torch.distributions.StudentT

class StudentT extends Distribution

new StudentT(df: number | Tensor, options?: StudentTOptions)

readonlydf(Tensor): – Degrees of freedom.
readonlyloc(Tensor): – Location parameter.
readonlyscale(Tensor): – Scale parameter.
readonlyarg_constraints(unknown)
readonlysupport(unknown)
readonlyhas_rsample(unknown)
readonlymean(Tensor)
readonlymode(Tensor)
readonlyvariance(Tensor)

Student's t-distribution: continuous distribution for heavy-tailed normal-like data.

Parameterized by degrees of freedom (df), location (loc), and scale. Generalizes normal distribution with heavier tails for modeling outliers and robustness. Essential for:

Statistical inference with small samples (t-tests)
Robust modeling (heavier tails than normal)
Modeling data with outliers and extreme values
Bayesian modeling with student-t priors (robust)
Approximating normal with uncertainty
Heavy-tailed phenomena (financial returns, measurement errors)
Mixture models with normal components and heavy tails

Ratio distribution: StudentT(ν, μ, σ) = μ + σ * Z / √(V/ν) where Z ~ Normal(0, 1) and V ~ Chi2(ν) independently

\begin{aligned} \text{PDF: } f(x) = \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi\nu}\Gamma(\frac{\nu}{2})} \left(1 + \frac{(x-\mu)^2}{\nu\sigma^2}\right)^{-\frac{\nu+1}{2}} \\ \text{Mean: } \mathbb{E}[X] = \mu \quad \nu > 1 \\ \text{Variance: } \text{Var}(X) = \sigma^2 \frac{\nu}{\nu - 2} \quad \nu > 2 \\ \text{Mode: } \mu \\ \text{Entropy: } H(X) = \frac{1}{2}\ln\left(\nu\sigma^2\pi\frac{\Gamma(\frac{\nu+1}{2})^2}{\Gamma(\frac{\nu}{2})^2}\right) + \left(\frac{\nu+1}{2}\right)\left(\psi\left(\frac{\nu+1}{2}\right) - \psi\left(\frac{\nu}{2}\right)\right) \end{aligned}

Heavy tails: Probability of extreme values higher than normal
Robust inference: Better for data with outliers than normal
Limiting case: t(∞) = Normal (converges to normal with large df)
t-tests: Student's t-test uses this distribution for inference
Moment existence: Mean exists only for df 1, variance for df 2
Symmetric: Always symmetric around location parameter μ

Mean undefined: For df ≤ 1, mean doesn't exist
Variance undefined: For df ≤ 2, variance is infinite
Small df: df 1 has undefined mean; df = 1 is Cauchy (undefined mean/variance)
Extreme values: Heavy tails mean occasional very large deviations
Numerical stability: Very small df can cause numerical issues

Examples

// Standard t-distribution: df=3, mean=0, std=1
const t_dist = new torch.distributions.StudentT(3);
t_dist.sample();  // Heavy-tailed, robust to outliers

// T-test with small sample: df = n - 1
const sample_size = 20;
const df = sample_size - 1;  // 19 degrees of freedom
const t_critical = torch.tensor([1.729]);  // For α=0.1, two-tailed
const test_dist = new torch.distributions.StudentT(df);

// Robust regression: use t-distribution instead of normal
// Student-t likelihood is more robust to outliers than normal
const df = 2;  // Low df for heavy tails
const loc = torch.tensor([0]);  // Model prediction
const scale = torch.tensor([1]);  // Model uncertainty
const likelihood = new torch.distributions.StudentT(df, { loc, scale });

// Different degrees of freedom: varying tail heaviness
const dfs = torch.tensor([1, 2, 5, 10, 30]);  // Low to high
const dist = new torch.distributions.StudentT(dfs);
const samples = dist.sample([1000]);  // Compare tail behavior

// Bayesian robust regression: hierarchical model
// Data might have outliers, use heavy-tailed likelihood
const data = torch.tensor([...]);  // Measurements with potential outliers
const mu_pred = torch.tensor([...]);  // Model predictions
const sigma = torch.tensor([...]);  // Measurement uncertainty
const df = 4;  // Moderate tail heaviness
const likelihood = new torch.distributions.StudentT(df, { loc: mu_pred, scale: sigma });
const log_likelihood = likelihood.log_prob(data).sum();

// Location-scale variant: shifted and scaled t-distribution
const location = 10;
const scale = 2;
const df = 5;
const shifted_t = new torch.distributions.StudentT(df, { loc: location, scale: scale });
// Mean = 10, heavier tails than Normal(10, 2)

// Comparison: normal vs t-distribution
const df_values = torch.tensor([1, 2, 5, Infinity]);  // Cauchy to Normal
const dists = df_values.map(df => new torch.distributions.StudentT(df));
// Lower df -> more extreme values more probable