/optimisers.jl). Default is `Momentum(0.01)`

MCIntegrationSVI(;ϵ::T=1e-5,nMC::Integer=1000,optimiser=Momentum(0.0001))

Stochastic Variational Inference solver by approximating gradients via Monte Carlo integration

Argument

-`nMinibatch::Integer` : Number of samples per mini-batches

Keyword arguments

- `ϵ::T` : convergence criteria, which can be user defined
- `nMC::Int` : Number of samples per data point for the integral evaluation
- `natural::Bool` : Use natural gradients
- `optimiser` : Optimiser used for the variational updates. Should be an Optimiser object from the [Flux.jl](https://github.com/FluxML/Flux.jl) library, see list here [Optimisers](https://fluxml.ai/Flux.jl/stable/training/optimisers/) and on [this list](https://github.com/theogf/AugmentedGaussianProcesses.jl/tree/master/src/inference/optimisers.jl). Default is `Momentum()` (ρ=(τ+iter)^-κ)

train!(model::AbstractGP;iterations::Integer=100,callback=0,convergence=0)

Function to train the given GP model.

Keyword Arguments

there are options to change the number of max iterations,

• iterations::Int : Number of iterations (not necessarily epochs!)for training
• callback::Function : Callback function called at every iteration. Should be of type function(model,iter) ... end
• convergence::Function : Convergence function to be called every iteration, should return a scalar and take the same arguments as callback

train!(model::AbstractGP;iterations::Integer=100,callback=0,conv_function=0)

Function to train the given GP model.

Keyword Arguments

there are options to change the number of max iterations,

• iterations::Int : Number of iterations (not necessarily epochs!)for training
• callback::Function : Callback function called at every iteration. Should be of type function(model,iter) ... end
• conv_function::Function : Convergence function to be called every iteration, should return a scalar and take the same arguments as callback

Compute the mean of the predicted latent distribution of f on X_test for the variational GP model

Return also the diagonal variance if covf=true and the full covariance if fullcov=true

predict_y(model::AbstractGP,X_test::AbstractMatrix)

Return - the predictive mean of X_test for regression - the sign of X_test for classification - the most likely class for multi-class classification - the expected number of events for an event likelihood

proba_y(model::AbstractGP,X_test::AbstractMatrix)

Return the probability distribution p(ytest|model,Xtest) :

- Tuple of vectors of mean and variance for regression
- Vector of probabilities of y_test = 1 for binary classification
- Dataframe with columns and probability per class for multi-class classification

ZeroMean

ZeroMean()

Construct a mean prior set to 0 and which cannot be updated.

ConstantMean

ConstantMean(c::T=1.0;opt=ADMA(0.01))

Construct a prior mean with constant c Optionally set an optimiser opt (ADAM(0.01) by default)

EmpiricalMean julia` function EmpiricalMean(c::V=1.0;opt=ADAM(0.01)) where {V<:AbstractVector{<:Real}} Construct a empirical mean with values c Optionally give an optimiser opt (ADAM(0.01) by default)