| Name | ghq JSON |
| Version |
0.0.3
JSON |
| download |
| home_page | |
| Summary | Gauss-Hermite quadrature in JAX |
| upload_time | 2024-02-10 16:17:43 |
| maintainer | |
| docs_url | None |
| author | |
| requires_python | ~=3.9 |
| license | Apache-2.0 |
| keywords |
jax
integration
quadrature
|
| VCS |
 |
| bugtrack_url |
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| requirements |
No requirements were recorded.
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# ghq
Gauss-Hermite quadrature in JAX
# Installation
```
pip install ghq
```
# Numerical Integration
## Univariate Gaussian integrals
Gauss-Hermite quadrature is a method for numerically integrating functions of the form:
$$
\int_{-\infty}^{\infty} f(x) \mathbf{N}(x \mid \mu, \sigma^2) dx \approx \sum_{i=1}^\mathsf{degree} w_i f(x_i),
$$
where $\lbrace x_i, w_i \rbrace_{i=1}^\mathsf{degree}$ are chosen [deterministically](https://en.wikipedia.org/wiki/Gauss%E2%80%93Hermite_quadrature). The integral approximation can be executed easily with `ghq`:
```python
ghq.univariate(f, mu, sigma, degree=32)
```
where `f: Callable[[float], Array]` is a JAX vectorisable function and `degree` is an optional argument controlling the number of function evalutions, increasing `degree` increases the accuracy of the integral. (Note that the output of `f` can be multivariate but the input is univariate).
## Univariate unbounded integrals
More generally, we can use an importance-sampling-like approach to integrate functions of the form:
$$
\int_{-\infty}^{\infty} f(x) dx = \int_{-\infty}^{\infty} \frac{f(x)}{\mathbf{N}(x \mid \mu, \sigma^2)} \mathbf{N}(x \mid \mu, \sigma^2) dx \approx \sum_{i=1}^\mathsf{degree} w_i \frac{f(x_i)}{\mathbf{N}(x_i \mid \mu, \sigma^2)},
$$
and with `ghq`:
```python
ghq.univariate_importance(f, mu, sigma, degree=32)
```
## Univariate half-bounded integrals
Consider lower-bounded integrals of the form:
$$
\int_{a}^{\infty} f(x) dx =\int_{-\infty}^{\infty} f(a + e^y) e^y dy \approx \sum_{i=1}^\mathsf{degree} w_i \frac{f(a + e^{y_i}) e^{y_i}}{\mathbf{N}(y_i \mid \mu, \sigma^2)},
$$
where we use the transformation $y = \log(x - a)$ to map the lower-bounded integral to an unbounded integral. This can be approximated with `ghq`:
```python
ghq.univariate_importance(f, mu, sigma, degree=32, lower=a)
```
or for upper-bounded integrals over $[-\infty, b)$ using transformation $y = \log(b - x)$:
```python
ghq.univariate_importance(f, mu, sigma, degree=32, upper=b)
```
## Univariate bounded integrals
For doubly-bounded integrals in $[a, b)$ we have
$$
\int_{a}^{b} f(x) dx = \int_{-\infty}^{\infty} f\left(a + (b-a)\text{logit}^{-1}(y)\right) (b-a) \text{logit}^{-1}(y) \left(1-\text{logit}^{-1}(y)\right) dy,
$$
where we use the transfomation $y=\text{logit}(\frac{x-a}{b-a})$ with $\text{logit}(u)=\log\frac{u}{1-u}$ and $\text{logit}^{-1}(v) = \frac{1}{1+e^{-v}}$.
In `ghq` we have:
```python
ghq.univariate_importance(f, mu, sigma, degree=32, lower=a, upper=b).
```
The [Stan reference manual](https://mc-stan.org/docs/reference-manual/variable-transforms.html) provides an excellent reference for transformations of variables.
## Multivariate Gaussian integrals
$$
\int f(x) \mathbf{N}(x \mid \mu, \Sigma) dx,
$$
in `ghq` is:
```python
ghq.multivariate(f, mu, Sigma, degree=32)
```
where `f: Callable[[Array], Array]` is a function that takes a multivariate input.
Beware though that multivariate Gauss-Hermite quadrature has complexity
$O(\text{degree}^d)$ where $d$ is the dimension of the integral, so it is not feasible
for high-dimensional integrals.
## Multivariate unbounded integrals
Coming soon...
Raw data
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