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pfjax

Modules:

Name Description
experimental
loglik_full
mcmc

MCMC algorithms for state space models.
models
mvn_bridge

Multivariate normal bridge proposals.
neg_loglik
particle_filter

Particle filters which approximate the score and fisher information.
particle_resamplers

Particle resamplers.
particle_smooth
sde

Generic methods for SDEs.
simulate
test
utils

Utility functions.

Classes:

Name Description
BaseModel

Base model for particle filters.

Functions:

Name Description
particle_filter_rb

Rao-Blackwellized particle filter.

BaseModel

Bases: object

Base model for particle filters.

This class sets up a PF model from small set of methods:

  • The derived class should provide methods state_lpdf(), meas_lpdf(), state_sample() and meas_sample() in order calculate the complete data likelihood pfjax.loglik_full() and simulate data from the model via pfjax.simulate().

  • To use the "basic" particle filter pfjax.particle_filter(), the derived class must provide methods pf_init() and pf_step().

  • To use the Rao-Blackwellized particle filter pfjax.particle_filter_rb(), the derived class must provide methods pf_init(), step_sample(), and step_lpdf().

  • If pf_init() is missing, the base class will automatically construct it from prior_lpdf(), init_sample(), and init_lpdf().

  • if pf_step() is missing, the base class will automatically construct it from state_lpdf(), meas_lpdf(), step_sample(), and step_lpdf().

  • If in either of the above step_sample() and step_lpdf() are missing, the base class assumes a bootstrap particle filter and sets step_sample = state_sample and step_lpdf = state_lpdf.

  • If in either of the above init_sample() and init_lpdf() are missing, the base calss sets init_sample = prior_sample and init_lpdf = prior_lpdf.

Notes:

  • pf_step() and pf_init() could be computed more efficiently for bootstrap sampling. Perhaps this could be specified with an argument bootstrap to the constructor.

  • In general, nothing is stopping the user from creating e.g., pf_step() which is inconsistent with step_sample(), etc.

Parameters:

Name Type Description Default
bootstrap

Boolean for whether or not to create a bootstrap particle filter.

 required

Methods:

Name Description
step_sample

Sample from default proposal distribution
step_lpdf

Calculate log-density of the default proposal distribution
init_sample

Sample from default initial proposal distribution
init_lpdf

Calculate log-density of the default proposal distribution
pf_step

Particle filter update.
pf_init

Initial step of particle filter.
Source code in src/pfjax/models/base_model.py 
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class BaseModel(object):
    r"""
    Base model for particle filters.

    This class sets up a PF model from small set of methods:

    - The derived class should provide methods `state_lpdf()`, `meas_lpdf()`, `state_sample()` and `meas_sample()` in order calculate the complete data likelihood `pfjax.loglik_full()` and simulate data from the model via `pfjax.simulate()`.

    - To use the "basic" particle filter `pfjax.particle_filter()`, the derived class must provide methods `pf_init()` and `pf_step()`.

    - To use the Rao-Blackwellized particle filter `pfjax.particle_filter_rb()`, the derived class must provide methods `pf_init()`, `step_sample()`, and `step_lpdf()`.

    - If `pf_init()` is missing, the base class will automatically construct it from `prior_lpdf()`, `init_sample()`, and `init_lpdf()`.

    - if `pf_step()` is missing, the base class will automatically construct it from `state_lpdf()`, `meas_lpdf()`, `step_sample()`, and `step_lpdf()`.

    - If in either of the above `step_sample()` and `step_lpdf()` are missing, the base class assumes a bootstrap particle filter and sets `step_sample = state_sample` and `step_lpdf = state_lpdf`.

    - If in either of the above `init_sample()` and `init_lpdf()` are missing, the base calss sets `init_sample = prior_sample` and `init_lpdf = prior_lpdf`.

    **Notes:**

    - `pf_step()` and `pf_init()` could be computed more efficiently for bootstrap sampling.  Perhaps this could be specified with an argument `bootstrap` to the constructor.  

    - In general, nothing is stopping the user from creating e.g., `pf_step()` which is inconsistent with `step_sample()`, etc.

