def apt_maf_loss_atomic_proposal(net, svi=False, combined_loss=False):
"""Define loss function for training with a atomic proposal. Assumes a
uniform proposal distribution over each sample parameter and an externally
provided set of alternatives.
net: MAF-based conditional density net
svi : bool
Whether to use SVI version of the mdn or not
"""
assert net.density == 'maf'
assert not svi, 'SVI not supported for MAFs'
# define symbolic variable to hold params that will be inferred
# params : n_batch x n_outputs
# all_thetas : (n_batch * (n_atoms + 1) x n_outputs
# lprs : (n_atoms + 1) x n_batch
# stats : n_batch x n_inputs
# x_nl : (n_batch * (n_atoms + 1)) x n_inputs
theta_all = tensorN(2, name='params_nl', dtype=dtype)
x_nl = tensorN(2, name='stats_nl', dtype=dtype)
lprs = tensorN(2, name='lprs', dtype=dtype) # log tilde_p / p
n_batch = tt.shape(lprs)[1]
n_atoms = tt.shape(lprs)[0] - 1
# compute MAF log-densities for true and other atoms
lprobs = theano.clone(output=net.lprobs,
replace={
net.params: theta_all,
net.stats: x_nl
},
share_inputs=True)
lprobs = tt.reshape(lprobs, newshape=(n_atoms + 1, n_batch), ndim=2)
# compute nonnormalized log posterior probabilities
atomic_ppZ = lprobs - lprs
# compute posterior probability of true params in atomic task
atomic_pp = atomic_ppZ[0, :].squeeze() - \
MyLogSumExp(atomic_ppZ, axis=0).squeeze()
# collect the extra input variables that have to be provided for each
# training data point, and calculate the loss by averaging over samples
trn_inputs = [theta_all, x_nl, lprs]
if combined_loss: # add prior loss on prior samples
l_ml = lprobs[0, :].squeeze() # direct posterior evaluation
is_prior_sample = tensorN(1, name='prop_mask', dtype=dtype)
trn_inputs.append(is_prior_sample)
loss = -tt.mean(atomic_pp + is_prior_sample * l_ml)
else:
loss = -tt.mean(atomic_pp)
return loss, trn_inputs
def init_mdn(self, svi=False, n_components=1, rank=None,
mdn_actfun=lnl.tanh, homoscedastic=False, min_precisions=None,
**unused_kwargs):
"""
:param svi: bool
Whether to use SVI version or not
:param n_components: int
:param rank: int
:param homoscedastic: bool
:param unused_kwargs: dict
:param mdn_actfun: lasagne nonlinearity
activation function for hidden units
:param min_precisions: minimum values for diagonal elements of precision
matrix for all components (usually taken to be prior precisions)
:return: None
"""
self.svi, self.n_components, self.rank, self.mdn_actfun,\
self.homoscedastic, self.min_precisions = \
svi, n_components, rank, mdn_actfun, homoscedastic, min_precisions
for key in unused_kwargs.keys():
print("MDN ignoring unused input {0}".format(key))
# hidden layers
for l in range(len(self.n_hiddens)):
self.layer['hidden_' + str(l + 1)] = dl.FullyConnectedLayer(
last(self.layer), n_units=self.n_hiddens[l],
actfun=self.mdn_actfun,
svi=self.svi, name='h' + str(l + 1))
last_hidden = last(self.layer)
# mixture layers
self.layer['mixture_weights'] = dl.MixtureWeightsLayer(last_hidden,
n_units=self.n_components, actfun=lnl.softmax, svi=self.svi,
name='weights')
self.layer['mixture_means'] = dl.MixtureMeansLayer(last_hidden,
n_components=self.n_components, n_dim=self.n_outputs, svi=self.svi,
name='means')
if self.homoscedastic:
PrecisionsLayer = dl.MixtureHomoscedasticPrecisionsLayer
else:
