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  5. Wishart

torch.distributions.Wishart

class Wishart extends Distribution
new Wishart(df: number | Tensor, options: { covariance_matrix?: Tensor; precision_matrix?: Tensor; scale_tril?: Tensor; } & DistributionOptions)
readonlydf(Tensor)
– Degrees of freedom (must be >= dimension of the matrix).
readonlyarg_constraints(unknown)
readonlysupport(unknown)
readonlyhas_rsample(unknown)
readonlyscale_tril(Tensor)
– Get the Cholesky factor of the scale matrix.
readonlycovariance_matrix(Tensor)
– Get the covariance/scale matrix.
readonlymean(Tensor)
readonlymode(Tensor)
readonlyvariance(Tensor)

Wishart distribution: distribution over positive-definite matrices.

Parameterized by df (degrees of freedom) and scale matrix S. Essential for:

  • Conjugate prior for covariance/precision matrices in multivariate normal
  • Bayesian multivariate regression
  • Inverse Wishart priors (for covariance matrices)
  • Covariance matrix estimation with uncertainty
  • Gaussian graphical models and structure learning
  • Hierarchical Bayesian models with matrix-valued parameters

The probability density function involves positive-definite matrices. Support: Space of d×d positive-definite symmetric matrices

PDF: f(W)∝∣W∣(df−d−1)/2exp⁡(−12tr(S−1W))Mean: E[W]=df⋅SMode: (df−d−1)Sfor df>d+1Expectation of inverse: E[W−1]=1df−d−1S−1Inverse Wishart: IW(ν,Ψ)=Wd−1(ν,Ψ−1)\begin{aligned} \text{PDF: } f(W) \propto |W|^{(df-d-1)/2} \exp\left(-\frac{1}{2}\text{tr}(S^{-1}W)\right) \\ \text{Mean: } \mathbb{E}[W] = df \cdot S \\ \text{Mode: } (df - d - 1) S \quad \text{for } df > d + 1 \\ \text{Expectation of inverse: } \mathbb{E}[W^{-1}] = \frac{1}{df - d - 1} S^{-1} \\ \text{Inverse Wishart: } \text{IW}(\nu, \Psi) = W_d^{-1}(\nu, \Psi^{-1}) \end{aligned}PDF: f(W)∝∣W∣(df−d−1)/2exp(−21​tr(S−1W))Mean: E[W]=df⋅SMode: (df−d−1)Sfor df>d+1Expectation of inverse: E[W−1]=df−d−11​S−1Inverse Wishart: IW(ν,Ψ)=Wd−1​(ν,Ψ−1)​

Examples

const m = new Wishart(
  5,                      // degrees of freedom (must be >= dim)
  { scale_tril: torch.eye(3) }
);
m.sample();  // sample a 3x3 positive definite matrix
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