# BayesianQuadrature

Package for implementing different methods of Bayesian quadrature.
Bayesian quadrature consists in estimating the integral `I = ∫ f(x) p(x) dx`

by using Gaussian Processes where `p(x)`

is assumed to be Gaussian.
More precisely we replace `f(x)`

by a GP by estimating `f`

for multiple samples `x_i`

.
We then get a posterior distribution for the integral : `p(I|{x_i}) = N(m, S)`

.

Given a Bayesian problem `p(x|y) = p(y|x) p_0(x) / p(y)`

you can estimate `p(y)`

by calling :

```
using BayesianQuadrature
using Distributions
using KernelFunctions
p_0 = MvNormal(ones(2)) # As for now the prior must be a MvNormal
log_f(x) = logpdf(MvNormal(0.5 * ones(2)), x) # The logarithm of the Integrand log_f, the log-likelihood function typically
model = BayesModel(p_0, log_f) # Combine both to create the model
bquad = BayesQuad(SEKernel(); l=0.1, σ=1.0) # Will simply approximate p(y|x) with a GP (only works with SEKernel for now
sampler = PriorSampling() # Will sample from the prior p_0
p_I, _ = bquad(model, sampler; nsamples=100) # Returns a Normal distribution
@show p_I # Normal{Float64}(μ=0.07063602778449946, σ=0.0028050929209120458)
```