PrecisionsLayer = dl.MixturePrecisionsLayer
# why is homoscedastic an input to the layer init?
self.layer['mixture_precisions'] = PrecisionsLayer(last_hidden,
n_components=self.n_components, n_dim=self.n_outputs, svi=self.svi,
name='precisions', rank=self.rank, homoscedastic=self.homoscedastic,
min_precisions=min_precisions)
last_mog = [self.layer['mixture_weights'],
self.layer['mixture_means'],
self.layer['mixture_precisions']]
# mixture parameters
# a : weights, matrix with shape (batch, n_components)
# ms : means, list of len n_components with (batch, n_dim, n_dim)
# Us : precision factors, n_components list with (batch, n_dim, n_dim)
# ldetUs : log determinants of precisions, n_comp list with (batch, )
self.a, self.ms, precision_out = ll.get_output(last_mog,
deterministic=False)
self.Us = precision_out['Us']
self.ldetUs = precision_out['ldetUs']
self.comps = {
**{'a': self.a},
**{'m' + str(i): self.ms[i] for i in range(self.n_components)},
**{'U' + str(i): self.Us[i] for i in range(self.n_components)}}
# log probability of y given the mixture distribution
# lprobs_comps : log probs per component, list of len n_components with (batch, )
# probs : log probs of mixture, (batch, )
self.lprobs_comps = [-0.5 * tt.sum(tt.sum((self.params - m).dimshuffle(
[0, 'x', 1]) * U, axis=2)**2, axis=1) + ldetU
for m, U, ldetU in zip(self.ms, self.Us, self.ldetUs)]
self.lprobs = (MyLogSumExp(tt.stack(self.lprobs_comps, axis=1) + tt.log(self.a), axis=1)
- (0.5 * self.n_outputs * np.log(2 * np.pi))).squeeze()
# the quantities from above again, but with deterministic=True
# --- in the svi case, this will disable injection of randomness;
# the mean of weights is used instead
self.da, self.dms, dprecision_out = ll.get_output(last_mog,
deterministic=True)
self.dUs = dprecision_out['Us']
self.dldetUs = dprecision_out['ldetUs']
self.dcomps = {
**{'a': self.da},
**{'m' + str(i): self.dms[i] for i in range(self.n_components)},
**{'U' + str(i): self.dUs[i] for i in range(self.n_components)}}
self.dlprobs_comps = [-0.5 * tt.sum(tt.sum((self.params - m).dimshuffle(
[0, 'x', 1]) * U, axis=2)**2, axis=1) + ldetU
for m, U, ldetU in zip(self.dms, self.dUs, self.dldetUs)]
self.dlprobs = (MyLogSumExp(tt.stack(self.dlprobs_comps, axis=1) + tt.log(self.da), axis=1) \
- (0.5 * self.n_outputs * np.log(2 * np.pi))).squeeze()
# parameters of network
self.aps = ll.get_all_params(last_mog) # all parameters
self.mps = ll.get_all_params(last_mog, mp=True) # means
self.sps = ll.get_all_params(last_mog, sp=True) # log stds
# weight and bias parameter sets as separate lists
self.mps_wp = ll.get_all_params(last_mog, mp=True, wp=True)
self.sps_wp = ll.get_all_params(last_mog, sp=True, wp=True)
self.mps_bp = ll.get_all_params(last_mog, mp=True, bp=True)
self.sps_bp = ll.get_all_params(last_mog, sp=True, bp=True)
def apt_loss_MoG_proposal(mdn, prior, n_proposal_components=None, svi=False):
"""Define loss function for training with a MoG proposal, allowing
the proposal distribution to be different for each sample. The proposal
means, precisions and weights are passed along with the stats and params.
This function computes a symbolic expression but does not actually compile
or calculate that expression.
This loss does not include any regularization or prior terms, which must be
added separately before compiling.
Parameters
----------
mdn : NeuralNet
Mixture density network.