    Args:
        bootstrap: Boolean for whether or not to create a bootstrap particle filter.

    """

    def __init__(self, bootstrap):
        self._bootstrap = bootstrap

    def step_sample(self, key, x_prev, y_curr, theta):
        r"""
        Sample from default proposal distribution

        ::

            q(x_curr | x_prev, y_curr, theta) = p(x_curr | x_prev, theta)

        Args:
            key: PRNG key.
            x_prev: State variable at previous time `t-1`.
            y_curr: Measurement variable at current time `t`.
            theta: Parameter value.

        Returns:
            Sample of the state variable `x_curr` at current time `t`.
        """
        if self._bootstrap:
            return self.state_sample(key=key, x_prev=x_prev, theta=theta)
        else:
            pass

    def step_lpdf(self, x_curr, x_prev, y_curr, theta):
        r"""
        Calculate log-density of the default proposal distribution

        ::

            q(x_curr | x_prev, y_curr, theta) = p(x_curr | x_prev, theta)

        Args:
            x_curr: State variable at current time `t`.
            x_prev: State variable at previous time `t-1`.
            y_curr: Measurement variable at current time `t`.
            theta: Parameter value.

        Returns:
            Log-density of the state variable `x_curr` at current time `t`.
        """
        if self._bootstrap:
            return self.state_lpdf(x_curr=x_curr, x_prev=x_prev, theta=theta)
        else:
            pass

    def init_sample(self, key, y_init, theta):
        r"""
        Sample from default initial proposal distribution

        ::

            q(x_init | y_init, theta) = p(x_init | theta)

        Args:
            key: PRNG key.
            y_init: Measurement variable at initial time `t = 0`.
            theta: Parameter value.

        Returns:
            Sample of the state variable `x_init` at initial time `t = 0`.
        """
        if self._bootstrap:
            return self.prior_sample(key=key, theta=theta)
        else:
            pass

    def init_lpdf(self, x_init, y_init, theta):
        r"""
        Calculate log-density of the default proposal distribution

        ::

            q(x_curr | x_prev, y_curr, theta) = p(x_curr | x_prev, theta)

        Args:
            x_init: State variable at initial time `t = 0`.
            y_init: Measurement variable at initial time `t = 0`.
            theta: Parameter value.

        Returns:
            Log-density of the state variable `x_init` at initial time `t = 0`.
        """
        if self._bootstrap:
            return self.prior_lpdf(x_init=x_init, theta=theta)
        else:
            pass

    def pf_step(self, key, x_prev, y_curr, theta):
        r"""
        Particle filter update.

        Returns a draw from proposal distribution

        ::

            x_curr ~ q(x_curr) = q(x_curr | x_prev, y_curr, theta)

        and the log weight

        ::

            logw = log p(y_curr | x_curr, theta) + log p(x_curr | x_prev, theta) - log q(x_curr)

        Args:
            key: PRNG key.
            x_prev: State variable at previous time `t-1`.
            y_curr: Measurement variable at current time `t`.
            theta: Parameter value.

        Returns:
            Tuple:

            - **x_curr** - A sample from the proposal distribution at current time `t`.
            - **logw** - The log-weight of `x_curr`.
        """

        if self._bootstrap:
            x_curr = self.state_sample(key=key, x_prev=x_prev, theta=theta)
            logw = self.meas_lpdf(y_curr=y_curr, x_curr=x_curr, theta=theta)
        else:
            x_curr = self.step_sample(key=key, x_prev=x_prev,
                                      y_curr=y_curr, theta=theta)
            lp_prop = self.step_lpdf(x_curr=x_curr,
                                     x_prev=x_prev, y_curr=y_curr, theta=theta)
            lp_targ = self.state_lpdf(
                x_curr=x_curr, x_prev=x_prev, theta=theta
            ) + self.meas_lpdf(y_curr=y_curr, x_curr=x_curr, theta=theta)
            logw = lp_targ - lp_prop
        return x_curr, logw

    def pf_init(self, key, y_init, theta):
        r"""
        Initial step of particle filter.

