# Path: blackjax/mcmc/integrators.py
# class IntegratorState(NamedTuple):
#     """State of the trajectory integration.
# 
#     We keep the gradient of the potential energy to speedup computations.
#     """
# 
#     position: PyTree
#     momentum: PyTree
#     potential_energy: float
#     potential_energy_grad: PyTree
# 
# Path: blackjax/mcmc/proposal.py
# class Proposal(NamedTuple):
#     """Proposal for the next chain step.
#     state:
#         The trajectory state corresponding to this proposal.
#     energy:
#         The potential energy corresponding to the state.
#     weight:
#         Weight of the proposal. It is equal to the logarithm of the sum of the canonical
#         densities of each state :math:`e^{-H(z)}` along the trajectory.
#     sum_log_p_accept:
#         cumulated Metropolis-Hastings acceptance probabilty across entire trajectory.
#     """
# 
#     state: IntegratorState
#     energy: float
#     weight: float
#     sum_log_p_accept: float
# 
# def progressive_biased_sampling(rng_key, proposal, new_proposal):
#     """Baised proposal sampling.
# 
#     Unlike uniform sampling, biased sampling favors new proposals. It thus
#     biases the transition away from the trajectory's initial state.
# 
#     """
#     p_accept = jnp.clip(jnp.exp(new_proposal.weight - proposal.weight), a_max=1)
#     do_accept = jax.random.bernoulli(rng_key, p_accept)
#     new_weight = jnp.logaddexp(proposal.weight, new_proposal.weight)
#     new_sum_log_p_accept = jnp.logaddexp(
#         proposal.sum_log_p_accept, new_proposal.sum_log_p_accept
#     )
# 
#     return jax.lax.cond(
#         do_accept,
#         lambda _: Proposal(
#             new_proposal.state,
#             new_proposal.energy,
#             new_weight,
#             new_sum_log_p_accept,
#         ),
#         lambda _: Proposal(
#             proposal.state,
#             proposal.energy,
#             new_weight,
#             new_sum_log_p_accept,
#         ),
#         operand=None,
#     )
# 
# def progressive_uniform_sampling(rng_key, proposal, new_proposal):
#     # Using expit to compute exp(w1) / (exp(w0) + exp(w1))
#     p_accept = jax.scipy.special.expit(new_proposal.weight - proposal.weight)
#     do_accept = jax.random.bernoulli(rng_key, p_accept)
#     new_weight = jnp.logaddexp(proposal.weight, new_proposal.weight)
#     new_sum_log_p_accept = jnp.logaddexp(
#         proposal.sum_log_p_accept, new_proposal.sum_log_p_accept
#     )
# 
#     return jax.lax.cond(
#         do_accept,
#         lambda _: Proposal(
#             new_proposal.state,
#             new_proposal.energy,
#             new_weight,
#             new_sum_log_p_accept,
#         ),
#         lambda _: Proposal(
#             proposal.state,
#             proposal.energy,
#             new_weight,
#             new_sum_log_p_accept,
#         ),
#         operand=None,
#     )
# 
# def proposal_generator(
#     kinetic_energy: Callable, divergence_threshold: float
# ) -> Tuple[Callable, Callable]:
#     def new(state: IntegratorState) -> Proposal:
#         energy = state.potential_energy + kinetic_energy(state.momentum)
#         return Proposal(state, energy, 0.0, -np.inf)
# 
#     def update(initial_energy: float, state: IntegratorState) -> Tuple[Proposal, bool]:
#         """Generate a new proposal from a trajectory state.
# 
#         The trajectory state records information about the position in the state
#         space and corresponding potential energy. A proposal also carries a
#         weight that is equal to the difference between the current energy and
#         the previous one. It thus carries information about the previous state
#         as well as the current state.
# 
#         Parameters
#         ----------
#         initial_energy:
#             The initial energy.
#         state:
#             The new state.
# 
#         """
#         new_energy = state.potential_energy + kinetic_energy(state.momentum)
# 
#         delta_energy = initial_energy - new_energy
#         delta_energy = jnp.where(jnp.isnan(delta_energy), -jnp.inf, delta_energy)
#         is_transition_divergent = jnp.abs(delta_energy) > divergence_threshold
# 
#         # The weight of the new proposal is equal to H0 - H(z_new)
#         weight = delta_energy
#         # Acceptance statistic min(e^{H0 - H(z_new)}, 1)
#         sum_log_p_accept = jnp.minimum(delta_energy, 0.0)
# 
#         return (
#             Proposal(
#                 state,
#                 new_energy,
#                 weight,
#                 sum_log_p_accept,
#             ),
#             is_transition_divergent,
#         )
# 
#     return new, update
# 
# Path: blackjax/types.py



