The estimated variance is sometimes to small, so the Normal distribution can't be constructed.

Error message

ERROR: LoadError: ArgumentError: Normal: the condition σ >= zero(σ) is not satisfied.
Stacktrace:
 [1] macro expansion
   @ ~/.julia/packages/Distributions/cNe2C/src/utils.jl:6 [inlined]
 [2] #Normal#112
   @ ~/.julia/packages/Distributions/cNe2C/src/univariate/continuous/normal.jl:37 [inlined]
 [3] Normal(μ::Float64, σ::Float64)
   @ Distributions ~/.julia/packages/Distributions/cNe2C/src/univariate/continuous/normal.jl:37
 [4] quadrature(bquad::BayesQuad{SqExponentialKernel, Float64, Float64}, model::BayesModel{ZeroMeanDiagNormal{Tuple{Base.OneTo{Int64}}}, typeof(log_f)}, samples::Vector{Vector{Float64}})
   @ BayesianQuadrature ~/.julia/packages/BayesianQuadrature/eNnqB/src/bayesquads/bayesquad.jl:58
 [5] (::BayesQuad{SqExponentialKernel, Float64, Float64})(rng::MersenneTwister, model::BayesModel{ZeroMeanDiagNormal{Tuple{Base.OneTo{Int64}}}, typeof(log_f)}, sampler::PriorSampling; x_init::Vector{Any}, nsamples::Int64, callback::Nothing)
   @ BayesianQuadrature ~/.julia/packages/BayesianQuadrature/eNnqB/src/interface.jl:10
 [6] top-level scope
   @ ~/Documents/uni/master/master-thesis-code-base/src/min_example_normal.jl:13
 [7] include(fname::String)
   @ Base.MainInclude ./client.jl:444
 [8] top-level scope
   @ REPL[8]:1

Example

using BayesianQuadrature
using Distributions
using KernelFunctions
using Random

rng = Random.MersenneTwister(42)

p_0 = MvNormal(ones(2)) # As for now the prior must be a MvNormal
log_f(x) = logpdf(MvNormal(0.5 * ones(2)), x) 
m = BayesModel(p_0, log_f) 
bquad = BayesQuad(SEKernel(); l=10.0, σ=1.0) 
sampler = PriorSampling() 
p_I, _ = bquad(rng, m, sampler; nsamples=200) # Returns a Normal distribution
@show p_I 

Maybe move the one from the README to the docs. Add other examples

This issue is used to trigger TagBot; feel free to unsubscribe.

If you haven't already, you should update your TagBot.yml to include issue comment triggers.
Please see this post on Discourse for instructions and more details.

If you'd like for me to do this for you, comment TagBot fix on this issue.
I'll open a PR within a few hours, please be patient!

Soss, Turing, etc...
However, we would need first an (adapted) auto-weighting of the prior

I would suggest to rename BMC.
Bayesian Monte Carlo, the simplest approach for Bayesian Quadrature. Assumes that the prior is Gaussian and that the integrand is as well
https://github.com/theogf/BayesianQuadrature.jl/blob/d621e84566cbcafa7cbe13a6036873479b19471d/src/integrators/bmc.jl#LL1-L7

It is true that it is motivated by the example from the paper and it should be referenced (I suggest in the readme). It is misleading to use the name BMC in the quadrature part (BMC refers more on how to sample the states and than use BQ).

I would refer instead to the distribution (or even kernel-distribution pair), where the distribution is the one we are importance re-weighting "against" (e.g. Gaussian in the simple case (SE kernel - Gaussian)).

See also 4.2 Tractable and Intractable Kernel Means
Briol, F. X., Oates, C. J., Girolami, M., Osborne, M. A., & Sejdinovic, D. (2019). Probabilistic integration: A role in statistical computation?. Statistical Science, 34(1), 1-22.

Remark: The idea is having kernel-distribution pairs with a closed form kernel mean.
the case (SE kernel - Gaussian) is called Bayes–Hermite quadrature
O’Hagan, A. (1991). Bayes–Hermite quadrature.J. Statist. Plann. Inference, 29:245–260.

So Bayesian quadrature is quite unlike other quadrature packages, therefore we need to think how to approach it.

On top of the likelihood/prior input, there are two important playing parameters:

• The method to select new samples
• The method to evaluate the different integrals (vanilla or with transformation)

I think for simplicity it is better to pass these two separately as there is no real benefit to embed the first one in the second.

Implementation of real valued integrands with the Type GeneralModel. (BayesModel assumes positive integrands see #13)

We should have an automatic fallback for when the prior is not Gaussian:
f(x) * p(x) -> (f(x) * p(x)) / q(x) * q(x) where q(x) is MvNormal
We can of course have the default q(x) = N(0, I) but there are easy heuristics to get a better proposal given p(x)

We could probably find some inspiration for the API in this other implementation of BayesianQuadrature : http://www.probabilistic-numerics.org/en/latest/api/quad.html

We should allow the possibility to give a series of evaluated samples (x, y) before even sampling.

Since the math are a bit heavy we should have proper docs since the README is too limited

I feel a callback function would be nice to have. I think we can simply do it by passing it to the AbstractMCMC.sample method. I will make a PR for it.

How should samples be stored? Since we need to go iteratively over them and do not necessarily know their type in advance it is not possible to allow a type stable storage in either the sampler or the integrator.
One package to look at for inspiration is https://github.com/TuringLang/AbstractMCMC.jl

Generally one is concerned about expensive functions, therefore not saving the function evaluations does not seem optimal.

Using the logarithm of the integrand in the BayesModel provides a more stable way of computing functions regarding the integrand (e.g. logjoint)