def dot(self, x, dims_to_omit=None, return_numpy=False, obs_mode=False):
""" Dot product of a this distribution with `x`
@NOTE see `spm_dot` in core.maths
@TODO create better workaround for `obs_mode`
The dimensions in `dims_to_omit` will not be summed across during the dot product
Parameters
----------
- `x` [1D np.ndarray || Categorical]
The array to perform the dot product with
- `dims_to_omit` [list of ints] (optional)
Which dimensions to omit
- `return_numpy` [bool] (optional)
Whether to return `np.ndarray` or `Categorical` - defaults to `Categorical`
- 'obs_mode' [bool] (optional)
Whether to perform the inner product of `x` with the leading dimension of self
@NOTE We call this `obs_mode` because it's often used to get the likelihood
of an observation (leading dimension) under different settings of
hidden states (lagging dimensions)
"""
x = utils.to_numpy(x)
# perform dot product on each sub-array
if self.IS_AOA:
y = np.empty(self.n_arrays, dtype=object)
for i in range(self.n_arrays):
y[i] = maths.spm_dot(self[i].values, x, dims_to_omit, obs_mode)
else:
y = maths.spm_dot(self.values, x, dims_to_omit, obs_mode)
if return_numpy:
return y
else:
return Categorical(values=y)
def run_fpi(A,
obs,
n_observations,
n_states,
prior=None,
num_iter=10,
dF=1.0,
dF_tol=0.001):
"""
Update marginal posterior beliefs about hidden states using variational fixed point iteration (FPI).
Parameters
----------
- 'A' [numpy nd.array (matrix or tensor or array-of-arrays)]:
Observation likelihood of the generative model, mapping from hidden states to observations.
Used to invert generative model to obtain marginal likelihood over hidden states, given
the observation
- 'obs' [numpy 1D array or array of arrays (with 1D numpy array entries)]:
The observation (generated by the environment). If single modality, this can be a
1D array (one-hot vector representation).
If multi-modality, this can be an array of arrays (whose entries are 1D one-hot vectors).
- 'n_observations' [int or list of ints]
- 'n_states' [int or list of ints]
- 'prior' [numpy 1D array, array of arrays (with 1D numpy array entries) or None]:
Prior beliefs of the agent, to be integrated with the marginal likelihood to obtain posterior.
If absent, prior is set to be a uniform distribution over hidden states (identical to the
initialisation of the posterior)
-'num_iter' [int]:
Number of variational fixed-point iterations to run.
-'dF' [float]:
Starting free energy gradient (dF/dt) before updating in the course of gradient descent.
-'dF_tol' [float]:
Threshold value of the gradient of the variational free energy (dF/dt), to be checked at
each iteration. If
dF <= dF_tol, the iterations are halted pre-emptively and the final
marginal posterior belief(s) is(are) returned
Returns
----------
-'qs' [numpy 1D array or array of arrays (with 1D numpy array entries):
Marginal posterior beliefs over hidden states (single- or multi-factor)
achieved via variational fixed point iteration (mean-field)
"""
# get model dimensions
n_modalities = len(n_observations)
n_factors = len(n_states)
"""
=========== Step 1 ===========
Loop over the observation modalities and use assumption of independence
among observation modalitiesto multiply each modality-specific likelihood
onto a single joint likelihood over hidden factors [size n_states]
"""
likelihood = get_joint_likelihood(A, obs, n_states)
likelihood = np.log(likelihood + 1e-16)
"""
=========== Step 2 ===========
Create a flat posterior (and prior if necessary)
"""
qs = np.empty(n_factors, dtype=object)
for factor in range(n_factors):
qs[factor] = np.ones(n_states[factor]) / n_states[factor]
"""
If prior is not provided, initialise prior to be identical to posterior
(namely, a flat categorical distribution). Take the logarithm of it (required for
FPI algorithm below).
"""
if prior is None:
prior = np.empty(n_factors, dtype=object)
for factor in range(n_factors):
prior[factor] = np.log(
np.ones(n_states[factor]) / n_states[factor] + 1e-16)
"""
=========== Step 3 ===========
Initialize initial free energy
"""
prev_vfe = calc_free_energy(qs, prior, n_factors)
"""
=========== Step 4 ===========
If we have a single factor, we can just add prior and likelihood,
otherwise we run FPI
"""
if n_factors == 1:
qL = spm_dot(likelihood, qs, [0])
return softmax(qL + prior[0])
else:
"""
=========== Step 5 ===========
Run the FPI scheme
"""
curr_iter = 0
while curr_iter < num_iter and dF >= dF_tol:
# Initialise variational free energy
vfe = 0
# arg_list = [likelihood, list(range(n_factors))]
# arg_list = arg_list + list(chain(*([qs_i,[i]] for i, qs_i in enumerate(qs)))) + [list(range(n_factors))]
# LL_tensor = np.einsum(*arg_list)
qs_all = qs[0]
for factor in range(n_factors - 1):
qs_all = qs_all[..., None] * qs[factor + 1]
LL_tensor = likelihood * qs_all
for factor, qs_i in enumerate(qs):
# qL = np.einsum(LL_tensor, list(range(n_factors)), 1.0/qs_i, [factor], [factor])
qL = np.einsum(LL_tensor, list(range(n_factors)),
[factor]) / qs_i
qs[factor] = softmax(qL + prior[factor])
# List of orders in which marginal posteriors are sequentially multiplied into the joint likelihood:
# First order loops over factors starting at index = 0, second order goes in reverse
# factor_orders = [range(n_factors), range((n_factors - 1), -1, -1)]
# iteratively marginalize out each posterior marginal from the joint log-likelihood
# except for the one associated with a given factor
# for factor_order in factor_orders:
# for factor in factor_order:
# qL = spm_dot(likelihood, qs, [factor])
# qs[factor] = softmax(qL + prior[factor])
# calculate new free energy
vfe = calc_free_energy(qs, prior, n_factors, likelihood)
# stopping condition - time derivative of free energy
dF = np.abs(prev_vfe - vfe)
prev_vfe = vfe
curr_iter += 1
return qs
def run_fpi_faster(A,
obs,
n_observations,
n_states,
prior=None,
num_iter=10,
dF=1.0,
dF_tol=0.001):
"""
Update marginal posterior beliefs about hidden states
using a new version of variational fixed point iteration (FPI).
