code: def gaussian_tensordot(
      x: Gaussian, y: Gaussian, dims: int = 0
  ) -> Gaussian:
      """
      Computes the integral over two gaussians:
  
          `(x @ y)(a,c) = log(integral(exp(x(a,b) + y(b,c)), b))`,
  
      where `x` is a gaussian over variables (a,b), `y` is a gaussian over variables
      (b,c), (a,b,c) can each be sets of zero or more variables, and `dims` is the size of b.
  
      :param x: a Gaussian instance
      :param y: a Gaussian instance
      :param dims: number of variables to contract
      """
      assert isinstance(x, Gaussian)
      assert isinstance(y, Gaussian)
      na = x.dim() - dims
      nb = dims
      nc = y.dim() - dims
      assert na >= 0
      assert nb >= 0
      assert nc >= 0
  
      Paa, Pba, Pbb = (
          x.precision[..., :na, :na],
          x.precision[..., na:, :na],
          x.precision[..., na:, na:],
      )
      Qbb, Qbc, Qcc = (
          y.precision[..., :nb, :nb],
          y.precision[..., :nb, nb:],
          y.precision[..., nb:, nb:],
      )
      xa, xb = (
          x.info_vec[..., :na],
          x.info_vec[..., na:],
      )  # x.precision @ x.mean
      yb, yc = (
          y.info_vec[..., :nb],
          y.info_vec[..., nb:],
      )  # y.precision @ y.mean
  
      precision = pad(Paa, (0, nc, 0, nc)) + pad(Qcc, (na, 0, na, 0))
      info_vec = pad(xa, (0, nc)) + pad(yc, (na, 0))
      log_normalizer = x.log_normalizer + y.log_normalizer
      if nb > 0:
          B = pad(Pba, (0, nc)) + pad(Qbc, (na, 0))
          b = xb + yb
  
          # Pbb + Qbb needs to be positive definite, so that we can malginalize out `b` (to have a finite integral)
          L = safe_cholesky(Pbb + Qbb)
          LinvB = triangular_solve(B, L, upper=False)
          LinvBt = LinvB.transpose(-2, -1)
          Linvb = triangular_solve(b.unsqueeze(-1), L, upper=False)
  
          precision = precision - matmul(LinvBt, LinvB)
          # NB: precision might not be invertible for getting mean = precision^-1 @ info_vec
          if na + nc > 0:
              info_vec = info_vec - matmul(LinvBt, Linvb).squeeze(-1)
          logdet = torch.diagonal(L, dim1=-2, dim2=-1).log().sum(-1)
          diff = (
              0.5 * nb * math.log(2 * math.pi)
              + 0.5 * Linvb.squeeze(-1).pow(2).sum(-1)
              - logdet
          )
          log_normalizer = log_normalizer + diff
  
      return Gaussian(log_normalizer, info_vec, precision)
  
imports: import math
  from typing import Optional, Tuple, Union
  import torch
  from torch.distributions.utils import lazy_property
  from torch.nn.functional import pad
  from pyro.distributions.util import broadcast_shape
  from pyro.ops.tensor_utils import matmul, matvecmul, safe_cholesky, triangular_solve
label_desc: Use image node_id 4 for calling a global function with example usage: gaussian_tensordot(x, y, dims) and returns: Gaussian