code: def importance_corrected_td_errors(
      r_t: Array,
      discount_t: Array,
      rho_tm1: Array,
      lambda_: Array,
      values: Array,
      stop_target_gradients: bool = False,
  ) -> Array:
      """Computes the multistep td errors with per decision importance sampling.
  
      Given a trajectory of length `T+1`, generated under some policy π, for each
      time-step `t` we can estimate a multistep temporal difference error δₜ(ρ,λ),
      by combining rewards, discounts, and state values, according to a mixing
      parameter `λ` and importance sampling ratios ρₜ = π(aₜ|sₜ) / μ(aₜ|sₜ):
  
        td-errorₜ = ρₜ δₜ(ρ,λ)
        δₜ(ρ,λ) = δₜ + ρₜ₊₁ λₜ₊₁ γₜ₊₁ δₜ₊₁(ρ,λ),
  
      where δₜ = rₜ₊₁ + γₜ₊₁ vₜ₊₁ - vₜ is the one step, temporal difference error
      for the agent's state value estimates. This is equivalent to computing
      the λ-return with λₜ = ρₜ (e.g. using the `lambda_returns` function from
      above), and then computing errors as  td-errorₜ = ρₜ(Gₜ - vₜ).
  
      See "A new Q(λ) with interim forward view and Monte Carlo equivalence"
      by Sutton et al. (http://proceedings.mlr.press/v32/sutton14.html).
  
      Args:
        r_t: sequence of rewards rₜ for timesteps t in [1, T].
        discount_t: sequence of discounts γₜ for timesteps t in [1, T].
        rho_tm1: sequence of importance ratios for all timesteps t in [0, T-1].
        lambda_: mixing parameter; scalar or have per timestep values in [1, T].
        values: sequence of state values under π for all timesteps t in [0, T].
        stop_target_gradients: bool indicating whether or not to apply stop gradient
          to targets.
  
      Returns:
        Off-policy estimates of the multistep td errors.
      """
      chex.assert_rank([r_t, discount_t, rho_tm1, values], [1, 1, 1, 1])
      chex.assert_type([r_t, discount_t, rho_tm1, values], float)
      chex.assert_equal_shape([r_t, discount_t, rho_tm1, values[1:]])
  
      v_tm1 = values[:-1]  # Predictions to compute errors for.
      v_t = values[1:]  # Values for bootstrapping.
      rho_t = jnp.concatenate(
          (rho_tm1[1:], jnp.array([1.0]))
      )  # Unused dummy value.
      lambda_ = (
          jnp.ones_like(discount_t) * lambda_
      )  # If scalar, make into vector.
  
      # Compute the one step temporal difference errors.
      one_step_delta = r_t + discount_t * v_t - v_tm1
  
      # Work backwards to compute `delta_{T-1}`, ..., `delta_0`.
      def _body(acc, xs):
          deltas, discounts, rho_t, lambda_ = xs
          acc = deltas + discounts * rho_t * lambda_ * acc
          return acc, acc
  
      _, errors = jax.lax.scan(
          _body,
          0.0,
          (one_step_delta, discount_t, rho_t, lambda_),
          reverse=True,
      )
  
      errors = rho_tm1 * errors
      return jax.lax.select(
          stop_target_gradients,
          jax.lax.stop_gradient(errors + v_tm1) - v_tm1,
          errors,
      )
  
file_path: rlax/rlax/_src/multistep.py
imports: from typing import Union
  import chex
  import jax
  import jax.numpy
  from rlax._src import base
label_desc: Use image node_id 4 for calling a global function with example usage: importance_corrected_td_errors(r_t, discount_t, rho_tm1, lambda_, values, stop_target_gradients) and returns: jax, acc,  acc