svi : bool
Whether to use SVI version of the mdn or not
Returns
-------
loss : theano scalar
Loss function
trn_inputs : list
Tensors to be provided to the loss function during training
"""
assert mdn.density == 'mog'
uniform_prior = isinstance(prior, dd.Uniform)
if not uniform_prior and not isinstance(prior, dd.Gaussian):
raise NotImplemented # prior must be Gaussian or uniform
ncprop = mdn.n_components if n_proposal_components is None \
else n_proposal_components # default to component count of posterior
nbatch = mdn.params.shape[0]
# a mixture weights, ms means, P=U^T U precisions, QFs are tensorQF(P, m)
a, ms, Us, ldetUs, Ps, ldetPs, Pms, QFs = \
mdn.get_mog_tensors(return_extras=True, svi=svi)
# convert mixture vars from lists of tensors to single tensors, of sizes:
las = tt.log(a)
Ps = tt.stack(Ps, axis=3).dimshuffle(0, 3, 1, 2)
Pms = tt.stack(Pms, axis=2).dimshuffle(0, 2, 1)
ldetPs = tt.stack(ldetPs, axis=1)
QFs = tt.stack(QFs, axis=1)
# as: (batch, mdn.n_components)
# Ps: (batch, mdn.n_components, n_outputs, n_outputs)
# Pm: (batch, mdn.n_components, n_outputs)
# ldetPs: (batch, mdn.n_components)
# QFs: (batch, mdn.n_components)
# Define symbolic variables, that hold for each sample's MoG proposal:
# precisions times means (batch, ncprop, n_outputs)
# precisions (batch, ncprop, n_outputs, n_outputs)
# log determinants of precisions (batch, ncprop)
# log mixture weights (batch, ncprop)
# quadratic forms QF = m^T P m (batch, ncprop)
prop_Pms = tensorN(3, name='prop_Pms', dtype=dtype)
prop_Ps = tensorN(4, name='prop_Ps', dtype=dtype)
prop_ldetPs = tensorN(2, name='prop_ldetPs', dtype=dtype)
prop_las = tensorN(2, name='prop_las', dtype=dtype)
prop_QFs = tensorN(2, name='prop_QFs', dtype=dtype)
# calculate corrections to precisions (P_0s) and precisions * means (Pm_0s)
P_0s = prop_Ps
Pm_0s = prop_Pms
if not uniform_prior: # Gaussian prior
P_0s = P_0s - prior.P
Pm_0s = Pm_0s - prior.Pm
# To calculate the proposal posterior, we multiply all mixture component
# pdfs from the true posterior by those from the proposal. The resulting
# new mixture is ordered such that all product terms involving the first
# component of the true posterior appear first. The shape of pp_Ps is
# (batch, mdn.n_components, ncprop, n_outputs, n_outputs)
pp_Ps = Ps.dimshuffle(0, 1, 'x', 2, 3) + P_0s.dimshuffle(0, 'x', 1, 2, 3)
pp_Ss = invert_each(pp_Ps) # covariances of proposal posterior components
pp_ldetPs = det_each(pp_Ps, log=True) # log determinants
# precision times mean for each proposal posterior component:
pp_Pms = Pms.dimshuffle(0, 1, 'x', 2) + Pm_0s.dimshuffle(0, 'x', 1, 2)
# mean of proposal posterior components:
pp_ms = (pp_Ss * pp_Pms.dimshuffle(0, 1, 2, 'x', 3)).sum(axis=4)
# quadratic form defined by each pp_P evaluated at each pp_m
pp_QFs = (pp_Pms * pp_ms).sum(axis=3)
# normalization constants for integrals of Gaussian product-quotients
# (for Gaussian proposals) or Gaussian products (for uniform priors)
# Note we drop a "constant" (for each combination of sample, proposal
# component posterior component) term of
#
# 0.5 * (tensorQF(prior.P, prior.m) - prior_ldetP)
#
# since we're going to normalize the pp mixture coefficients sum to 1
pp_lZs = 0.5 * ((ldetPs - QFs).dimshuffle(0, 1, 'x') +
(prop_ldetPs - prop_QFs).dimshuffle(0, 'x', 1) -
(pp_ldetPs - pp_QFs))
# calculate non-normalized log mixture coefficients of the proposal
# posterior by adding log posterior weights a to normalization coefficients
# Z. These do not yet sum to 1 in the linear domain
pp_las_nonnormed = \
las.dimshuffle(0, 1, 'x') + prop_las.dimshuffle(0, 'x', 1) + pp_lZs
# reshape tensors describing proposal posterior components so that there's
# only one dimension that ranges over components
ncpp = ncprop * mdn.n_components # number of proposal posterior components
pp_las_nonnormed = pp_las_nonnormed.reshape((nbatch, ncpp))
pp_ldetPs = pp_ldetPs.reshape((nbatch, ncpp))
pp_ms = pp_ms.reshape((nbatch, ncpp, mdn.n_outputs))
pp_Ps = pp_Ps.reshape((nbatch, ncpp, mdn.n_outputs, mdn.n_outputs))
# normalize log mixture weights so they sum to 1 in the linear domain
pp_las = pp_las_nonnormed - MyLogSumExp(pp_las_nonnormed, axis=1)
mog_LL_inputs = \
[(pp_ms[:, i, :], pp_Ps[:, i, :, :], pp_ldetPs[:, i])
for i in range(ncpp)] # list (over comps) of tuples (over vars)
# 2 tensor inputs, lists (over comps) of tensors:
mog_LL_inputs = [mdn.params, pp_las, *zip(*mog_LL_inputs)]
loss = -tt.mean(mog_LL(*mog_LL_inputs))
# collect extra input variables to be provided for each training data point
trn_inputs = [
mdn.params, mdn.stats, prop_Pms, prop_Ps, prop_ldetPs, prop_las,
prop_QFs
]
return loss, trn_inputs
def apt_mdn_loss_atomic_proposal(mdn, svi=False, combined_loss=False):
"""Define loss function for training with a atomic proposal. Assumes a
uniform proposal distribution over each sample parameter and an externally
provided set of alternatives.
