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"""Probability models (Chapter 12-13)"""
import
copy
import
random
from
collections
import
defaultdict
from
functools
import
reduce
import
numpy
as
np
from
utils4e
import
product
,
probability
,
extend
# ______________________________________________________________________________
# Chapter 12 Qualifying Uncertainty
# 12.1 Acting Under Uncertainty
def
DTAgentProgram
(
belief_state
):
"""A decision-theoretic agent. [Figure 12.1]"""
def
program
(
percept
):
belief_state
.
observe
(
program
.
action
,
percept
)
program
.
action
=
max
(
belief_state
.
actions
(),
key
=
belief_state
.
expected_outcome_utility
)
return
program
.
action
program
.
action
=
None
return
program
# ______________________________________________________________________________
# 12.2 Basic Probability Notation
class
ProbDist
:
"""A discrete probability distribution. You name the random variable
in the constructor, then assign and query probability of values.
>>> P = ProbDist('Flip'); P['H'], P['T'] = 0.25, 0.75; P['H']
0.25
>>> P = ProbDist('X', {'lo': 125, 'med': 375, 'hi': 500})
>>> P['lo'], P['med'], P['hi']
(0.125, 0.375, 0.5)
"""
def
__init__
(
self
,
varname
=
'?'
,
freqs
=
None
):
"""If freqs is given, it is a dictionary of values - frequency pairs,
then ProbDist is normalized."""
self
.
prob
=
{}
self
.
varname
=
varname
self
.
values
=
[]
if
freqs
:
for
(
v
,
p
)
in
freqs
.
items
():
self
[
v
]
=
p
self
.
normalize
()
def
__getitem__
(
self
,
val
):
"""Given a value, return P(value)."""
try
:
return
self
.
prob
[
val
]
except
KeyError
:
return
0
def
__setitem__
(
self
,
val
,
p
):
"""Set P(val) = p."""
if
val
not
in
self
.
values
:
self
.
values
.
append
(
val
)
self
.
prob
[
val
]
=
p
def
normalize
(
self
):
"""Make sure the probabilities of all values sum to 1.
Returns the normalized distribution.
Raises a ZeroDivisionError if the sum of the values is 0."""
total
=
sum
(
self
.
prob
.
values
())
if
not
np
.
isclose
(
total
,
1.0
):
for
val
in
self
.
prob
:
self
.
prob
[
val
]
/=
total
return
self
def
show_approx
(
self
,
numfmt
=
'{:.3g}'
):
"""Show the probabilities rounded and sorted by key, for the
sake of portable doctests."""
return
', '
.
join
([(
'{}: '
+
numfmt
).
format
(
v
,
p
)
for
(
v
,
p
)
in
sorted
(
self
.
prob
.
items
())])
def
__repr__
(
self
):
return
"P({})"
.
format
(
self
.
varname
)
# ______________________________________________________________________________
# 12.3 Inference Using Full Joint Distributions
class
JointProbDist
(
ProbDist
):
"""A discrete probability distribute over a set of variables.
>>> P = JointProbDist(['X', 'Y']); P[1, 1] = 0.25
>>> P[1, 1]
0.25
>>> P[dict(X=0, Y=1)] = 0.5
>>> P[dict(X=0, Y=1)]
0.5"""
def
__init__
(
self
,
variables
):
self
.
prob
=
{}
self
.
variables
=
variables
self
.
vals
=
defaultdict
(
list
)
def
__getitem__
(
self
,
values
):
"""Given a tuple or dict of values, return P(values)."""
values
=
event_values
(
values
,
self
.
variables
)
return
ProbDist
.
__getitem__
(
self
,
values
)
def
__setitem__
(
self
,
values
,
p
):
"""Set P(values) = p. Values can be a tuple or a dict; it must
have a value for each of the variables in the joint. Also keep track
of the values we have seen so far for each variable."""
values
=
event_values
(
values
,
self
.
variables
)
self
.
prob
[
values
]
=
p
for
var
,
val
in
zip
(
self
.
variables
,
values
):
if
val
not
in
self
.
vals
[
var
]:
self
.
vals
[
var
].
append
(
val
)
def
values
(
self
,
var
):
"""Return the set of possible values for a variable."""
return
self
.
vals
[
var
]
def
__repr__
(
self
):
return
"P({})"
.
format
(
self
.
variables
)
def
event_values
(
event
,
variables
):
"""Return a tuple of the values of variables in event.
