Specify different measures, kernels and domains for n-dimensional integration using bayesquad

[panweihit]

Hi,

I use the example in #512. I have two naive questions.

1. I am trying to use a different measure and kernel inprobnum's bayesquad function. Did I make something wrong in the argument?

from probnum.quad import bayesquad
from probnum.quad.integration_measures LebesgueMeasure
from probnum.randprocs.kernels import ExpQuad

bayesquad(f, 2, domain=domain, measure=LebesgueMeasure, kernel=ExpQuad)

2. Here the domain for the two inputs are both

domain=np.array([-3, 3]) 

Can I specify different domains, such as

domain = np.array([[-3, 3],[0, 1]])

Expect your reply. Thanks!

[mmahsereci]

hey, could you let me know which version or commit of probnum you are using? The quad package changed quite a bit lately and the answer depends a bit on the version you are using. Thanks!

[panweihit]

Just to make sure, I just re-tried pip install git+https://github.com/probabilistic-numerics/probnum.git

It should be the latest version.

[mmahsereci]

OK, that should be the latest version then, that's great. You can check it also by running

import probnum
probnum.__version__
>>> '0.1.22.dev10'

Regarding your questions:

1. In order to use the Lebesgue measure, you can simply provide the domain. If all lower bounds and all upper bounds of the box-domain are the same, you only need to provide a tuple of two floats containing the lower and upper bound.

import numpy as np
from probnum.quad import bayesquad
from probnum.randprocs.kernels import ExpQuad

def fun(x):
    return np.sum(x **2, axis=1)

input_dim = 2
rng = np.random.default_rng()
domain = (-3, 3)  # if input_dim > 1, domain will be expanded to ([-3, -3, ...], [3, 3, ...])
kernel=ExpQuad(input_shape=(input_dim,))

bayesquad(
    fun=fun, 
    input_dim=input_dim, 
    domain=domain, 
    kernel=kernel,
    rng=rng,
)

2. If lower and upper bounds differ between dimensions, domain needs to contain an array containing all lower bounds, and an array containing all upper bounds, i.e., ([lower_1, lower_2, ...], [upper_1, upper_2, ...]).

input_dim = 2
rng = np.random.default_rng()
domain = ([-3, 0], [3, 1])  # Domain contains bounds ([lower_1, lower_2, ...], [upper_1, upper_2, ...])
kernel=ExpQuad(input_shape=(input_dim,))

bayesquad(
    fun=fun, 
    input_dim=input_dim, 
    domain=domain, 
    kernel=kernel,
    rng=rng,
)

3. You can also provide the Lebesgue measure directly

from probnum.quad.integration_measures import LebesgueMeasure

input_dim = 2
rng = np.random.default_rng()
domain = ([-3, 0], [3, 1]) 
measure = LebesgueMeasure(input_dim=input_dim, domain=domain)
kernel=ExpQuad(input_shape=(input_dim,))

bayesquad(
    fun=fun, 
    input_dim=input_dim, 
    measure=measure,
    kernel=kernel,
    rng=rng,
)

4. You can swap out the Lebesgue measure for the Gaussian measure, e.g.

from probnum.quad.integration_measures import GaussianMeasure

input_dim = 2
rng = np.random.default_rng()
measure = GaussianMeasure(input_dim=input_dim, mean=0.0, cov=1.0)
kernel=ExpQuad(input_shape=(input_dim,))

bayesquad(
    fun=fun, 
    input_dim=input_dim, 
    measure=measure,
    kernel=kernel,
    rng=rng,
)

Please also note that bayesquad does not yet support hyper-parameters optimization (fitting the kernel parameters), apart from setting the global scale parameter to the maximum marginal likelihood estimate. Hence you should currently only use it if you can supply the ExpQuad kernel with some appropriate lengthscale for your problem. You can provide it like so:

input_dim = 2
rng = np.random.default_rng()
domain = ([-3, 0], [3, 1])
kernel=ExpQuad(input_shape=(input_dim,), lengthscale=0.3)  # appropriate lengthscale

bayesquad(
    fun=fun, 
    input_dim=input_dim, 
    domain=domain, 
    kernel=kernel,
    rng=rng,
)

Can you check if the code runs for you? And does that answer your questions?

[panweihit]

It works very well! Thanks a lot!

I would assume that the hyper-parameters optimization feature is on the way, so this feature request is unnecessary.

[mmahsereci]

It works very well! Thanks a lot!

Glad to hear! Thanks for the quick feedback.

I would assume that the hyper-parameters optimization feature is on the way, so this feature request is unnecessary.

Yes, it's planned (a first step is this PR #581 ). It may still take a while though and I cannot promise a date unfortunately.

[mmahsereci]

Closing this issue as it seems resolved. @panweihit feel free to re-open, or open a new issue an case there are further questions.