@NOTE (Conor, 26.02.2020):
This method uses a faster algorithm than the traditional 'spm_dot' approach. Instead of
separately computing a conditional joint log likelihood of an outcome, under the
posterior probabilities of a certain marginal, instead all marginals are multiplied into one
joint tensor that gives the joint likelihood of an observation under all hidden states,
that is then sequentially (and *parallelizably*) marginalized out to get each marginal posterior.
This method is less RAM-intensive, admits heavy parallelization, and runs (about 2x) faster.
@NOTE (Conor, 28.02.2020):
After further testing, discovered interesting differences between this version and the
original version. It appears that the
original version (simple 'run_FPI') shows mean-field biases or 'explaining away'
effects, whereas this version spreads probabilities more 'fairly' among possibilities.
To summarize: it actually matters what order you do the summing across the joint likelihood tensor.
In this verison, all marginals are multiplied into the likelihood tensor before summing out,
whereas in the previous version, marginals are recursively multiplied and summed out.
@NOTE (Conor, 24.04.2020): I would expect that the factor_order approach used above would help
ameliorate the effects of the mean-field bias. I would also expect that the use of a factor_order
below is unnnecessary, since the marginalisation w.r.t. each factor is done only after all marginals
are multiplied into the larger tensor.
Parameters
----------
- 'A' [numpy nd.array (matrix or tensor or array-of-arrays)]:
Observation likelihood of the generative model, mapping from hidden states to observations.
Used to invert generative model to obtain marginal likelihood over hidden states,
given the observation
- 'obs' [numpy 1D array or array of arrays (with 1D numpy array entries)]:
The observation (generated by the environment). If single modality, this can be a 1D array
(one-hot vector representation). If multi-modality, this can be an array of arrays
(whose entries are 1D one-hot vectors).
- 'n_observations' [int or list of ints]
- 'n_states' [int or list of ints]
- 'prior' [numpy 1D array, array of arrays (with 1D numpy array entries) or None]:
Prior beliefs of the agent, to be integrated with the marginal likelihood to obtain posterior.
If absent, prior is set to be a uniform distribution over hidden states
(identical to the initialisation of the posterior)
-'num_iter' [int]:
Number of variational fixed-point iterations to run.
-'dF' [float]:
Starting free energy gradient (dF/dt) before updating in the course of gradient descent.
-'dF_tol' [float]:
Threshold value of the gradient of the variational free energy (dF/dt),
to be checked at each iteration. If dF <= dF_tol, the iterations are halted pre-emptively
and the final marginal posterior belief(s) is(are) returned
Returns
----------
-'qs' [numpy 1D array or array of arrays (with 1D numpy array entries):
Marginal posterior beliefs over hidden states (single- or multi-factor) achieved
via variational fixed point iteration (mean-field)
"""
# get model dimensions
n_modalities = len(n_observations)
n_factors = len(n_states)
"""
=========== Step 1 ===========
Loop over the observation modalities and use assumption of independence
among observation modalities to multiply each modality-specific likelihood
onto a single joint likelihood over hidden factors [size n_states]
"""
# likelihood = np.ones(tuple(n_states))
# if n_modalities is 1:
# likelihood *= spm_dot(A, obs, obs_mode=True)
# else:
# for modality in range(n_modalities):
# likelihood *= spm_dot(A[modality], obs[modality], obs_mode=True)
likelihood = get_joint_likelihood(A, obs, n_states)
likelihood = np.log(likelihood + 1e-16)
"""
=========== Step 2 ===========
Create a flat posterior (and prior if necessary)
"""
qs = np.empty(n_factors, dtype=object)
for factor in range(n_factors):
qs[factor] = np.ones(n_states[factor]) / n_states[factor]
"""
If prior is not provided, initialise prior to be identical to posterior
(namely, a flat categorical distribution). Take the logarithm of it
(required for FPI algorithm below).
"""
if prior is None:
prior = np.empty(n_factors, dtype=object)
for factor in range(n_factors):
prior[factor] = np.log(
np.ones(n_states[factor]) / n_states[factor] + 1e-16)
"""
=========== Step 3 ===========
Initialize initial free energy
"""
prev_vfe = calc_free_energy(qs, prior, n_factors)
"""
=========== Step 4 ===========
If we have a single factor, we can just add prior and likelihood,
otherwise we run FPI
"""
if n_factors == 1:
qL = spm_dot(likelihood, qs, [0])
return softmax(qL + prior[0])
else:
"""
=========== Step 5 ===========
Run the revised fixed-point iteration scheme
"""
curr_iter = 0
while curr_iter < num_iter and dF >= dF_tol:
# Initialise variational free energy
vfe = 0
# List of orders in which marginal posteriors are sequentially
# multiplied into the joint likelihood: First order loops over
# factors starting at index = 0, second order goes in reverse
factor_orders = [range(n_factors), range((n_factors - 1), -1, -1)]
for factor_order in factor_orders:
# reset the log likelihood
L = likelihood.copy()
# multiply each marginal onto a growing single joint distribution
for factor in factor_order:
s = np.ones(np.ndim(L), dtype=int)
s[factor] = len(qs[factor])
L *= qs[factor].reshape(tuple(s))
# now loop over factors again, and this time divide out the
# appropriate marginal before summing out.
# !!! KEY DIFFERENCE BETWEEN THIS AND 'VANILLA' FPI,
# WHERE THE ORDER OF THE MARGINALIZATION MATTERS !!!
for f in factor_order:
s = np.ones(np.ndim(L), dtype=int)
s[factor] = len(qs[factor]) # type: ignore
# divide out the factor we multiplied into X already
temp = L * (1.0 / qs[factor]).reshape(
tuple(s)) # type: ignore
dims2sum = tuple(np.where(np.arange(n_factors) != f)[0])
qL = np.sum(temp, dims2sum)
temp = L * (1.0 / qs[factor]).reshape(
tuple(s)) # type: ignore
qs[factor] = softmax(qL + prior[factor]) # type: ignore
# calculate new free energy
vfe = calc_free_energy(qs, prior, n_factors, likelihood)
# stopping condition - time derivative of free energy
dF = np.abs(prev_vfe - vfe)
prev_vfe = vfe
curr_iter += 1
return qs
def calc_pA_info_gain(pA, qo_pi, qs_pi):
"""
Compute expected Dirichlet information gain about parameters pA under a policy
Parameters
----------
pA [numpy nd-array, array-of-arrays (where each entry is a numpy nd-array), or
Dirichlet (either single-factor of AoA)]:
Prior dirichlet parameters parameterizing beliefs about the likelihood
mapping from hidden states to observations,
with different modalities (if there are multiple) stored in different arrays.