mdn: NeuralNet
Mixture density network.
svi : bool
Whether to use SVI version of the mdn or not
"""
assert mdn.density == 'mog'
# a is mixture weights, ms are means, U^T U are precisions
a, ms, Us, ldetUs = mdn.get_mog_tensors(svi=svi)
# define symbolic variable to hold params that will be inferred
# theta_all : (n_batch * (n_atoms + 1) x n_outputs
# lprs : n_batch x (n_atoms+1)
theta_all = tensorN(3, name='params_nl',
dtype=dtype) # true (row 1), atoms
lprs = tensorN(2, name='lprs', dtype=dtype) # log tilde_p / p
# calculate Mahalanobis distances distances wrt U'U for every theta,x pair
# diffs : [ n_batch x (n_atoms+1) x n_outputs for each component ]
# Ms : [ n_batch x (n_atoms+1) for each component ]
# Ms[k][n,i] = (theta[i] - m[k][n])' U[k][n]' U[k][n] (theta[i] - m[k][n])
dthetas = [theta_all - m.dimshuffle([0, 'x', 1])
for m in ms] # theta[i] - m[k][n]
Ms = [
tt.sum(tt.sum(dtheta.dimshuffle([0, 1, 'x', 2]) *
U.dimshuffle([0, 'x', 1, 2]),
axis=3)**2,
axis=2) for dtheta, U in zip(dthetas, Us)
]
# compute (unnormalized) log-densities, weighted by log prior ratios
Ms = [-0.5 * M + lprs for M in Ms]
# compute per-component log-densities and log-normalizers
lprobs_comps = [M[:, 0] + ldetU for M, ldetU in zip(Ms, ldetUs)]
lZ_comps = [
MyLogSumExp(M, axis=1).squeeze() + ldetU
for M, ldetU in zip(Ms, ldetUs)
] # sum over all proposal thetas
# compute overall log-densities and log-normalizers across components
lq = MyLogSumExp(tt.stack(lprobs_comps, axis=1) + tt.log(a), axis=1)
lZ = MyLogSumExp(tt.stack(lZ_comps, axis=1) + tt.log(a), axis=1)
lprobs = lq.squeeze() - lZ.squeeze()
# collect the extra input variables that have to be provided for each
# training data point
trn_inputs = [theta_all, mdn.stats, lprs]
if combined_loss: # add prior loss on prior samples
l_ml = lq.squeeze() # direct posterior evalution
is_prior_sample = tensorN(1, name='prop_mask', dtype=dtype)
trn_inputs.append(is_prior_sample)
loss = -tt.mean(lprobs + is_prior_sample * l_ml)
else:
loss = -tt.mean(lprobs) # average over samples
return loss, trn_inputs
def apt_loss_gaussian_proposal(mdn, prior, svi=False):
"""Define loss function for training with a Gaussian proposal, allowing
the proposal distribution to be different for each sample. The proposal
mean and precision are passed along with the stats and params.
This function computes a symbolic expression but does not actually compile
or calculate that expression.
This loss does not include any regularization or prior terms, which must be
added separately before compiling.