        Returns a draw from the proposal distribution

        ::

            x_init ~ q(x_init) = q(x_init | y_init, theta)

        and calculates the log weight

        ::

            logw = log p(y_init | x_init, theta) + log p(x_init | theta) - log q(x_init)

        Args:
            key: PRNG key.
            y_init: Measurement variable at initial time `t = 0`.
            theta: Parameter value.

        Returns:
            Tuple:

            - **x_init** - A sample from the proposal distribution at initial tme `t = 0`.
            - **logw** - The log-weight of `x_init`.
        """
        if self._bootstrap:
            x_curr = self.prior_sample(key=key, theta=theta)
            logw = self.meas_lpdf(y_curr=y_init, x_curr=x_curr, theta=theta)
        else:
            x_curr = self.init_sample(key=key, theta=theta)
            lp_prop = self.init_lpdf(x_curr=x_curr, theta=theta)
            lp_targ = self.prior_lpdf(x_curr=x_curr, theta=theta) + \
                self.meas_lpdf(y_curr=y_init, x_curr=x_curr, theta=theta)
            logw = lp_targ - lp_prop
        return x_curr, logw

step_sample(key, x_prev, y_curr, theta)

Sample from default proposal distribution

::

q(x_curr | x_prev, y_curr, theta) = p(x_curr | x_prev, theta)

Parameters:

Name Type Description Default
key

PRNG key.

 required
x_prev

State variable at previous time t-1.

 required
y_curr

Measurement variable at current time t.

 required
theta

Parameter value.

 required

Returns:

Type Description

Sample of the state variable x_curr at current time t.
Source code in src/pfjax/models/base_model.py 
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def step_sample(self, key, x_prev, y_curr, theta):
    r"""
    Sample from default proposal distribution

    ::

        q(x_curr | x_prev, y_curr, theta) = p(x_curr | x_prev, theta)

    Args:
        key: PRNG key.
        x_prev: State variable at previous time `t-1`.
        y_curr: Measurement variable at current time `t`.
        theta: Parameter value.

    Returns:
        Sample of the state variable `x_curr` at current time `t`.
    """
    if self._bootstrap:
        return self.state_sample(key=key, x_prev=x_prev, theta=theta)
    else:
        pass

step_lpdf(x_curr, x_prev, y_curr, theta)

Calculate log-density of the default proposal distribution

::

q(x_curr | x_prev, y_curr, theta) = p(x_curr | x_prev, theta)

Parameters:

Name Type Description Default
x_curr

State variable at current time t.

 required
x_prev

State variable at previous time t-1.

 required
y_curr

Measurement variable at current time t.

 required
theta

Parameter value.

 required

Returns:

Type Description

Log-density of the state variable x_curr at current time t.
Source code in src/pfjax/models/base_model.py 
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def step_lpdf(self, x_curr, x_prev, y_curr, theta):
    r"""
    Calculate log-density of the default proposal distribution

    ::

        q(x_curr | x_prev, y_curr, theta) = p(x_curr | x_prev, theta)

    Args:
        x_curr: State variable at current time `t`.
        x_prev: State variable at previous time `t-1`.
        y_curr: Measurement variable at current time `t`.
        theta: Parameter value.

    Returns:
        Log-density of the state variable `x_curr` at current time `t`.
    """
    if self._bootstrap:
        return self.state_lpdf(x_curr=x_curr, x_prev=x_prev, theta=theta)
    else:
        pass

init_sample(key, y_init, theta)

Sample from default initial proposal distribution

::

q(x_init | y_init, theta) = p(x_init | theta)

Parameters:

Name Type Description Default
key

PRNG key.

 required
y_init

Measurement variable at initial time t = 0.

 required
theta

Parameter value.

 required

Returns:

Type Description

Sample of the state variable x_init at initial time t = 0.
Source code in src/pfjax/models/base_model.py 
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def init_sample(self, key, y_init, theta):
    r"""
    Sample from default initial proposal distribution

    ::

        q(x_init | y_init, theta) = p(x_init | theta)

    Args:
        key: PRNG key.
        y_init: Measurement variable at initial time `t = 0`.
        theta: Parameter value.