qo_pi [numpy 1D array, array-of-arrays (where each entry is a numpy 1D array),
Categorical (either single-factor or AoA), or list]:
Expected observations. If a list, each entry of the list is the posterior
predictive for a given timepoint of an expected trajectory
qs_pi [numpy 1D array, array-of-arrays (where each entry is a numpy 1D array),
Categorical (either single-factor or AoA), or list]:
Posterior predictive density over hidden states. If a list, each entry of
the list is the posterior predictive for a given timepoint of an expected trajectory
Returns
-------
infogain_pA [scalar]:
Surprise (about dirichlet parameters) expected under the policy in question
"""
if isinstance(qo_pi, list):
n_steps = len(qo_pi)
for t in range(n_steps):
qo_pi[t] = utils.to_numpy(qo_pi[t], flatten=True)
else:
n_steps = 1
qo_pi = [utils.to_numpy(qo_pi, flatten=True)]
if isinstance(qs_pi, list):
for t in range(n_steps):
qs_pi[t] = utils.to_numpy(qs_pi[t], flatten=True)
else:
n_steps = 1
qs_pi = [utils.to_numpy(qs_pi, flatten=True)]
if isinstance(pA, Dirichlet):
if pA.IS_AOA:
num_modalities = pA.n_arrays
else:
num_modalities = 1
wA = pA.expectation_of_log()
else:
if utils.is_arr_of_arr(pA):
num_modalities = len(pA)
wA = np.empty(num_modalities, dtype=object)
for modality in range(num_modalities):
wA[modality] = spm_wnorm(pA[modality])
else:
num_modalities = 1
wA = spm_wnorm(pA)
pA = utils.to_numpy(pA)
pA_infogain = 0
if num_modalities == 1:
wA = wA * (pA > 0).astype("float")
for t in range(n_steps):
pA_infogain = -qo_pi[t].dot(spm_dot(wA, qs_pi[t])[:, np.newaxis])
else:
for modality in range(num_modalities):
wA_modality = wA[modality] * (pA[modality] > 0).astype("float")
for t in range(n_steps):
pA_infogain -= qo_pi[t][modality].dot(spm_dot(wA_modality, qs_pi[t])[:, np.newaxis])
return pA_infogain
def run_mmp(lh_seq,
B,
policy,
prev_actions=None,
prior=None,
num_iter=10,
grad_descent=False,
tau=0.25,
last_timestep=False,
save_vfe_seq=False):
"""
Marginal message passing scheme for updating posterior beliefs about multi-factor hidden states over time,
conditioned on a particular policy.
Parameters:
--------------
`lh_seq`[numpy object array]:
Likelihoods of hidden state factors given a sequence of observations over time. This is logged beforehand
`B`[numpy object array]:
Transition likelihood of the generative model, mapping from hidden states at T to hidden states at T+1. One B matrix per modality (e.g. `B[f]` corresponds to f-th factor's B matrix)
This is used in inference to compute the 'forward' and 'backward' messages conveyed between beliefs about temporally-adjacent timepoints.
`policy` [2-D numpy.ndarray]:
Matrix of shape (policy_len, num_control_factors) that indicates the indices of each action (control state index) upon timestep t and control_factor f in the element `policy[t,f]` for a given policy.
`prev_actions` [None or 2-D numpy.ndarray]:
If provided, should be a matrix of previous actions of shape (infer_len, num_control_factors) taht indicates the indices of each action (control state index) taken in the past (up until the current timestep).
`prior`[None or numpy object array]:
If provided, this a numpy object array with one sub-array per hidden state factor, that stores the prior beliefs about initial states (at t = 0, relative to `infer_len`).
`num_iter`[Int]:
Number of variational iterations
`grad_descent` [Bool]:
Flag for whether to use gradient descent (predictive coding style)
`tau` [Float]:
Decay constant for use in `grad_descent` version
`last_timestep` [Bool]:
Flag for whether we are at the last timestep of belief updating
`save_vfe_seq` [Bool]:
Flag for whether to save the sequence of variational free energies over time (for this policy). If `False`, then VFE is integrated across time/iterations.
Returns:
--------------
`qs_seq`[list]: the sequence of beliefs about the different hidden state factors over time, one multi-factor posterior belief per timestep in `infer_len`
`F`[Float or list, depending on setting of save_vfe_seq]
"""
# window
past_len = len(lh_seq)
future_len = policy.shape[0]
if last_timestep:
infer_len = past_len + future_len - 1
else:
infer_len = past_len + future_len
future_cutoff = past_len + future_len - 2
# dimensions
_, num_states, _, num_factors = get_model_dimensions(A=None, B=B)
B = to_arr_of_arr(B)
# beliefs
qs_seq = obj_array(infer_len)
for t in range(infer_len):
qs_seq[t] = obj_array_uniform(num_states)
# last message
qs_T = obj_array_zeros(num_states)
# prior
if prior is None:
prior = obj_array_uniform(num_states)
# transposed transition
trans_B = obj_array(num_factors)
for f in range(num_factors):
trans_B[f] = spm_norm(np.swapaxes(B[f], 0, 1))
# full policy
if prev_actions is None:
prev_actions = np.zeros((past_len, policy.shape[1]))
policy = np.vstack((prev_actions, policy))
# initialise variational free energy of policy (accumulated over time)
if save_vfe_seq:
F = []
F.append(0.0)
else:
F = 0.0
for itr in range(num_iter):
for t in range(infer_len):
for f in range(num_factors):
# likelihood
if t < past_len:
lnA = spm_log(spm_dot(lh_seq[t], qs_seq[t], [f]))
else:
lnA = np.zeros(num_states[f])
# past message
if t == 0:
lnB_past = spm_log(prior[f])
else:
past_msg = B[f][:, :, int(policy[t - 1,
f])].dot(qs_seq[t - 1][f])
lnB_past = spm_log(past_msg)
# future message
if t >= future_cutoff:
lnB_future = qs_T[f]
else:
future_msg = trans_B[f][:, :, int(policy[t, f])].dot(
qs_seq[t + 1][f])
lnB_future = spm_log(future_msg)
# inference
if grad_descent:
lnqs = spm_log(qs_seq[t][f])
coeff = 1 if (t >= future_cutoff) else 2
err = (coeff * lnA + lnB_past + lnB_future) - coeff * lnqs
err -= err.mean()
lnqs = lnqs + tau * err
qs_seq[t][f] = softmax(lnqs)
if (t == 0) or (t == (infer_len - 1)):
F += +0.5 * lnqs.dot(0.5 * err)
else:
F += lnqs.dot(
0.5 * (err - (num_factors - 1) * lnA / num_factors)
) # @NOTE: not sure why Karl does this in SPM_MDP_VB_X, we should look into this
else:
qs_seq[t][f] = softmax(lnA + lnB_past + lnB_future)
if not grad_descent:
if save_vfe_seq:
if t < past_len:
F.append(
F[-1] +
calc_free_energy(qs_seq[t],
prior,
num_factors,
likelihood=spm_log(lh_seq[t]))[0])
else:
F.append(
F[-1] +
calc_free_energy(qs_seq[t], prior, num_factors)[0])
else:
if t < past_len:
F += calc_free_energy(qs_seq[t],
prior,
num_factors,
likelihood=spm_log(lh_seq[t]))
else:
F += calc_free_energy(qs_seq[t], prior, num_factors)
return qs_seq, F
def get_expected_obs(qs_pi, A, return_numpy=False):
"""
Given a posterior predictive density Qs_pi and an observation likelihood model A,
get the expected observations given the predictive posterior.