Parameters
----------
mdn : NeuralNet
Mixture density network.
svi : bool
Whether to use SVI version of the mdn or not
Returns
-------
loss : theano scalar
Loss function
trn_inputs : list
Tensors to be provided to the loss function during training
"""
assert mdn.density == 'mog'
uniform_prior = isinstance(prior, dd.Uniform)
if not uniform_prior and not isinstance(prior, dd.Gaussian):
raise NotImplemented # prior must be Gaussian or uniform
# a mixture weights, ms means, P=U^T U precisions, QFs are tensorQF(P, m)
a, ms, Us, ldetUs, Ps, ldetPs, Pms, QFs = \
mdn.get_mog_tensors(return_extras=True, svi=svi)
# define symbolic variables to hold for each sample's Gaussian proposal:
# means (batch, n_outputs)
# precisions (batch, n_outputs, n_outputs)
prop_m = tensorN(2, name='prop_m', dtype=dtype)
prop_P = tensorN(3, name='prop_P', dtype=dtype)
# calculate corrections to precision (P_0) and precision * mean (Pm_0)
P_0 = prop_P
Pm_0 = tt.sum(prop_P * prop_m.dimshuffle(0, 'x', 1), axis=2)
if not uniform_prior: # Gaussian prior
P_0 = P_0 - prior.P
Pm_0 = Pm_0 - prior.Pm
# precisions of proposal posterior components:
pp_Ps = [P + P_0 for P in Ps]
# covariances of proposal posterior components:
pp_Ss = [invert_each(P) for P in pp_Ps]
# log determinant of each proposal posterior component's precision:
pp_ldetPs = [det_each(P, log=True) for P in pp_Ps]
# precision times mean for each proposal posterior component:
pp_Pms = [Pm + Pm_0 for Pm in Pms]
# mean of proposal posterior components:
pp_ms = [tt.batched_dot(S, Pm) for S, Pm in zip(pp_Ss, pp_Pms)]
# quadratic form defined by each pp_P evaluated at each pp_m
pp_QFs = [tt.sum(m * P, axis=1) for m, P in zip(pp_ms, pp_Pms)]
# normalization constants for integrals of Gaussian product-quotients
# (for Gaussian proposals) or Gaussian products (for uniform priors)
# Note we drop a "constant" (for each sample, w.r.t trained params) term of
#
# 0.5 * (prop_ldetP - prior_ldetP +
# tensorQF(prior.P, prior.m) - tensorQF(prop_P, prop_m))
#
# since we're going to normalize the pp mixture coefficients sum to 1
pp_lZs = [
0.5 * (ldetP - pp_ldetP - QF + pp_QF)
for ldetP, pp_ldetP, QF, pp_QF in zip(ldetPs, pp_ldetPs, QFs, pp_QFs)
]
# calculate log mixture coefficients of proposal posterior in two steps:
# 1) add log posterior weights a to normalization coefficients Z
# 2) normalize to sum to 1 in the linear domain, but stay in the log domain
pp_las = tt.stack(pp_lZs, axis=1) + tt.log(a)
pp_las = pp_las - MyLogSumExp(pp_las, axis=1)
loss = -tt.mean(mog_LL(mdn.params, pp_las, pp_ms, pp_Ps, pp_ldetPs))
# collect extra input variables to be provided for each training data point
trn_inputs = [mdn.params, mdn.stats, prop_m, prop_P]
return loss, trn_inputs
def apt_loss_gaussian_proposal(mdn,
prior,
svi=False,
add_prior_precision=True,
Ptol=1e-7):
"""Define loss function for training with a Gaussian proposal, allowing
the proposal distribution to be different for each sample. The proposal
mean and precision are passed along with the stats and params.
This function computes a symbolic expression but does not actually compile
or calculate that expression.
This loss does not include any regularization or prior terms, which must be
added separately before compiling.
Parameters
----------
mdn : NeuralNet
Mixture density network.
svi : bool
Whether to use SVI version of the mdn or not
Returns
-------
loss : theano scalar
Loss function
trn_inputs : list
Tensors to be provided to the loss function during training
prior: delfi distribution
Prior distribution on parameters
"""
assert mdn.density == 'mog'
uniform_prior = isinstance(prior, dd.Uniform)
if not uniform_prior and not isinstance(prior, dd.Gaussian):
raise NotImplemented # prior must be Gaussian or uniform
# a mixture weights, ms means, P=U^T U precisions, QFs are tensorQF(P, m)
a, ms, Us, ldetUs, Ps, ldetPs, Pms, QFs = \
mdn.get_mog_tensors(return_extras=True, svi=svi)
# define symbolic variables to hold for each sample's Gaussian proposal:
# means (batch, n_outputs)
# precisions (batch, n_outputs, n_outputs)
prop_m = tensorN(2, name='prop_m', dtype=dtype)
prop_P = tensorN(3, name='prop_P', dtype=dtype)
# calculate corrections to precision (P_0) and precision * mean (Pm_0)
P_0 = prop_P
Pm_0 = tt.sum(prop_P * prop_m.dimshuffle(0, 'x', 1), axis=2)
if not uniform_prior and not add_prior_precision: # Gaussian prior
P_0 = P_0 - prior.P
Pm_0 = Pm_0 - prior.Pm
# precisions of proposal posterior components (before numerical conditioning step)
pp_Ps = [P + P_0 for P in Ps]
# get square roots of diagonal entries of posterior proposal precision components, which are equal to the L2 norms
# of the Cholesky factor columns for the same matrix. we'll use these to improve the numerical conditioning of pp_Ps
ds = [
tt.sqrt(tt.sum(pp_P * np.eye(mdn.n_outputs), axis=2)) for pp_P in pp_Ps
]