from typing import Callable, NamedTuple, Tuple
from blackjax.mcmc.integrators import IntegratorState
from blackjax.mcmc.proposal import (
    Proposal,
    progressive_biased_sampling,
    progressive_uniform_sampling,
    proposal_generator,
)
from blackjax.types import PRNGKey, PyTree
import jax
import jax.numpy as jnp
    max_num_expansions: int = 10,
    rate: int = 2,
) -> Callable:
    """Sample a trajectory and update the proposal sequentially
    until the termination criterion is met.

    The trajectory is sampled with the following procedure:
    1. Pick a direction at random;
    2. Integrate `num_step` steps in this direction;
    3. If the integration has stopped prematurely, do not update the proposal;
    4. Else if the trajectory is performing a U-turn, return current proposal;
    5. Else update proposal, `num_steps = num_steps ** rate` and repeat from (1).

    Parameters
    ----------
    trajectory_integrator
        A function that runs the symplectic integrators and returns a new proposal
        and the integrated trajectory.
    uturn_check_fn
        Function used to check the U-Turn criterion.
    step_size
        The step size used by the symplectic integrator.
    max_num_expansions
        The maximum number of trajectory expansions until the proposal is
        returned.
    rate
        The rate of the geometrical expansion. Typically 2 in NUTS, this is why
        the literature often refers to "tree doubling".

    """