    Returns:
        Sample of the state variable `x_init` at initial time `t = 0`.
    """
    if self._bootstrap:
        return self.prior_sample(key=key, theta=theta)
    else:
        pass

init_lpdf(x_init, y_init, theta)

Calculate log-density of the default proposal distribution

::

q(x_curr | x_prev, y_curr, theta) = p(x_curr | x_prev, theta)

Parameters:

Name Type Description Default
x_init

State variable at initial time t = 0.

 required
y_init

Measurement variable at initial time t = 0.

 required
theta

Parameter value.

 required

Returns:

Type Description

Log-density of the state variable x_init at initial time t = 0.
Source code in src/pfjax/models/base_model.py 
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def init_lpdf(self, x_init, y_init, theta):
    r"""
    Calculate log-density of the default proposal distribution

    ::

        q(x_curr | x_prev, y_curr, theta) = p(x_curr | x_prev, theta)

    Args:
        x_init: State variable at initial time `t = 0`.
        y_init: Measurement variable at initial time `t = 0`.
        theta: Parameter value.

    Returns:
        Log-density of the state variable `x_init` at initial time `t = 0`.
    """
    if self._bootstrap:
        return self.prior_lpdf(x_init=x_init, theta=theta)
    else:
        pass

pf_step(key, x_prev, y_curr, theta)

Particle filter update.

Returns a draw from proposal distribution

::

x_curr ~ q(x_curr) = q(x_curr | x_prev, y_curr, theta)

and the log weight

::

logw = log p(y_curr | x_curr, theta) + log p(x_curr | x_prev, theta) - log q(x_curr)

Parameters:

Name Type Description Default
key

PRNG key.

 required
x_prev

State variable at previous time t-1.

 required
y_curr

Measurement variable at current time t.

 required
theta

Parameter value.

 required

Returns:

Name Type Description
Tuple
  • x_curr - A sample from the proposal distribution at current time t.
  • logw - The log-weight of x_curr.
Source code in src/pfjax/models/base_model.py 
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def pf_step(self, key, x_prev, y_curr, theta):
    r"""
    Particle filter update.

    Returns a draw from proposal distribution

    ::

        x_curr ~ q(x_curr) = q(x_curr | x_prev, y_curr, theta)

    and the log weight

    ::

        logw = log p(y_curr | x_curr, theta) + log p(x_curr | x_prev, theta) - log q(x_curr)

    Args:
        key: PRNG key.
        x_prev: State variable at previous time `t-1`.
        y_curr: Measurement variable at current time `t`.
        theta: Parameter value.

    Returns:
        Tuple:

        - **x_curr** - A sample from the proposal distribution at current time `t`.
        - **logw** - The log-weight of `x_curr`.
    """

    if self._bootstrap:
        x_curr = self.state_sample(key=key, x_prev=x_prev, theta=theta)
        logw = self.meas_lpdf(y_curr=y_curr, x_curr=x_curr, theta=theta)
    else:
        x_curr = self.step_sample(key=key, x_prev=x_prev,
                                  y_curr=y_curr, theta=theta)
        lp_prop = self.step_lpdf(x_curr=x_curr,
                                 x_prev=x_prev, y_curr=y_curr, theta=theta)
        lp_targ = self.state_lpdf(
            x_curr=x_curr, x_prev=x_prev, theta=theta
        ) + self.meas_lpdf(y_curr=y_curr, x_curr=x_curr, theta=theta)
        logw = lp_targ - lp_prop
    return x_curr, logw

pf_init(key, y_init, theta)

Initial step of particle filter.

Returns a draw from the proposal distribution

::

x_init ~ q(x_init) = q(x_init | y_init, theta)

and calculates the log weight

::

logw = log p(y_init | x_init, theta) + log p(x_init | theta) - log q(x_init)

Parameters:

Name Type Description Default
key

PRNG key.

 required
y_init

Measurement variable at initial time t = 0.

 required
theta

Parameter value.

 required

Returns:

Name Type Description
Tuple
  • x_init - A sample from the proposal distribution at initial tme t = 0.
  • logw - The log-weight of x_init.
Source code in src/pfjax/models/base_model.py 
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def pf_init(self, key, y_init, theta):
    r"""
    Initial step of particle filter.