Parameters
----------
qs_pi [numpy 1D array, array-of-arrays (where each entry is a numpy 1D array), Categorical
(either single-factor or AoA), or list]:
Posterior predictive density over hidden states. If a list, each entry of the list is the
posterior predictive for a given timepoint of an expected trajectory
A [numpy nd-array, array-of-arrays (where each entry is a numpy nd-array), or Categorical
(either single-factor of AoA)]:
Observation likelihood mapping from hidden states to observations, with different modalities
(if there are multiple) stored in different arrays
return_numpy [Boolean]:
True/False flag to determine whether output of function is a numpy array or a Categorical
Returns
-------
qo_pi [numpy 1D array, array-of-arrays (where each entry is a numpy 1D array), Categorical
(either single-factor or AoA), or list]:
Expected observations under the given policy. If a list, a list of the expected observations
over the time horizon of policy evaluation, where
each entry is the expected observations at a given timestep.
"""
# initialise expected observations
qo_pi = []
A = utils.to_numpy(A)
if isinstance(qs_pi, list):
n_steps = len(qs_pi)
for t in range(n_steps):
qs_pi[t] = utils.to_numpy(qs_pi[t], flatten=True)
else:
n_steps = 1
qs_pi = [utils.to_numpy(qs_pi, flatten=True)]
if utils.is_arr_of_arr(A):
num_modalities = len(A)
for t in range(n_steps):
qo_pi_t = np.empty(num_modalities, dtype=object)
qo_pi.append(qo_pi_t)
# get expected observations over time
for t in range(n_steps):
for modality in range(num_modalities):
qo_pi[t][modality] = spm_dot(A[modality], qs_pi[t])
else:
# get expected observations over time
for t in range(n_steps):
qo_pi.append(spm_dot(A, qs_pi[t]))
if return_numpy:
if n_steps == 1:
return qo_pi[0]
else:
return qo_pi
else:
if n_steps == 1:
return utils.to_categorical(qo_pi[0])
else:
for t in range(n_steps):
qo_pi[t] = utils.to_categorical(qo_pi[t])
return qo_pi
def get_expected_states(qs, B, policy, return_numpy=False):
"""
Given a posterior density qs, a transition likelihood model B, and a policy,
get the state distribution expected under that policy's pursuit
Parameters
----------
- `qs` [numpy 1D array, array-of-arrays (where each entry is a numpy 1D array), or
Categorical (either single-factor or AoA)]:
Current posterior beliefs about hidden states
- `B` [numpy nd-array, array-of-arrays (where each entry is a numpy nd-array), or Categorical
(either single-factor of AoA)]:
Transition likelihood mapping from states at t to states at t + 1, with different actions
(per factor) stored along the lagging dimension
- `policy` [np.arrays]:
np.array of size (policy_len x n_factors) where each value corrresponds to a control state
- `return_numpy` [Boolean]:
True/False flag to determine whether output of function is a numpy array or a Categorical
Returns
-------
- `qs_pi` [ list of np.arrays with len n_steps, where in case of multiple hidden state factors,
each np.array in the list is a 1 x n_factors array-of-arrays, otherwise a list of 1D numpy arrays]:
Expected states under the given policy - also known as the 'posterior predictive density'
"""
n_steps = policy.shape[0]
n_factors = policy.shape[1]
qs = utils.to_numpy(qs, flatten=True)
B = utils.to_numpy(B)
# initialise beliefs over expected states
qs_pi = []
if utils.is_arr_of_arr(B):
for t in range(n_steps):
qs_pi_t = np.empty(n_factors, dtype=object)
qs_pi.append(qs_pi_t)
# initialise expected states after first action using current posterior (t = 0)
for control_factor, control in enumerate(policy[0, :]):
qs_pi[0][control_factor] = spm_dot(B[control_factor][:, :, control], qs[control_factor])
# get expected states over time
if n_steps > 1:
for t in range(1, n_steps):
for control_factor, control in enumerate(policy[t, :]):
qs_pi[t][control_factor] = spm_dot(
B[control_factor][:, :, control], qs_pi[t - 1][control_factor]
)
else:
# initialise expected states after first action using current posterior (t = 0)
qs_pi.append(spm_dot(B[:, :, policy[0, 0]], qs))
# then loop over future timepoints
if n_steps > 1:
for t in range(1, n_steps):
qs_pi.append(spm_dot(B[:, :, policy[t, 0]], qs_pi[t - 1]))
if return_numpy:
if len(qs_pi) == 1:
return qs_pi[0]
else:
return qs_pi
else:
if len(qs_pi) == 1:
return utils.to_categorical(qs_pi[0])
else:
for t in range(n_steps):
qs_pi[t] = utils.to_categorical(qs_pi[t])
return qs_pi
def run_mmp_old(
A,
B,
obs_t,
policy,
curr_t,
t_horizon,
T,
qs_bma=None,
prior=None,
num_iter=10,
dF=1.0,
dF_tol=0.001,
previous_actions=None,
use_gradient_descent=False,
tau=0.25,
):
"""
Optimise marginal posterior beliefs about hidden states using marginal message-passing scheme (MMP) developed
by Thomas Parr and colleagues, see https://github.com/tejparr/nmpassing
Parameters
----------
- 'A' [numpy nd.array (matrix or tensor or array-of-arrays)]:
Observation likelihood of the generative model, mapping from hidden states to observations.
Used in inference to get the likelihood of an observation, under different hidden state configurations.