# Path: blackjax/mcmc/integrators.py
# class IntegratorState(NamedTuple):
#     """State of the trajectory integration.
# 
#     We keep the gradient of the potential energy to speedup computations.
#     """
# 
#     position: PyTree
#     momentum: PyTree
#     potential_energy: float
#     potential_energy_grad: PyTree
# 
# Path: blackjax/mcmc/proposal.py
# class Proposal(NamedTuple):
#     """Proposal for the next chain step.
#     state:
#         The trajectory state corresponding to this proposal.
#     energy:
#         The potential energy corresponding to the state.
#     weight:
#         Weight of the proposal. It is equal to the logarithm of the sum of the canonical
#         densities of each state :math:`e^{-H(z)}` along the trajectory.
#     sum_log_p_accept:
#         cumulated Metropolis-Hastings acceptance probabilty across entire trajectory.
#     """
# 
#     state: IntegratorState
#     energy: float
#     weight: float
#     sum_log_p_accept: float
# 
# def progressive_biased_sampling(rng_key, proposal, new_proposal):
#     """Baised proposal sampling.
# 
#     Unlike uniform sampling, biased sampling favors new proposals. It thus
#     biases the transition away from the trajectory's initial state.
# 
#     """
#     p_accept = jnp.clip(jnp.exp(new_proposal.weight - proposal.weight), a_max=1)
#     do_accept = jax.random.bernoulli(rng_key, p_accept)
#     new_weight = jnp.logaddexp(proposal.weight, new_proposal.weight)
#     new_sum_log_p_accept = jnp.logaddexp(
#         proposal.sum_log_p_accept, new_proposal.sum_log_p_accept
#     )
# 
#     return jax.lax.cond(
#         do_accept,
#         lambda _: Proposal(
#             new_proposal.state,
#             new_proposal.energy,
#             new_weight,
#             new_sum_log_p_accept,
#         ),
#         lambda _: Proposal(
#             proposal.state,
#             proposal.energy,
#             new_weight,
#             new_sum_log_p_accept,
#         ),
#         operand=None,
#     )
# 
# def progressive_uniform_sampling(rng_key, proposal, new_proposal):
#     # Using expit to compute exp(w1) / (exp(w0) + exp(w1))
#     p_accept = jax.scipy.special.expit(new_proposal.weight - proposal.weight)
#     do_accept = jax.random.bernoulli(rng_key, p_accept)
#     new_weight = jnp.logaddexp(proposal.weight, new_proposal.weight)
#     new_sum_log_p_accept = jnp.logaddexp(
#         proposal.sum_log_p_accept, new_proposal.sum_log_p_accept
#     )
# 
#     return jax.lax.cond(
#         do_accept,
#         lambda _: Proposal(
#             new_proposal.state,
#             new_proposal.energy,
#             new_weight,
#             new_sum_log_p_accept,
#         ),
#         lambda _: Proposal(
#             proposal.state,
#             proposal.energy,
#             new_weight,
#             new_sum_log_p_accept,
#         ),
#         operand=None,
#     )
# 
# def proposal_generator(
#     kinetic_energy: Callable, divergence_threshold: float
# ) -> Tuple[Callable, Callable]:
#     def new(state: IntegratorState) -> Proposal:
#         energy = state.potential_energy + kinetic_energy(state.momentum)
#         return Proposal(state, energy, 0.0, -np.inf)
# 
#     def update(initial_energy: float, state: IntegratorState) -> Tuple[Proposal, bool]:
#         """Generate a new proposal from a trajectory state.
# 
#         The trajectory state records information about the position in the state
#         space and corresponding potential energy. A proposal also carries a
#         weight that is equal to the difference between the current energy and
#         the previous one. It thus carries information about the previous state
#         as well as the current state.
# 
#         Parameters
#         ----------
#         initial_energy:
#             The initial energy.
#         state:
#             The new state.
# 
#         """
#         new_energy = state.potential_energy + kinetic_energy(state.momentum)
# 
#         delta_energy = initial_energy - new_energy
#         delta_energy = jnp.where(jnp.isnan(delta_energy), -jnp.inf, delta_energy)
#         is_transition_divergent = jnp.abs(delta_energy) > divergence_threshold
# 
#         # The weight of the new proposal is equal to H0 - H(z_new)
#         weight = delta_energy
#         # Acceptance statistic min(e^{H0 - H(z_new)}, 1)
#         sum_log_p_accept = jnp.minimum(delta_energy, 0.0)
# 
#         return (
#             Proposal(
#                 state,
#                 new_energy,
#                 weight,
#                 sum_log_p_accept,
#             ),
#             is_transition_divergent,
#         )
# 
#     return new, update
# 
# Path: blackjax/types.py



# Path: blackjax/mcmc/trajectory.py
from typing import Callable, NamedTuple, Tuple
from blackjax.mcmc.integrators import IntegratorState
from blackjax.mcmc.proposal import (
    Proposal,
    progressive_biased_sampling,
    progressive_uniform_sampling,
    proposal_generator,
)
from blackjax.types import PRNGKey, PyTree
import jax
import jax.numpy as jnp

    max_num_expansions: int = 10,
    rate: int = 2,
) -> Callable:
    """Sample a trajectory and update the proposal sequentially
    until the termination criterion is met.

    The trajectory is sampled with the following procedure:
    1. Pick a direction at random;
    2. Integrate `num_step` steps in this direction;
    3. If the integration has stopped prematurely, do not update the proposal;
    4. Else if the trajectory is performing a U-turn, return current proposal;
    5. Else update proposal, `num_steps = num_steps ** rate` and repeat from (1).

    Parameters
    ----------
    trajectory_integrator
        A function that runs the symplectic integrators and returns a new proposal
        and the integrated trajectory.
    uturn_check_fn
        Function used to check the U-Turn criterion.
    step_size
        The step size used by the symplectic integrator.
    max_num_expansions
        The maximum number of trajectory expansions until the proposal is
        returned.
    rate
        The rate of the geometrical expansion. Typically 2 in NUTS, this is why
        the literature often refers to "tree doubling".

    """

    proposal_sampler = progressive_biased_sampling