torch.distributions.HalfNormal

class HalfNormal extends Distribution

new HalfNormal(scale: number | Tensor, options?: DistributionOptions)

readonlyscale(Tensor): – Scale parameter (standard deviation of the underlying normal).
readonlyarg_constraints(unknown)
readonlysupport(unknown)
readonlyhas_rsample(unknown)
readonlymean(Tensor)
readonlymode(Tensor)
readonlyvariance(Tensor)

Half-Normal distribution: folded normal distribution for positive-only values.

Parameterized by scale σ. The half-normal is obtained by taking the absolute value of a Normal(0, σ) distribution: if X = |Y| where Y ~ N(0, σ), then X ~ HalfNormal(σ). Support is (0, ∞) only. Critical for Bayesian modeling because it's the standard weakly-informative prior for scale parameters. Essential for:

Prior distributions for scale, variance, and standard deviation parameters
Weakly informative Bayesian priors (less heavy-tailed than HalfCauchy)
Modeling absolute deviations and measurement magnitudes
Reliability analysis and tolerance limit specifications
Quality control and precision specifications
Bayesian hierarchical models (shrinkage priors)
Folded normal models and reflected distributions

Relationship to Normal: HalfNormal(σ) = |Normal(0, σ)|. Conceptually, it's like cutting a normal distribution in half at the mean and flipping one half onto the other.

Why useful as prior: More stable than HalfCauchy (lighter tails) while still being weakly informative. Good default for scale parameters when you have weak prior information.

\begin{aligned} \text{PDF: } f(x) = \sqrt{\frac{2}{\pi}} \frac{1}{\sigma} \exp\left(-\frac{x^2}{2\sigma^2}\right) \quad x > 0 \\ \text{CDF: } F(x) = \text{erf}\left(\frac{x}{\sigma\sqrt{2}}\right) \\ \text{Mean: } \mathbb{E}[X] = \sigma\sqrt{\frac{2}{\pi}} \approx 0.798\sigma \\ \text{Variance: } \text{Var}(X) = \sigma^2\left(1 - \frac{2}{\pi}\right) \approx 0.363\sigma^2 \\ \text{Mode: } 0 \quad \text{(at the boundary)} \\ \text{Entropy: } H(X) = \frac{1}{2}\ln\left(\frac{\pi e \sigma^2}{2}\right) \end{aligned}

Absolute value interpretation: Equivalent to |Normal(0, σ)|
Mean ≈ 0.798σ: Less than the scale parameter due to folding
Variance ≈ 0.363σ²: Less than Normal(0,σ) due to positive support constraint
Mode at 0: Highest density at boundary; PDF → 0 as x → ∞
Posterior form: Posterior for scale in Normal model has similar form (related conjugacy)
Light tails vs HalfCauchy: Decays exponentially (HalfCauchy decays polynomially)
Weakly informative: Good default prior for scale parameters without strong beliefs

Scale must be positive: scale ≤ 0 causes errors
Support is positive: No negative values; only x 0
Heavy-tailed at 0: PDF has peak at 0, not spread out like some positive distributions
Related to normal precision: Not the same as inverse gamma (which is for precision)

Examples

// Standard half-normal: scale=1
const hn = new torch.distributions.HalfNormal(1);
const samples = hn.sample([1000]);  // 1000 positive samples
// Typical values: 0.5-1.5, rarely > 3

// Bayesian prior for scale parameter
// Prior for standard deviation of measurement noise
const measurement_scale = 0.1;  // express weak prior knowledge
const prior = new torch.distributions.HalfNormal(measurement_scale);
const prior_samples = prior.sample([10000]);  // 10000 prior samples
n * // Use in: const noise_std = prior_samples\n *
// Different scale values affect spread
const narrow = new torch.distributions.HalfNormal(0.5);  // tighter, more concentrated
const wide = new torch.distributions.HalfNormal(2.0);    // wider, more dispersed
// narrow.mean ≈ 0.4, wide.mean ≈ 1.6\n *
// Batched distributions: different scale parameters
const scales = torch.tensor([0.1, 0.5, 1.0, 2.0, 5.0]);
const dist = new torch.distributions.HalfNormal(scales);  // [5] batch shape
const samples = dist.sample();  // [5] shaped samples, each from different scale
const means = dist.mean;  // ~[0.08, 0.4, 0.8, 1.6, 4.0]

// Hierarchical model: half-normal prior for group-level variation
// Each group has its own scale, drawn from half-normal hyperprior
const num_groups = 5;
const group_scale_prior = new torch.distributions.HalfNormal(1);
const group_scales = group_scale_prior.sample([num_groups]);  // 5 group scales
// Then: X_i_j ~ Normal(0, group_scales[i]) for observations in group i\n *
// Quantiles and CDF
const hn = new torch.distributions.HalfNormal(1);
const cdf_val = hn.cdf(torch.tensor([0.5, 1.0, 2.0]));  // cumulative probability at these points
const q = hn.icdf(torch.tensor([0.5, 0.9, 0.95]));      // quantiles (median, 90%, 95%)