- 'B' [numpy.ndarray (tensor or array-of-arrays)]:
Transition likelihood of the generative model, mapping from hidden states at t to hidden states at t+1.
Used in inference to get expected future (or past) hidden states, given past (or future) hidden
states (or expectations thereof).
- 'obs_t' [list of length t_horizon of numpy 1D array or array of arrays (with 1D numpy array entries)]:
Sequence of observations sampled from beginning of time horizon the current timestep t.
The first observation (the start of the time horizon) is either the first timestep of the generative
process or the first timestep of the policy horizon (whichever is closer to 'curr_t' in time).
The observations over time are stored as a list of numpy arrays, where in case of multi-modalities
each numpy array is an array-of-arrays, with one 1D numpy.ndarray for each modality.
In the case of a single modality, each observation is a single 1D numpy.ndarray.
- 'policy' [2D np.ndarray]:
Array of actions constituting a single policy. Policy is a shape
(n_steps, n_control_factors) numpy.ndarray, the values of which indicate actions along a given control
factor (column index) at a given timestep (row index).
- 'curr_t' [int]:
Current timestep (relative to the 'absolute' time of the generative process).
- 't_horizon'[int]:
Temporal horizon of inference for states and policies.
- 'T' [int]:
Temporal horizon of the generative process (absolute time)
- `qs_bma` [numpy 1D array, array of arrays (with 1D numpy array entries) or None]:
- 'prior' [numpy 1D array, array of arrays (with 1D numpy array entries) or None]:
Prior beliefs of the agent at the beginning of the time horizon, to be integrated
with the marginal likelihood to obtain posterior at the first timestep.
If absent, prior is set to be a uniform distribution over hidden states (identical to the
initialisation of the posterior.
-'num_iter' [int]:
Number of variational iterations to run. (optional)
-'dF' [float]:
Starting free energy gradient (dF/dt) before updating in the course of gradient descent. (optional)
-'dF_tol' [float]:
Threshold value of the gradient of the variational free energy (dF/dt), to be checked
at each iteration. If dF <= dF_tol, the iterations are halted pre-emptively and the final
marginal posterior belief(s) is(are) returned. (optional)
-'previous_actions' [numpy.ndarray with shape (num_steps, n_control_factors) or None]:
Array of previous actions, which can be used to constrain the 'past' messages in inference
to only consider states of affairs that were possible under actions that are known to have been taken.
The first dimension of previous-arrays (previous_actions.shape[0]) encodes how far back in time the agent is
considering. The first timestep of this either corresponds to either the first timestep of the generative
process or the first timestep of the policy horizon (whichever is sooner in time). (optional)
-'use_gradient_descent' [bool]:
Flag to indicate whether to use gradient descent to optimise posterior beliefs.
-'tau' [float]:
Learning rate for gradient descent (only used if use_gradient_descent is True)
Returns
----------
-'qs' [list of length T of numpy 1D arrays or array of arrays (with 1D numpy array entries):
Marginal posterior beliefs over hidden states (single- or multi-factor) achieved
via marginal message pasing
-'qss' [list of lists of length T of numpy 1D arrays or array of arrays (with 1D numpy array entries):
Marginal posterior beliefs about hidden states (single- or multi-factor) held at
each timepoint, *about* each timepoint of the observation
sequence
-'F' [2D np.ndarray]:
Variational free energy of beliefs about hidden states, indexed by time point and variational iteration
-'F_pol' [float]:
Total free energy of the policy under consideration.
"""
# get temporal window for inference
min_time = max(0, curr_t - t_horizon)
max_time = min(T, curr_t + t_horizon)
window_idxs = np.array([t for t in range(min_time, max_time)])
window_len = len(window_idxs)
# TODO: needs a better name - the point at which we ignore future messages
future_cutoff = window_len - 1
inference_len = window_len + 1
obs_seq_len = len(obs_t)
# get relevant observations, given our current time point
if curr_t == 0:
obs_range = [0]
else:
min_obs_idx = max(0, curr_t - t_horizon)
max_obs_idx = curr_t + 1
obs_range = range(min_obs_idx, max_obs_idx)
# get model dimensions
# TODO: make a general function in `utils` for extracting model dimensions
if utils.is_arr_of_arr(obs_t[0]):
num_obs = [obs.shape[0] for obs in obs_t[0]]
else:
num_obs = [obs_t[0].shape[0]]
if utils.is_arr_of_arr(B):
num_states = [b.shape[0] for b in B]
else:
num_states = [B[0].shape[0]]
B = utils.to_arr_of_arr(B)
num_modalities = len(num_obs)
num_factors = len(num_states)
"""
=========== Step 1 ===========
Calculate likelihood
Loop over modalities and use assumption of independence among observation modalities
to combine each modality-specific likelihood into a single joint likelihood over hidden states
"""
# likelihood of observations under configurations of hidden states (over time)
likelihood = np.empty(len(obs_range), dtype=object)
for t, obs in enumerate(obs_range):
# likelihood_t = np.ones(tuple(num_states))
# if num_modalities == 1:
# likelihood_t *= spm_dot(A[0], obs_t[obs], obs_mode=True)
# else:
# for modality in range(num_modalities):
# likelihood_t *= spm_dot(A[modality], obs_t[obs][modality], obs_mode=True)
likelihood_t = get_joint_likelihood(A, obs_t, num_states)
# The Thomas Parr MMP version, you log the likelihood first
# likelihood[t] = np.log(likelihood_t + 1e-16)
# Karl SPM version, logging doesn't happen until *after* the dotting with the posterior
likelihood[t] = likelihood_t
"""
=========== Step 2 ===========
Initialise a flat posterior (and prior if necessary)
If a prior is not provided, initialise a uniform prior
"""
qs = [np.empty(num_factors, dtype=object) for i in range(inference_len)]
for t in range(inference_len):
