ghq 0.0.2

Gauss-Hermite quadrature in JAX

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. The integral approximation can be executed easily with ghq:

ghq.univariate(f, mu, sigma, degree=32)

where f: Callable[[float], float] 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.

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:

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:

ghq.univariate_importance(f, mu, sigma, degree=32, lower=a)

or for upper-bounded integrals over $[-\infty, b)$ using transformation $y = \log(b - x)$:

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:

ghq.univariate_importance(f, mu, sigma, degree=32, lower=a, upper=b).

The Stan reference manual provides an excellent reference for transformations of variables.

Multivariate integrals

Coming soon...