# Compute Symbolic Closed-form Posteriors¶

Author

Brandon T. Willard

Date

2019-11-24

import numpy as np

import theano
import theano.tensor as tt

import pymc3 as pm

from functools import partial

from unification import var

from kanren import run
from kanren.graph import reduceo, walko

from symbolic_pymc.theano.printing import tt_pprint
from symbolic_pymc.theano.pymc3 import model_graph

from symbolic_pymc.relations.theano.conjugates import conjugate

theano.config.cxx = ''
theano.config.compute_test_value = 'ignore'

a_tt = tt.vector('a')
R_tt = tt.matrix('R')
F_t_tt = tt.matrix('F')
V_tt = tt.matrix('V')

a_tt.tag.test_value = np.r_[1., 0.]
R_tt.tag.test_value = np.diag([10., 10.])
F_t_tt.tag.test_value = np.c_[-2., 1.]
V_tt.tag.test_value = np.diag([0.5])

y_tt = tt.as_tensor_variable(np.r_[-3.])
y_tt.name = 'y'

with pm.Model() as model:

# A normal prior
beta_rv = pm.MvNormal('beta', a_tt, R_tt, shape=(2,))

# An observed random variable using the prior as a regression parameter
E_y_rv = F_t_tt.dot(beta_rv)
Y_rv = pm.MvNormal('Y', E_y_rv, V_tt, observed=y_tt)

# Create a graph for the model
fgraph = model_graph(model, output_vars=[Y_rv])

def conjugate_graph(graph):
"""Apply conjugate relations throughout a graph."""

def fixedp_conjugate_walko(x, y):
return reduceo(partial(walko, conjugate), x, y)

expr_graph, = run(1, var('q'),
fixedp_conjugate_walko(graph, var('q')))

fgraph_opt = expr_graph.eval_obj
fgraph_opt_tt = fgraph_opt.reify()
return fgraph_opt_tt

fgraph_conj = conjugate_graph(fgraph.outputs[0])


## Before¶

>>> print(tt_pprint(fgraph))
F in R**(N^F_0 x N^F_1), a in R**(N^a_0), R in R**(N^R_0 x N^R_1)
V in R**(N^V_0 x N^V_1)
beta ~ N(a, R) in R**(N^beta_0), Y ~ N((F * beta), V) in R**(N^Y_0)
Y = [-3.]


## After¶

>>> print(tt_pprint(fgraph_conj))
a in R**(N^a_0), R in R**(N^R_0 x N^R_1), F in R**(N^F_0 x N^F_1)
c in R**(N^c_0 x N^c_1), d in R**(N^d_0 x N^d_1)
V in R**(N^V_0 x N^V_1), e in R**(N^e_0 x N^e_1)
b ~ N((a + (((R * F.T) * c) * ([-3.] - (F * a)))), (R - ((((R * F.T) * d) * (V + (F * (R * F.T)))) * ((R * F.T) * e).T))) in R**(N^b_0)
b