# if t == window_len:
# # final message is zeros - has no effect on inference
# # TODO: this may be redundant now that we skip last step
# for f in range(num_factors):
# qs[t][f] = np.zeros(num_states[f])
# else:
# for f in range(num_factors):
# qs[t][f] = np.ones(num_states[f]) / num_states[f]
for f in range(num_factors):
qs[t][f] = np.ones(num_states[f]) / num_states[f]
if prior is None:
prior = np.empty(num_factors, dtype=object)
for f in range(num_factors):
prior[f] = np.ones(num_states[f]) / num_states[f]
"""
=========== Step 3 ===========
Create a normalized transpose of the transition distribution `B_transposed`
Used for computing backwards messages 'from the future'
"""
B_transposed = np.empty(num_factors, dtype=object)
for f in range(num_factors):
B_transposed[f] = np.zeros_like(B[f])
for u in range(B[f].shape[2]):
B_transposed[f][:, :, u] = spm_norm(B[f][:, :, u].T)
# zero out final message
# TODO: may be redundant now we skip final step
last_message = np.empty(num_factors, dtype=object)
for f in range(num_factors):
last_message[f] = np.zeros(num_states[f])
# if previous actions not given, zero out to stop any influence on inference
if previous_actions is None:
previous_actions = np.zeros((1, policy.shape[1]))
full_policy = np.vstack((previous_actions, policy))
# print(full_policy.shape)
"""
=========== Step 3 ===========
Loop over time indices of time window, updating posterior as we go
This includes past time steps and future time steps
"""
qss = [[] for i in range(num_iter)]
free_energy = np.zeros((len(qs), num_iter))
free_energy_pol = 0.0
# print(obs_seq_len)
print('Full policy history')
print('------------------')
print(full_policy)
for n in range(num_iter):
for t in range(inference_len):
lnB_past_tensor = np.empty(num_factors, dtype=object)
for f in range(num_factors):
# if t == 0 and n == 0:
# print(f"qs at time t = {t}, factor f = {f}, iteration i = {n}: {qs[t][f]}")
"""
=========== Step 3.a ===========
Calculate likelihood
"""
if t < len(obs_range):
# if t < len(obs_seq_len):
# Thomas Parr MMP version
# lnA = spm_dot(likelihood[t], qs[t], [f])
# Karl SPM version
lnA = np.log(spm_dot(likelihood[t], qs[t], [f]) + 1e-16)
else:
lnA = np.zeros(num_states[f])
if t == 1 and n == 0:
# pass
print(
f"lnA at time t = {t}, factor f = {f}, iteration i = {n}: {lnA}"
)
# print(f"lnA at time t = {t}, factor f = {f}, iteration i = {n}: {lnA}")
"""
=========== Step 3.b ===========
Calculate past message
"""
if t == 0 and window_idxs[0] == 0:
lnB_past = np.log(prior[f] + 1e-16)
else:
# Thomas Parr MMP version
# lnB_past = 0.5 * np.log(B[f][:, :, full_policy[t - 1, f]].dot(qs[t - 1][f]) + 1e-16)
# Karl SPM version
if t == 1 and n == 0 and f == 1:
print('past action:')
print('-------------')
print(full_policy[t - 1, :])
print(B[f][:, :, 0])
print(B[f][:, :, 1])
print(qs[t - 1][f])
lnB_past = np.log(
B[f][:, :, full_policy[t - 1, f]].dot(qs[t - 1][f]) +
1e-16)
# if t == 0:
# print(
# f"qs_t_1 at time t = {t}, factor f = {f}, iteration i = {n}: {qs[t - 1][f]}"
# )
if t == 1 and n == 0:
print(
f"lnB_past at time t = {t}, factor f = {f}, iteration i = {n}: {lnB_past}"
)
"""
=========== Step 3.c ===========
Calculate future message
"""
if t >= future_cutoff:
# TODO: this is redundant - not used in code
lnB_future = last_message[f]
else:
# Thomas Parr MMP version
# B_future = B_transposed[f][:, :, int(full_policy[t, f])].dot(qs[t + 1][f])
# lnB_future = 0.5 * np.log(B_future + 1e-16)
# Karl Friston SPM version
B_future = B_transposed[f][:, :,
int(full_policy[t, f])].dot(
qs[t + 1][f])
lnB_future = np.log(B_future + 1e-16)
# Thomas Parr MMP version
# lnB_past_tensor[f] = 2 * lnBpast
# Karl SPM version
lnB_past_tensor[f] = lnB_past
"""
=========== Step 3.d ===========
Update posterior
"""
if use_gradient_descent:
lns = np.log(qs[t][f] + 1e-16)
# Thomas Parr MMP version
# error = (lnA + lnBpast + lnBfuture) - lns
# Karl SPM version
if t >= future_cutoff:
error = lnA + lnB_past - lns
else:
error = (2 * lnA + lnB_past + lnB_future) - 2 * lns
# print(f"prediction error at time t = {t}, factor f = {f}, iteration i = {n}: {error}")
# print(f"OG {t} {f} {error}")
error -= error.mean()
lns = lns + tau * error
qs_t_f = softmax(lns)
free_energy_pol += 0.5 * qs[t][f].dot(error)
qs[t][f] = qs_t_f
else:
qs[t][f] = softmax(lnA + lnB_past + lnB_future)
# TODO: probably works anyways
# free_energy[t, n] = calc_free_energy(qs[t], lnB_past_tensor, num_factors, likelihood[t])
# free_energy_pol += F[t, n]
qss[n].append(qs)
return qs, qss, free_energy, free_energy_pol
def run_mmp(
A,
B,
obs_t,
policy,
curr_t,
t_horizon,
T,
qs_bma=None,
prior=None,
num_iter=10,
dF=1.0,
dF_tol=0.001,
previous_actions=None,
use_gradient_descent=False,
tau=0.25,
):
"""
Optimise marginal posterior beliefs about hidden states using marginal message-passing scheme (MMP) developed
by Thomas Parr and colleagues, see https://github.com/tejparr/nmpassing
Parameters
----------
- 'A' [numpy nd.array (matrix or tensor or array-of-arrays)]:
Observation likelihood of the generative model, mapping from hidden states to observations.
Used in inference to get the likelihood of an observation, under different hidden state configurations.
- 'B' [numpy.ndarray (tensor or array-of-arrays)]:
Transition likelihood of the generative model, mapping from hidden states at t to hidden states at t+1.
Used in inference to get expected future (or past) hidden states, given past (or future) hidden states (or expectations thereof).
- 'obs_t' [list of length t_horizon of numpy 1D array or array of arrays (with 1D numpy array entries)]:
Sequence of observations sampled from beginning of time horizon the current timestep t. The first observation (the start of the time horizon)
is either the first timestep of the generative process or the first timestep of the policy horizon (whichever is closer to 'curr_t' in time).
The observations over time are stored as a list of numpy arrays, where in case of multi-modalities each numpy array is an array-of-arrays, with
one 1D numpy.ndarray for each modality. In the case of a single modality, each observation is a single 1D numpy.ndarray.
