code: def multipath_pathfinder(
      *,
      target_density: Callable[[jnp.ndarray], jnp.ndarray],
      initial_values: jnp.ndarray,
      lbfgs_max_iters: int,  # L
      key,
      lbfgs_relative_tolerance: float = 1e-13,  # tau_rel
      lbfgs_inverse_hessian_history: int = 6,  # J
      monte_carlo_elbo_draws: int = 10,  # K
      num_pathfinder_draws: int = 10,  # M
      num_draws: int,  # R
  ):
      """Multi-pathfinder algorithm."""
      ps_ir_key, *keys = jax.random.split(key, len(initial_values) + 1)
      draws, q_probs, p_probs = [], [], []
      for key, initial_value in zip(keys, initial_values):
          draw, prob = pathfinder(
              target_density=target_density,
              initial_value=initial_value,
              lbfgs_max_iters=lbfgs_max_iters,
              key=key,
              lbfgs_relative_tolerance=lbfgs_relative_tolerance,
              lbfgs_inverse_hessian_history=lbfgs_inverse_hessian_history,
              monte_carlo_elbo_draws=monte_carlo_elbo_draws,
              num_draws=num_pathfinder_draws,
          )
          draws.append(draw)
          q_probs.append(prob)
          p_probs.append(target_density(draw))
      constant = jnp.log(len(initial_values))
      return ps_ir(
          draws=jnp.concatenate(draws),
          proposal_densities=jnp.concatenate(q_probs) - constant,
          target_densities=jnp.concatenate(p_probs) - constant,
          num_resampled=num_draws,
          key=ps_ir_key,
      )
  
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 2 for calling a global function with example usage: multipath_pathfinder() and returns: ps_ir