# Source code for funsor.joint

```# Copyright Contributors to the Pyro project.

import math
from collections import OrderedDict
from functools import reduce
from typing import Tuple, Union

from multipledispatch import dispatch

import funsor.ops as ops
from funsor.cnf import Contraction, GaussianMixture
from funsor.delta import Delta
from funsor.domains import Bint
from funsor.gaussian import Gaussian, align_gaussian
from funsor.ops import AssociativeOp
from funsor.tensor import Tensor, align_tensor
from funsor.terms import Funsor, Independent, Number, Reduce, Unary, eager, moment_matching, normalize

def eager_cat_homogeneous(name, part_name, *parts):
assert parts
output = parts[0].output
inputs = OrderedDict([(part_name, None)])
for part in parts:
assert part.output == output
assert part_name in part.inputs
inputs.update(part.inputs)

int_inputs = OrderedDict((k, v) for k, v in inputs.items() if v.dtype != "real")
real_inputs = OrderedDict((k, v) for k, v in inputs.items() if v.dtype == "real")
inputs = int_inputs.copy()
inputs.update(real_inputs)
discretes = []
info_vecs = []
precisions = []
for part in parts:
inputs[part_name] = part.inputs[part_name]
int_inputs[part_name] = inputs[part_name]
shape = tuple(d.size for d in int_inputs.values())
if isinstance(part, Gaussian):
discrete = None
gaussian = part
elif issubclass(type(part), GaussianMixture):  # TODO figure out why isinstance isn't working
discrete, gaussian = part.terms[0], part.terms[1]
discrete = ops.expand(align_tensor(int_inputs, discrete), shape)
else:
raise NotImplementedError("TODO")
discretes.append(discrete)
info_vec, precision = align_gaussian(inputs, gaussian)
info_vecs.append(ops.expand(info_vec, shape + (-1,)))
precisions.append(ops.expand(precision, shape + (-1, -1)))
if part_name != name:
del inputs[part_name]
del int_inputs[part_name]

dim = 0
info_vec = ops.cat(dim, *info_vecs)
precision = ops.cat(dim, *precisions)
inputs[name] = Bint[info_vec.shape[dim]]
int_inputs[name] = inputs[name]
result = Gaussian(info_vec, precision, inputs)
if any(d is not None for d in discretes):
for i, d in enumerate(discretes):
if d is None:
discretes[i] = ops.new_zeros(info_vecs[i], info_vecs[i].shape[:-1])
discrete = ops.cat(dim, *discretes)
result = result + Tensor(discrete, int_inputs)
return result

#################################
# patterns for moment-matching
#################################

def moment_matching_contract_default(*args):
return None

def moment_matching_contract_joint(red_op, bin_op, reduced_vars, discrete, gaussian):

approx_vars = frozenset(k for k in reduced_vars if k in gaussian.inputs
and gaussian.inputs[k].dtype != 'real')
exact_vars = reduced_vars - approx_vars

if exact_vars and approx_vars:
return Contraction(red_op, bin_op, exact_vars, discrete, gaussian).reduce(red_op, approx_vars)

if approx_vars and not exact_vars:
discrete += gaussian.log_normalizer
num_elements = reduce(ops.mul, [
gaussian.inputs[k].num_elements for k in approx_vars.difference(discrete.inputs)], 1)
if num_elements != 1:
new_discrete -= math.log(num_elements)

int_inputs = OrderedDict((k, d) for k, d in gaussian.inputs.items() if d.dtype != 'real')
probs = (discrete - new_discrete.clamp_finite()).exp()

old_loc = Tensor(ops.cholesky_solve(ops.unsqueeze(gaussian.info_vec, -1), gaussian._precision_chol).squeeze(-1),
int_inputs)
new_loc = (probs * old_loc).reduce(ops.add, approx_vars)
old_cov = Tensor(ops.cholesky_inverse(gaussian._precision_chol), int_inputs)
diff = old_loc - new_loc
outers = Tensor(ops.unsqueeze(diff.data, -1) * ops.unsqueeze(diff.data, -2), diff.inputs)
new_cov = ((probs * old_cov).reduce(ops.add, approx_vars) +

# Numerically stabilize by adding bogus precision to empty components.
mask = ops.unsqueeze(ops.unsqueeze((total.data == 0), -1), -1)
new_cov.data = new_cov.data + mask * ops.new_eye(new_cov.data, new_cov.data.shape[-1:])

new_precision = Tensor(ops.cholesky_inverse(ops.cholesky(new_cov.data)), new_cov.inputs)
new_info_vec = (new_precision.data @ ops.unsqueeze(new_loc.data, -1)).squeeze(-1)
new_inputs = new_loc.inputs.copy()
new_inputs.update((k, d) for k, d in gaussian.inputs.items() if d.dtype == 'real')
new_gaussian = Gaussian(new_info_vec, new_precision.data, new_inputs)
new_discrete -= new_gaussian.log_normalizer

return new_discrete + new_gaussian

return None

####################################################
# Patterns for normalizing
####################################################

def eager_reduce_exp(op, arg, reduced_vars):
if log_result is not normalize(Reduce, ops.logaddexp, arg.arg, reduced_vars):
return log_result.exp()
return None

[docs]@eager.register(Independent,
(Contraction[ops.NullOp, ops.AddOp, frozenset, Tuple[Delta, Union[Number, Tensor], Gaussian]],
Contraction[ops.NullOp, ops.AddOp, frozenset, Tuple[Delta, Union[Number, Tensor, Gaussian]]]),
str, str, str)
def eager_independent_joint(joint, reals_var, bint_var, diag_var):
if diag_var not in joint.terms[0].fresh:
return None

delta = Independent(joint.terms[0], reals_var, bint_var, diag_var)
new_terms = (delta,) + tuple(t.reduce(ops.add, bint_var) for t in joint.terms[1:])
return reduce(joint.bin_op, new_terms)
```