- 'policy' [2D np.ndarray]:
Array of actions constituting a single policy. Policy is a shape (n_steps, n_control_factors) numpy.ndarray, the values of which
indicate actions along a given control factor (column index) at a given timestep (row index).
- 'curr_t' [int]:
Current timestep (relative to the 'absolute' time of the generative process).
- 't_horizon'[int]:
Temporal horizon of inference for states and policies.
- 'T' [int]:
Temporal horizon of the generative process (absolute time)
- `qs_bma` [numpy 1D array, array of arrays (with 1D numpy array entries) or None]:
- 'prior' [numpy 1D array, array of arrays (with 1D numpy array entries) or None]:
Prior beliefs of the agent at the beginning of the time horizon, to be integrated with the marginal likelihood to obtain posterior at the first timestep.
If absent, prior is set to be a uniform distribution over hidden states (identical to the initialisation of the posterior.
-'num_iter' [int]:
Number of variational iterations to run. (optional)
-'dF' [float]:
Starting free energy gradient (dF/dt) before updating in the course of gradient descent. (optional)
-'dF_tol' [float]:
Threshold value of the gradient of the variational free energy (dF/dt), to be checked at each iteration. If
dF <= dF_tol, the iterations are halted pre-emptively and the final marginal posterior belief(s) is(are) returned. (optional)
-'previous_actions' [numpy.ndarray with shape (num_steps, n_control_factors) or None]:
Array of previous actions, which can be used to constrain the 'past' messages in inference to only consider states of affairs that were possible
under actions that are known to have been taken. The first dimension of previous-arrays (previous_actions.shape[0]) encodes how far back in time
the agent is considering. The first timestep of this either corresponds to either the first timestep of the generative process or the f
first timestep of the policy horizon (whichever is sooner in time). (optional)
-'use_gradient_descent' [bool]:
Flag to indicate whether to use gradient descent to optimise posterior beliefs.
-'tau' [float]:
Learning rate for gradient descent (only used if use_gradient_descent is True)
Returns
----------
-'qs' [list of length T of numpy 1D arrays or array of arrays (with 1D numpy array entries):
Marginal posterior beliefs over hidden states (single- or multi-factor) achieved via marginal message pasing
-'qss' [list of lists of length T of numpy 1D arrays or array of arrays (with 1D numpy array entries):
Marginal posterior beliefs about hidden states (single- or multi-factor) held at each timepoint, *about* each timepoint of the observation
sequence
-'F' [2D np.ndarray]:
Variational free energy of beliefs about hidden states, indexed by time point and variational iteration
-'F_pol' [float]:
Total free energy of the policy under consideration.
"""
# get model dimensions
time_window_idxs = np.array([
i
for i in range(max(0, curr_t - t_horizon), min(T, curr_t + t_horizon))
])
window_len = len(time_window_idxs)
print("t_horizon ", t_horizon)
print("window_len ", window_len)
if utils.is_arr_of_arr(obs_t[0]):
n_observations = [obs_array_i.shape[0] for obs_array_i in obs_t[0]]
else:
n_observations = [obs_t[0].shape[0]]
if utils.is_arr_of_arr(B):
n_states = [sub_B.shape[0] for sub_B in B]
else:
n_states = [B[0].shape[0]]
B = utils.to_arr_of_arr(B)
n_modalities = len(n_observations)
n_factors = len(n_states)
"""
=========== Step 1 ===========
Loop over the observation modalities and use assumption of independence among observation modalities
to multiply each modality-specific likelihood onto a single joint likelihood over hidden states [shape = n_states]
"""
# compute time-window, taking into account boundary conditions
if curr_t == 0:
obs_range = [0]
else:
obs_range = range(max(0, curr_t - t_horizon), curr_t + 1)
# likelihood of observations under configurations of hidden causes (over time)
likelihood = np.empty(len(obs_range), dtype=object)
for t in range(len(obs_range)):
# likelihood_t = np.ones(tuple(n_states))
# if n_modalities == 1:
# likelihood_t *= spm_dot(A, obs_t[obs_range[t]], obs_mode=True)
# else:
# for modality in range(n_modalities):
# likelihood_t *= spm_dot(A[modality], obs_t[obs_range[t]][modality], obs_mode=True)
likelihood_t = get_joint_likelihood(A, obs_t[obs_range[t]], n_states)
# print(f"likelihood (pre-logging) {likelihood_t}")
# likelihood[t] = np.log(likelihood_t + 1e-16) # The Thomas Parr MMP version, you log the likelihood first
likelihood[
t] = likelihood_t # Karl SPM version, logging doesn't happen until *after* the dotting with the posterior
"""
=========== Step 2 ===========
Create a flat posterior (and prior if necessary)
If prior is not provided, initialise prior to be identical to posterior
(namely, a flat categorical distribution). Also make a normalized version of
the transpose of the transition likelihood (for computing backwards messages 'from the future')
called `B_t`
"""
qs = [np.empty(n_factors, dtype=object) for i in range(window_len + 1)]
print(len(qs))
for t in range(window_len + 1):
if t == window_len:
for f in range(n_factors):
qs[t][f] = np.zeros(n_states[f])
else:
for f in range(n_factors):
qs[t][f] = np.ones(n_states[f]) / n_states[f]
if prior is None:
prior = np.empty(n_factors, dtype=object)
for f in range(n_factors):
prior[f] = np.ones(n_states[f]) / n_states[f]
if n_factors == 1:
B_t = np.zeros_like(B)
for u in range(B.shape[2]):
B_t[:, :, u] = spm_norm(B[:, :, u].T)
elif n_factors > 1:
B_t = np.empty(n_factors, dtype=object)
for f in range(n_factors):
B_t[f] = np.zeros_like(B[f])
for u in range(B[f].shape[2]):
B_t[f][:, :, u] = spm_norm(B[f][:, :, u].T)
# set final future message as all ones at the time horizon (no information from beyond the horizon)
last_message = np.empty(n_factors, dtype=object)
for f in range(n_factors):
last_message[f] = np.zeros(n_states[f])
"""
=========== Step 3 ===========
Loop over time indices of time window, which includes time before the policy horizon
as well as including the policy horizon
n_steps, n_factors [0 1 2 0;
1 2 0 1]
"""
if previous_actions is None:
previous_actions = np.zeros((1, policy.shape[1]))
full_policy = np.vstack((previous_actions, policy))
# print(f"full_policy shape {full_policy.shape}")
qss = [[] for i in range(num_iter)]
F = np.zeros((len(qs), num_iter))
F_pol = 0.0
print('length of qs:', len(qs))
# print(f"length obs_t {len(obs_t)}")
for n in range(num_iter):
for t in range(0, len(qs)):
# for t in range(0, len(qs)):
lnBpast_tensor = np.empty(n_factors, dtype=object)
for f in range(n_factors):
if t < len(
obs_t
): # this is because of Python indexing (when t == len(obs_t)-1, we're at t == curr_t)
print(t)
# if t <= len(obs_t):
# print(f"t index {t}")
# print(f"length likelihood {len(likelihood)}")
# print(f"length qs {len(qs)}")
# lnA = spm_dot(likelihood[t], qs[t], [f]) # the Thomas Parr MMP version
lnA = np.log(spm_dot(likelihood[t], qs[t], [f]) + 1e-16)
if t == 2 and f == 0:
print(f"lnA at time t = {t}, factor f = {f}: {lnA}")
else:
lnA = np.zeros(n_states[f])
if t == 0:
lnBpast = np.log(prior[f] + 1e-16)
else:
# lnBpast = 0.5 * np.log(
# B[f][:, :, full_policy[t - 1, f]].dot(qs[t - 1][f]) + 1e-16
# ) # the Thomas Parr MMP version
lnBpast = np.log(
B[f][:, :, full_policy[t - 1, f]].dot(qs[t - 1][f]) +
1e-16) # the Karl SPM version
if t == 2 and f == 0:
print(
f"lnBpast at time t = {t}, factor f = {f}: {lnBpast}")
# print(f"lnBpast at time t = {t}, factor f = {f}: {lnBpast}")
# this is never reached
if t >= len(
qs
) - 2: # if we're at the end of the inference chain (at the current moment), the last message is just zeros
lnBfuture = last_message[f]
print('At final timestep!')