    Returns a draw from the proposal distribution

    ::

        x_init ~ q(x_init) = q(x_init | y_init, theta)

    and calculates the log weight

    ::

        logw = log p(y_init | x_init, theta) + log p(x_init | theta) - log q(x_init)

    Args:
        key: PRNG key.
        y_init: Measurement variable at initial time `t = 0`.
        theta: Parameter value.

    Returns:
        Tuple:

        - **x_init** - A sample from the proposal distribution at initial tme `t = 0`.
        - **logw** - The log-weight of `x_init`.
    """
    if self._bootstrap:
        x_curr = self.prior_sample(key=key, theta=theta)
        logw = self.meas_lpdf(y_curr=y_init, x_curr=x_curr, theta=theta)
    else:
        x_curr = self.init_sample(key=key, theta=theta)
        lp_prop = self.init_lpdf(x_curr=x_curr, theta=theta)
        lp_targ = self.prior_lpdf(x_curr=x_curr, theta=theta) + \
            self.meas_lpdf(y_curr=y_init, x_curr=x_curr, theta=theta)
        logw = lp_targ - lp_prop
    return x_curr, logw

particle_filter_rb(model, key, y_meas, theta, n_particles, resampler=resample_multinomial, score=False, fisher=False, history=False)

Rao-Blackwellized particle filter.

Notes:

- Algorithm 2 of Poyiadjis et al 2011.

- Can optionally use an auxiliary particle filter if `model.pf_aux()` is provided.

Parameters:

Name Type Description Default
model

Object specifying the state-space model having the following methods:

  • pf_init : (key, y_init, theta) -> (x_particles, logw): For sampling and calculating log-weights for the initial latent variable.
  • step_sample : (key, x_prev, y_curr, theta) -> x_curr: Sampling from the proposal distribution for each subsequent latent variable.
  • step_lpdf : (x_curr, x_prev, y_curr, theta) -> logw: Calculate log-weights for each subsequent latent variable.
  • state_lpdf : (x_curr, x_prev, theta) -> lpdf: Calculates the log-density of the state model.
  • meas_lpdf : (y_curr, x_curr, theta) -> lpdf: Calculates the log-density of the measurement model.
  • pf_aux : (x_prev, y_curr, theta) -> logw: Optional method providing look-forward log-weights of the auxillary particle filter.
 required
key

PRNG key.

 required
y_meas

JAX array with leading dimension n_obs containing the measurement variables y_meas = (y_0, ..., y_T), where T = n_obs-1.

 required
theta

Parameter value.

 required
n_particles

Number of particles.

 required
resampler

Function used at step t to obtain sample of particles from p(x_{t} | y_{0:t}, theta) out of a sample of particles from p(x_{t-1} | y_{0:t-1}, theta). The argument signature is resampler(x_particles, logw, key), and the return value is a dictionary with mandatory element x_particles and optional elements that get carried to the next step t+1 via lax.scan().

 resample_multinomial
score

Whether or not to return an estimate of the score function at theta.

 False
fisher

Whether or not to return an estimate of the Fisher information at theta. If True returns score as well.

 False
history

Whether to output the history of the filter or only the last step.

 False

Returns:

Name Type Description
Dictionary
  • x_particles - JAX array containing the state variable particles at the last time point (leading dimension n_particles) or at all time points (leading dimensions (n_obs, n_particles) if history=True.
  • logw_bar - JAX array containing unnormalized log weights at the last time point (dimensions n_particles) or at all time points (dimensions (n_obs, n_particles)) ifhistory=True`.
  • loglik - The particle filter loglikelihood evaluated at theta.
  • score - Optional 1D JAX array of size n_theta = length(theta) containing the estimated score at theta.
  • fisher Optional JAX array of shape (n_theta, n_theta) containing the estimated observed fisher information at theta.
  • resample_out If history=True, a dictionary of additional outputs from resampler function. The leading dimension of each element of the dictionary has leading dimension n_obs-1, since these additional outputs do not apply to the first time point t=0.
Source code in src/pfjax/particle_filter.py 
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def particle_filter_rb(
    model,
    key,
    y_meas,
    theta,
    n_particles,
    resampler=resample_multinomial,
    score=False,
    fisher=False,
    history=False,
):
    r"""
    Rao-Blackwellized particle filter.

