class

StickBreakingTransform

extendsTransform

StickBreakingTransform()

source

Logistic stick-breaking bijection $\mathbb{R}^{K-1} \to \Delta^{K-1}$ .

A true bijection between unconstrained $(K-1)$ -vectors and the $K$ -simplex (i.e. one extra dimension is broken off as the residual stick). Unlike SoftmaxTransform it has the correct dimensionality and a tractable log-Jacobian determinant, making it the preferred choice for normalising flows over probability vectors and for unconstrained reparameterisations of lucid.distributions.Dirichlet priors. event_dim = 1.

Notes

Forward "stick breaking" (with $z_k = \sigma(x_k - \log(K-k))$ ):

y_k = z_k \prod_{j < k}(1 - z_j), \quad y_{K-1} = \prod_{j=0}^{K-2}(1 - z_j)

The last component is the residual stick remaining after the first $K - 1$ breaks. Each $y_k > 0$ and $\sum_k y_k = 1$ by construction.

Inverse (back-solve the stick lengths):

z_k = \frac{y_k}{1 - \sum_{j < k} y_j}, \qquad x_k = \mathrm{logit}(z_k) + \log(K - k)

Log Jacobian determinant:

\log|\det J| = \sum_{k=0}^{K-2} \bigl[\log y_k + \log(1 - \textstyle\sum_{j < k} y_j) + \log(1 - z_k)\bigr]

where the $-\log(K-k)$ shifts ensure the uniform Dirichlet corresponds to $\mathbf{x} = \mathbf{0}$ .

Examples

>>> import lucid
>>> from lucid.distributions.transforms import StickBreakingTransform
>>> T = StickBreakingTransform()
>>> y = T(lucid.tensor([0.0, 0.0]))  # maps to a Dirichlet(1,1,1) sample
>>> y.sum()
Tensor(1.0)

Methods (1)

log_abs_det_jacobian

→Tensor

log_abs_det_jacobian(x: Tensor, y: Tensor)

source

Standard simplex-to- $\mathbb{R}^{K-1}$ log-Jacobian.

$\log|\det J| = \sum_k \bigl[\log y_k + \log\!\bigl(1 - \sum_{j<k} y_j\bigr)\bigr]$ .