# print(f"lnBfuture at time t = {t}, factor f = {f}: {lnBfuture}")
else:
# if t == 0 and f == 0:
# print(B_t[f][:, :, int(full_policy[t, f])])
# print(qs[t + 1][f])
# lnBfuture = 0.5 * np.log(
# B_t[f][:, :, int(full_policy[t, f])].dot(qs[t + 1][f]) + 1e-16
# ) # the Thomas Parr MMP version
lnBfuture = np.log(
B_t[f][:, :, int(full_policy[t, f])].dot(
qs[t + 1][f]) + 1e-16
) # the Karl SPM version (without the 0.5 in front)
if t == 2 and f == 0:
print(
f"lnBfuture at time t = {t}, factor f = {f}: {lnBfuture}"
)
# if t == 0 and f == 0:
# print(f"lnBfuture at time t= {t}: {lnBfuture}")
# lnBpast_tensor[f] = 2 * lnBpast # the Thomas Parr MMP version
lnBpast_tensor[f] = lnBpast # the Karl version
if use_gradient_descent:
# gradients
lns = np.log(qs[t][f] + 1e-16) # current estimate
# e = (lnA + lnBpast + lnBfuture) - lns # prediction error, Thomas Parr version
if t >= len(qs) - 2:
e = lnA + lnBpast - lns
else:
e = (2 * lnA + lnBpast + lnBfuture
) - 2 * lns # prediction error, Karl SPM version
e -= e.mean() # Karl SPM version
print(
f"prediction error at time t = {t}, factor f = {f}: {e}"
)
lns += tau * e # increment the current (log) belief with the prediction error
qs_t_f = softmax(lns)
F_pol += 0.5 * qs[t][f].dot(e)
qs[t][f] = qs_t_f
else:
# free energy minimum for the factor in question
qs[t][f] = softmax(lnA + lnBpast + lnBfuture)
# F[t, n] = calc_free_energy(qs[t], lnBpast_tensor, n_factors, likelihood[t])
# F_pol += F[t, n]
qss[n].append(qs)
return qs, qss, F, F_pol
def run_mmp_v2(A,
B,
ll_seq,
policy,
prev_actions=None,
prior=None,
num_iter=10,
grad_descent=False,
tau=0.25):
# window
past_len = len(ll_seq)
future_len = policy.shape[0]
infer_len = past_len + future_len
future_cutoff = past_len + future_len - 2
# dimensions
_, num_states, _, num_factors = get_model_dimensions(A, B)
A = to_arr_of_arr(A)
B = to_arr_of_arr(B)
# beliefs
qs_seq = [np.empty(num_factors, dtype=object) for _ in range(infer_len)]
for t in range(infer_len):
for f in range(num_factors):
qs_seq[t][f] = np.ones(num_states[f]) / num_states[f]
# last message
qs_T = np.empty(num_factors, dtype=object)
for f in range(num_factors):
qs_T[f] = np.zeros(num_states[f])
# prior
if prior is None:
prior = np.empty(num_factors, dtype=object)
for f in range(num_factors):
prior[f] = np.ones(num_states[f]) / num_states[f]
# transposed transition
trans_B = np.empty(num_factors, dtype=object)
for f in range(num_factors):
trans_B[f] = np.zeros_like(B[f])
for u in range(B[f].shape[2]):
trans_B[f][:, :, u] = spm_norm(B[f][:, :, u].T)
# full policy
if prev_actions is None:
prev_actions = np.zeros((past_len, policy.shape[1]))
policy = np.vstack((prev_actions, policy))
for _ in range(num_iter):
for t in range(infer_len):
for f in range(num_factors):
# likelihood
if t < past_len:
lnA = np.log(spm_dot(ll_seq[t], qs_seq[t], [f]) + 1e-16)
else:
lnA = np.zeros(num_states[f])
# past message
if t == 0:
lnB_past = np.log(prior[f] + 1e-16)
else:
past_msg = B[f][:, :, int(policy[t - 1,
f])].dot(qs_seq[t - 1][f])
lnB_past = np.log(past_msg + 1e-16)
# future message
if t >= future_cutoff:
lnB_future = qs_T[f]
else:
future_msg = trans_B[f][:, :, int(policy[t, f])].dot(
qs_seq[t + 1][f])
lnB_future = np.log(future_msg + 1e-16)
# inference
if grad_descent:
lnqs = np.log(qs_seq[t][f] + 1e-16)
coeff = 1 if (t >= future_cutoff) else 2
err = (coeff * lnA + lnB_past + lnB_future) - coeff * lnqs
err -= err.mean()
lnqs = lnqs + tau * err
qs_seq[t][f] = softmax(lnqs)
else:
qs_seq[t][f] = softmax(lnA + lnB_past + lnB_future)
return qs_seq