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# Source code for torch.distributions.lkj_cholesky

    """
    This closely follows the implementation in NumPyro (https://github.com/pyro-ppl/numpyro).

    Original copyright notice:

    # Copyright: Contributors to the Pyro project.
    # SPDX-License-Identifier: Apache-2.0
    """

    import math

    import torch
    from torch.distributions import Beta, constraints
    from torch.distributions.distribution import Distribution
    from torch.distributions.utils import broadcast_all

    __all__ = ["LKJCholesky"]

    class LKJCholesky(Distribution):
        r"""
        LKJ distribution for lower Cholesky factor of correlation matrices.
        The distribution is controlled by ``concentration`` parameter :math:`\eta`
        to make the probability of the correlation matrix :math:`M` generated from
        a Cholesky factor proportional to :math:`\det(M)^{\eta - 1}`. Because of that,
        when ``concentration == 1``, we have a uniform distribution over Cholesky
        factors of correlation matrices::

            L ~ LKJCholesky(dim, concentration)
            X = L @ L' ~ LKJCorr(dim, concentration)

        Note that this distribution samples the
        Cholesky factor of correlation matrices and not the correlation matrices
        themselves and thereby differs slightly from the derivations in [1] for
        the `LKJCorr` distribution. For sampling, this uses the Onion method from
        [1] Section 3.

        Example::

            >>> # xdoctest: +IGNORE_WANT("non-deterinistic")
            >>> l = LKJCholesky(3, 0.5)
            >>> l.sample()  # l @ l.T is a sample of a correlation 3x3 matrix
            tensor([[ 1.0000,  0.0000,  0.0000],
                    [ 0.3516,  0.9361,  0.0000],
                    [-0.1899,  0.4748,  0.8593]])

        Args:
            dimension (dim): dimension of the matrices
            concentration (float or Tensor): concentration/shape parameter of the
                distribution (often referred to as eta)

        **References**

        [1] `Generating random correlation matrices based on vines and extended onion method` (2009),
        Daniel Lewandowski, Dorota Kurowicka, Harry Joe.
        Journal of Multivariate Analysis. 100. 10.1016/j.jmva.2009.04.008
        """
        arg_constraints = {"concentration": constraints.positive}
        support = constraints.corr_cholesky

        def __init__(self, dim, concentration=1.0, validate_args=None):
            if dim < 2:
                raise ValueError(
                    f"Expected dim to be an integer greater than or equal to 2. Found dim={dim}."
                )
            self.dim = dim
            (self.concentration,) = broadcast_all(concentration)
            batch_shape = self.concentration.size()
            event_shape = torch.Size((dim, dim))
            # This is used to draw vectorized samples from the beta distribution in Sec. 3.2 of [1].
            marginal_conc = self.concentration + 0.5 * (self.dim - 2)
            offset = torch.arange(
                self.dim - 1,
                dtype=self.concentration.dtype,
                device=self.concentration.device,
            )
            offset = torch.cat([offset.new_zeros((1,)), offset])
            beta_conc1 = offset + 0.5
            beta_conc0 = marginal_conc.unsqueeze(-1) - 0.5 * offset
            self._beta = Beta(beta_conc1, beta_conc0)
            super().__init__(batch_shape, event_shape, validate_args)

        def expand(self, batch_shape, _instance=None):
            new = self._get_checked_instance(LKJCholesky, _instance)
            batch_shape = torch.Size(batch_shape)
            new.dim = self.dim
            new.concentration = self.concentration.expand(batch_shape)
            new._beta = self._beta.expand(batch_shape + (self.dim,))
            super(LKJCholesky, new).__init__(
                batch_shape, self.event_shape, validate_args=False
            )
            new._validate_args = self._validate_args
            return new

        def sample(self, sample_shape=torch.Size()):
            # This uses the Onion method, but there are a few differences from [1] Sec. 3.2:
            # - This vectorizes the for loop and also works for heterogeneous eta.
            # - Same algorithm generalizes to n=1.
            # - The procedure is simplified since we are sampling the cholesky factor of
            #   the correlation matrix instead of the correlation matrix itself. As such,
            #   we only need to generate `w`.
            y = self._beta.sample(sample_shape).unsqueeze(-1)
            u_normal = torch.randn(
                self._extended_shape(sample_shape), dtype=y.dtype, device=y.device
            ).tril(-1)
            u_hypersphere = u_normal / u_normal.norm(dim=-1, keepdim=True)
            # Replace NaNs in first row
            u_hypersphere[..., 0, :].fill_(0.0)
            w = torch.sqrt(y) * u_hypersphere
            # Fill diagonal elements; clamp for numerical stability
            eps = torch.finfo(w.dtype).tiny
            diag_elems = torch.clamp(1 - torch.sum(w**2, dim=-1), min=eps).sqrt()
            w += torch.diag_embed(diag_elems)
            return w

        def log_prob(self, value):
            # See: https://mc-stan.org/docs/2_25/functions-reference/cholesky-lkj-correlation-distribution.html
            # The probability of a correlation matrix is proportional to
            #   determinant ** (concentration - 1) = prod(L_ii ^ 2(concentration - 1))
            # Additionally, the Jacobian of the transformation from Cholesky factor to
            # correlation matrix is:
            #   prod(L_ii ^ (D - i))
            # So the probability of a Cholesky factor is propotional to
            #   prod(L_ii ^ (2 * concentration - 2 + D - i)) = prod(L_ii ^ order_i)
            # with order_i = 2 * concentration - 2 + D - i
            if self._validate_args:
                self._validate_sample(value)
            diag_elems = value.diagonal(dim1=-1, dim2=-2)[..., 1:]
            order = torch.arange(2, self.dim + 1, device=self.concentration.device)
            order = 2 * (self.concentration - 1).unsqueeze(-1) + self.dim - order
            unnormalized_log_pdf = torch.sum(order * diag_elems.log(), dim=-1)
            # Compute normalization constant (page 1999 of [1])
            dm1 = self.dim - 1
            alpha = self.concentration + 0.5 * dm1
            denominator = torch.lgamma(alpha) * dm1
            numerator = torch.mvlgamma(alpha - 0.5, dm1)
            # pi_constant in [1] is D * (D - 1) / 4 * log(pi)
            # pi_constant in multigammaln is (D - 1) * (D - 2) / 4 * log(pi)
            # hence, we need to add a pi_constant = (D - 1) * log(pi) / 2
            pi_constant = 0.5 * dm1 * math.log(math.pi)
            normalize_term = pi_constant + numerator - denominator
            return unnormalized_log_pdf - normalize_term

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