code: def cov_estimate(
      *,
      optimization_path: Sequence[jnp.ndarray],
      optimization_path_grads: Sequence[jnp.ndarray],
      history: int,
  ):
      """Estimate covariance from an optimization path."""
      dim = optimization_path[0].shape[0]
      position_diffs = jnp.empty((dim, 0))
      gradient_diffs = jnp.empty((dim, 0))
      approximations: List[
          Tuple[jnp.ndarray, jnp.ndarray, jnp.ndarray]
      ] = []
      diagonal_estimate = jnp.ones(dim)
      for j in range(len(optimization_path) - 1):
          _, thin_factors, scaling_outer_product = bfgs_inverse_hessian(
              updates_of_position_differences=position_diffs,
              updates_of_gradient_differences=gradient_diffs,
          )
          position_diff = (
              optimization_path[j + 1] - optimization_path[j]
          )
          gradient_diff = (
              optimization_path_grads[j]
              - optimization_path_grads[j + 1]
          )
          b = position_diff @ gradient_diff
          gradient_diff_norm = gradient_diff**2
          new_diagonal_estimate = diagonal_estimate
          if b < 1e-12 * jnp.sum(gradient_diff_norm):
              position_diffs = jnp.column_stack(
                  (position_diffs[:, -history + 1 :], position_diff)
              )
              gradient_diffs = jnp.column_stack(
                  (gradient_diffs[:, -history + 1 :], gradient_diff)
              )
              a = gradient_diff @ (diagonal_estimate * gradient_diff)
              c = position_diff @ (position_diff / diagonal_estimate)
              new_diagonal_estimate = 1.0 / (
                  a / (b * diagonal_estimate)
                  + gradient_diff_norm / b
                  - (a * position_diff**2)
                  / (b * c * diagonal_estimate**2)
              )
          approximations.append(
              (diagonal_estimate, thin_factors, scaling_outer_product)
          )
          diagonal_estimate = new_diagonal_estimate
      return approximations
  
file_path: probability/discussion/pathfinder/pathfinder.py
imports: from typing import Tuple, List, Sequence, Callable
  import jax
  import jax.numpy
  import numpy
label_desc: Use image node_id 8 for calling a global function with example usage: cov_estimate() and returns: approximations