# normalize the estimate of each true posterior component according to the corresponding elements of d:
# first normalize the Cholesky factor of the true posterior component estimate...
Us_normed = [U / d.dimshuffle(0, 'x', 1) for U, d in zip(Us, ds)]
# then normalize the propsal. the resulting list is the same proposal, differently normalized for each component of
# the true posterior
P_0s_normed = [
P_0 / (d.dimshuffle(0, 'x', 1) * d.dimshuffle(0, 1, 'x')) for d in ds
]
pp_Ps_normed = [
tt.batched_dot(U_normed.dimshuffle(0, 2, 1), U_normed) + P_0_normed +
np.eye(mdn.n_outputs) * Ptol
for U_normed, P_0_normed in zip(Us_normed, P_0s_normed)
]
# lower Cholesky factors for normalized precisions of proposal posterior components
pp_Ls_normed = [cholesky_each(pp_P_normed) for pp_P_normed in pp_Ps_normed]
# log determinants of lower Cholesky factors for normalized precisions of proposal posterior components
pp_ldetLs_normed = [
tt.sum(tt.log(tt.sum(pp_L_normed * np.eye(mdn.n_outputs), axis=2)),
axis=1) for pp_L_normed in pp_Ls_normed
]
# precisions of proposal posterior components (now well-conditioned)
pp_Ps = [
d.dimshuffle(0, 1, 'x') * pp_P_normed * d.dimshuffle(0, 'x', 1)
for pp_P_normed, d in zip(pp_Ps_normed, ds)
]
# log determinants of proposal posterior precisions
pp_ldetPs = [
2.0 * (tt.sum(tt.log(d), axis=1) + pp_ldetL_normed)
for d, pp_ldetL_normed in zip(ds, pp_ldetLs_normed)
]
# covariances of proposal posterior components:
pp_Ss = [invert_each(P) for P in pp_Ps]
# precision times mean for each proposal posterior component:
pp_Pms = [Pm + Pm_0 for Pm in Pms]
# mean of proposal posterior components:
pp_ms = [tt.batched_dot(S, Pm) for S, Pm in zip(pp_Ss, pp_Pms)]
# quadratic form defined by each pp_P evaluated at each pp_m
pp_QFs = [tt.sum(m * Pm, axis=1) for m, Pm in zip(pp_ms, pp_Pms)]
# normalization constants for integrals of Gaussian product-quotients
# (for Gaussian proposals) or Gaussian products (for uniform priors)
# Note we drop a "constant" (for each sample, w.r.t trained params) term of
#
# 0.5 * (prop_ldetP - prior_ldetP + tensorQF(prior.P, prior.m) - tensorQF(prop_P, prop_m))
#
# since we're going to normalize the pp mixture coefficients sum to 1
pp_lZs = [
0.5 * (ldetP - pp_ldetP - QF + pp_QF)
for ldetP, pp_ldetP, QF, pp_QF in zip(ldetPs, pp_ldetPs, QFs, pp_QFs)
]
# calculate log mixture coefficients of proposal posterior in two steps:
# 1) add log posterior weights a to normalization coefficients Z
# 2) normalize to sum to 1 in the linear domain, but stay in the log domain
pp_las = tt.stack(pp_lZs, axis=1) + tt.log(a)
pp_las = pp_las - MyLogSumExp(pp_las, axis=1)
loss = -tt.mean(mog_LL(mdn.params, pp_las, pp_ms, pp_Ps, pp_ldetPs))
# collect extra input variables to be provided for each training data point
trn_inputs = [mdn.params, mdn.stats, prop_m, prop_P]
return loss, trn_inputs