    Notes:

        - Algorithm 2 of Poyiadjis et al 2011.

        - Can optionally use an auxiliary particle filter if `model.pf_aux()` is provided.

    Args:
        model: Object specifying the state-space model having the following methods:

            - `pf_init : (key, y_init, theta) -> (x_particles, logw)`: For sampling and calculating log-weights for the initial latent variable.
            - `step_sample : (key, x_prev, y_curr, theta) -> x_curr`: Sampling from the proposal distribution for each subsequent latent variable.
            - `step_lpdf : (x_curr, x_prev, y_curr, theta) -> logw`: Calculate log-weights for each subsequent latent variable.
            - `state_lpdf : (x_curr, x_prev, theta) -> lpdf`: Calculates the log-density of the state model.
            - `meas_lpdf : (y_curr, x_curr, theta) -> lpdf`: Calculates the log-density of the measurement model.
            - `pf_aux : (x_prev, y_curr, theta) -> logw`: Optional method providing look-forward log-weights of the auxillary particle filter.

        key: PRNG key.
        y_meas: JAX array with leading dimension `n_obs` containing the measurement variables `y_meas = (y_0, ..., y_T)`, where `T = n_obs-1`.
        theta: Parameter value.
        n_particles: Number of particles.
        resampler: Function used at step `t` to obtain sample of particles from `p(x_{t} | y_{0:t}, theta)` out of a sample of particles from `p(x_{t-1} | y_{0:t-1}, theta)`.   The argument signature is `resampler(x_particles, logw, key)`, and the return value is a dictionary with mandatory element `x_particles`  and optional elements that get carried to the next step `t+1` via `lax.scan()`.
        score: Whether or not to return an estimate of the score function at `theta`.
        fisher: Whether or not to return an estimate of the Fisher information at `theta`.  If `True` returns score as well.
        history: Whether to output the history of the filter or only the last step.

    Returns:
        Dictionary:

        - **x_particles** - JAX array containing the state variable particles at the last time point (leading dimension `n_particles`) or at all time points (leading dimensions `(n_obs, n_particles)` if `history=True`.
        - **logw_bar** - JAX array containing unnormalized log weights at the last time point (dimensions `n_particles`) or at all time points (dimensions (n_obs, n_particles)`) if `history=True`.
        - **loglik** - The particle filter loglikelihood evaluated at `theta`.
        - **score** - Optional 1D JAX array of size `n_theta = length(theta)` containing the estimated score at `theta`.
        - **fisher** Optional JAX array of shape `(n_theta, n_theta)` containing the estimated observed fisher information at `theta`.
        - **resample_out** If `history=True`, a dictionary of additional outputs from `resampler` function.  The leading dimension of each element of the dictionary has leading dimension `n_obs-1`, since these additional outputs do not apply to the first time point `t=0`.
    """
    n_obs = jtu.tree_leaves(y_meas)[0].shape[0]
    has_aux = callable(getattr(model, "pf_aux", None))
    has_acc = score or fisher

    def accumulate_deriv(x_prev, x_curr, y_curr, acc_prev, logw_aux):
        """
        Accumulator for weights and possibly derivative calculations.

        Parameters
        ----------
        acc_prev : dict
            Dictionary with elements `logw_bar`, and optionally `alpha` and `beta`.

        Returns:
            Dictionary with elements:
            - logw_targ: logw_bar + state_lpdf.
            - logw_prop: logw_aux + prop_lpdf.
            - alpha: optional current value of alpha_bar.
            - beta: optional current value of beta_bar.
        """
        logw_bar = acc_prev["logw_bar"]
        logw_targ = (
            model.state_lpdf(x_curr=x_curr, x_prev=x_prev, theta=theta) + logw_bar
        )
        # logw_prop: elements of q_theta(x_n | y_1:n)
        logw_prop = (
            model.step_lpdf(x_curr=x_curr, x_prev=x_prev, y_curr=y_curr, theta=theta)
            + logw_bar
            + logw_aux
        )
        acc_full = {"logw_targ": logw_targ, "logw_prop": logw_prop}
        if has_acc:
            grad_meas = jax.grad(model.meas_lpdf, argnums=2)
            grad_state = jax.grad(model.state_lpdf, argnums=2)
            alpha = (
                grad_meas(y_curr, x_curr, theta)
                + grad_state(x_curr, x_prev, theta)
                + acc_prev["alpha"]
            )
            acc_full["alpha"] = alpha
            if fisher:
                hess_meas = jax.jacfwd(
                    jax.jacrev(model.meas_lpdf, argnums=2), argnums=2
                )
                hess_state = jax.jacfwd(
                    jax.jacrev(model.state_lpdf, argnums=2), argnums=2
                )
                beta = (
                    jnp.outer(alpha, alpha)
                    + hess_meas(y_curr, x_curr, theta)
                    + hess_state(x_curr, x_prev, theta)
                    + acc_prev["beta"]
                )
                acc_full["beta"] = beta
        return acc_full

    def pf_prop(x_curr, x_prev, y_curr):
        """
        Calculate log-density of the proposal distribution `x_curr ~ q(x_t | x_t-1, y_t, theta)`.
        """
        return model.step_lpdf(
            x_curr=x_curr,
            x_prev=x_prev,
            y_curr=y_curr,
            theta=theta,
        )

    def pf_step(key, x_prev, y_curr):
        """
        Sample from the proposal distribution `x_curr ~ q(x_t | x_t-1, y_t, theta)`.
        """
        return model.step_sample(
            key=key,
            x_prev=x_prev,
            y_curr=y_curr,
            theta=theta,
        )

    def pf_init(key):
        return model.pf_init(
            key=key,
            y_init=utils.tree_subset(y_meas, 0),
            theta=theta,
        )

    def pf_aux(x_prev, y_curr):
        """
        Compute the log-density for auxillary sampling `logw_aux`.

        The auxillary log-density is given by model.pf_aux(). If this method is missing, `logw_aux` is set to 0.
        """
        if has_aux:
            logw_aux = model.pf_aux(
                x_prev=x_prev,
                y_curr=y_curr,
                theta=theta,
            )
        else:
            logw_aux = jnp.array(0.0)
        return logw_aux

    def pf_bar(x_prev, x_curr, y_curr, acc_prev, logw_aux):
        """
        Update calculations relating to logw_bar.

        This is the vmap version, which does the loop over x_prev here and the loop over x_curr in filter_step.

        Returns:
            Dictionary with elements:
            - logw_bar: rao-blackwellized weights.
            - alpha: optional current value of alpha.
            - beta: optional current value of beta.
        """
        # Calculate the accumulated values looping over x_prev
        acc_full = jax.vmap(
            accumulate_deriv,
            in_axes=(0, None, None, 0, 0),
        )(x_prev, x_curr, y_curr, acc_prev, logw_aux)
        # Calculate logw_tilde and logw_bar
        logw_targ, logw_prop = acc_full["logw_targ"], acc_full["logw_prop"]
        # logw_tilde: (19) of poyiadjis et al
        logw_tilde = jsp.special.logsumexp(logw_targ) + model.meas_lpdf(
            y_curr=y_curr, x_curr=x_curr, theta=theta
        )
        # logw_bar: (18) of poyiadjis et al
        logw_bar = logw_tilde - jsp.special.logsumexp(logw_prop)
        acc_curr = {"logw_bar": logw_bar}
        if has_acc:
            # weight the derivatives
            grad_curr = utils.tree_mean(
                tree=utils.rm_keys(acc_full, ["logw_targ", "logw_prop"]),
                logw=logw_targ,
            )
            if fisher:
                grad_curr["beta"] = grad_curr["beta"] - jnp.outer(
                    grad_curr["alpha"], grad_curr["alpha"]
                )
            acc_curr.update(grad_curr)
        return acc_curr

    # lax.scan stepping function
    def filter_step(carry, y_curr):
        # 1. sample particles from previous time point
        key, subkey = random.split(carry["key"])
        # auxiliary weights
        logw_aux = jax.vmap(
            pf_aux,
            in_axes=(0, None),
        )(carry["x_particles"], y_curr)
        # resampled particles
        resample_out = resampler(
            key=subkey,
            x_particles_prev=carry["x_particles"],
            logw=logw_aux + carry["logw_bar"],
        )
        # 2. sample particles for current timepoint
        key, *subkeys = random.split(key, num=n_particles + 1)
        x_particles = jax.vmap(
            pf_step,
            in_axes=(0, 0, None),
        )(jnp.array(subkeys), resample_out["x_particles"], y_curr)
        # 3. compute all double summations
        acc_prev = {"logw_bar": carry["logw_bar"]}
        if has_acc:
            acc_prev["alpha"] = carry["alpha"]
            if fisher:
                acc_prev["beta"] = carry["beta"]
        acc_curr = jax.vmap(
            pf_bar,
            in_axes=(None, 0, None, None, None),
        )(carry["x_particles"], x_particles, y_curr, acc_prev, logw_aux)
        # output
        res_carry = {
            "x_particles": x_particles,
            "loglik": carry["loglik"] + jsp.special.logsumexp(acc_curr["logw_bar"]),
            "key": key,
            # "resample_out": rm_keys(resample_out, "x_particles")
        }
        res_carry.update(acc_curr)
        if history:
            res_stack = {k: res_carry[k] for k in ["x_particles", "logw_bar"]}
            if set(["x_particles"]) < resample_out.keys():
                res_stack["resample_out"] = utils.rm_keys(
                    x=resample_out, keys="x_particles"
                )
        else:
            res_stack = None
        return res_carry, res_stack

    # lax.scan initial value
    key, *subkeys = random.split(key, num=n_particles + 1)
    # initial particles and weights
    x_particles, logw = jax.vmap(pf_init)(jnp.array(subkeys))
    # # dummy initialization for resampler
    # init_res = resampler(key, x_particles, logw)
    # init_res = tree_zeros(rm_keys(init_res, ["x_particles"]))
    filter_init = {
        "x_particles": x_particles,
        "loglik": jsp.special.logsumexp(logw),
        "logw_bar": logw,
        "key": key,
        # "resample_out": init_res
    }
    if has_acc:
        # dummy initialization for derivatives
        filter_init["alpha"] = _initialize_score(theta, n_particles)
        if fisher:
            filter_init["beta"] = _initialize_fisher(theta, n_particles)

    # lax.scan itself
    last, full = lax.scan(
        f=filter_step,
        init=filter_init,
        xs=utils.tree_subset(y_meas, jnp.arange(1, n_obs)),
    )

    # format output
    if history:
        # append initial values of x_particles and logw_bar
        full["x_particles"] = utils.tree_append_first(
            tree=full["x_particles"], first=filter_init["x_particles"]
        )
        full["logw_bar"] = utils.tree_append_first(
            tree=full["logw_bar"], first=filter_init["logw_bar"]
        )
        # full["x_particles"] = jnp.concatenate(
        #     [filter_init["x_particles"][None], full["x_particles"]]
        # )
        # full["logw_bar"] = jnp.concatenate(
        #     [filter_init["logw_bar"][None], full["logw_bar"]]
        # )
    else:
        full = last

    # calculate loglikelihood
    full["loglik"] = last["loglik"] - n_obs * jnp.log(n_particles)
    if has_acc:
        # logw_bar = last["logw_bar"]
        if not fisher:
            # calculate score only
            full["score"] = utils.tree_mean(last["alpha"], last["logw_bar"])
        else:
            # calculate score and fisher information
            prob = utils.logw_to_prob(last["logw_bar"])
            alpha = last["alpha"]
            beta = last["beta"]
            alpha, gamma = jax.vmap(lambda p, a, b: (p * a, p * (jnp.outer(a, a) + b)))(
                prob, alpha, beta
            )
            alpha = jnp.sum(alpha, axis=0)
            hess = jnp.sum(gamma, axis=0) - jnp.outer(alpha, alpha)
            full["score"] = alpha
            full["fisher"] = -hess

    return full