>>> event_values ({'A': 10, 'B': 9, 'C': 8}, ['C', 'A'])
(8, 10)
>>> event_values ((1, 2), ['C', 'A'])
(1, 2)
"""
if
isinstance
(
event
,
tuple
)
and
len
(
event
)
==
len
(
variables
):
return
event
else
:
return
tuple
([
event
[
var
]
for
var
in
variables
])
def
enumerate_joint_ask
(
X
,
e
,
P
):
"""Return a probability distribution over the values of the variable X,
given the {var:val} observations e, in the JointProbDist P. [Section 12.3]
>>> P = JointProbDist(['X', 'Y'])
>>> P[0,0] = 0.25; P[0,1] = 0.5; P[1,1] = P[2,1] = 0.125
>>> enumerate_joint_ask('X', dict(Y=1), P).show_approx()
'0: 0.667, 1: 0.167, 2: 0.167'
"""
assert
X
not
in
e
,
"Query variable must be distinct from evidence"
Q
=
ProbDist
(
X
)
# probability distribution for X, initially empty
Y
=
[
v
for
v
in
P
.
variables
if
v
!=
X
and
v
not
in
e
]
# hidden variables.
for
xi
in
P
.
values
(
X
):
Q
[
xi
]
=
enumerate_joint
(
Y
,
extend
(
e
,
X
,
xi
),
P
)
return
Q
.
normalize
()
def
enumerate_joint
(
variables
,
e
,
P
):
"""Return the sum of those entries in P consistent with e,
provided variables is P's remaining variables (the ones not in e)."""
if
not
variables
:
return
P
[
e
]
Y
,
rest
=
variables
[
0
],
variables
[
1
:]
return
sum
([
enumerate_joint
(
rest
,
extend
(
e
,
Y
,
y
),
P
)
for
y
in
P
.
values
(
Y
)])
# ______________________________________________________________________________
# 12.4 Independence
def
is_independent
(
variables
,
P
):
"""
Return whether a list of variables are independent given their distribution P
P is an instance of JoinProbDist
>>> P = JointProbDist(['X', 'Y'])
>>> P[0,0] = 0.25; P[0,1] = 0.5; P[1,1] = P[1,0] = 0.125
>>> is_independent(['X', 'Y'], P)
False
"""
for
var
in
variables
:
event_vars
=
variables
[:]
event_vars
.
remove
(
var
)
event
=
{}
distribution
=
enumerate_joint_ask
(
var
,
event
,
P
)
events
=
gen_possible_events
(
event_vars
,
P
)
for
e
in
events
:
conditional_distr
=
enumerate_joint_ask
(
var
,
e
,
P
)
if
conditional_distr
.
prob
!=
distribution
.
prob
:
return
False
return
True
def
gen_possible_events
(
vars
,
P
):
"""Generate all possible events of a collection of vars according to distribution of P"""
events
=
[]
def
backtrack
(
vars
,
P
,
temp
):
if
not
vars
:
events
.
append
(
temp
)
return
var
=
vars
[
0
]
for
val
in
P
.
values
(
var
):
temp
[
var
]
=
val
backtrack
([
v
for
v
in
vars
if
v
!=
var
],
P
,
copy
.
copy
(
temp
))
backtrack
(
vars
,
P
, {})
return
events
# ______________________________________________________________________________
# Chapter 13 Probabilistic Reasoning
# 13.1 Representing Knowledge in an Uncertain Domain
class
BayesNet
:
"""Bayesian network containing only boolean-variable nodes."""
def
__init__
(
self
,
node_specs
=
None
):
"""
Nodes must be ordered with parents before children.
:param node_specs: an nested iterable object, each element contains (variable name, parents name, cpt)
for each node
"""
self
.
nodes
=
[]
self
.
variables
=
[]
node_specs
=
node_specs
or
[]
for
node_spec
in
node_specs
:
self
.
add
(
node_spec
)
def
add
(
self
,
node_spec
):
"""
Add a node to the net. Its parents must already be in the
net, and its variable must not.
Initialize Bayes nodes by detecting the length of input node specs
"""
if
len
(
node_spec
)
>=
5
:
node
=
ContinuousBayesNode
(
*
node_spec
)
else
:
node
=
BayesNode
(
*
node_spec
)
assert
node
.
variable
not
in
self
.
variables
assert
all
((
parent
in
self
.
variables
)
for
parent
in
node
.
parents
)
self
.
nodes
.
append
(
node
)
self
.
variables
.
append
(
node
.
variable
)
for
parent
in
node
.
parents
:
self
.
variable_node
(
parent
).
children
.
append
(
node
)
def
variable_node
(
self
,
var
):
"""
Return the node for the variable named var.
>>> burglary.variable_node('Burglary').variable
'Burglary'
"""
for
n
in
self
.
nodes
:
if
n
.
variable
==
var
:
return
n
raise
Exception
(
"No such variable: {}"
.
format
(
var
))
def
variable_values
(
self
,
var
):
"""Return the domain of var."""
return
[
True
,
False
]
def
__repr__
(
self
):
return
'BayesNet({0!r})'
.
format
(
self
.
nodes
)
class
BayesNode
:
"""
A conditional probability distribution for a boolean variable,
P(X | parents). Part of a BayesNet.
"""
def
__init__
(
self
,
X
,
parents
,
cpt
):
"""
:param X: variable name,
:param parents: a sequence of variable names or a space-separated string. Representing the names of parent nodes
:param cpt: the conditional probability table, takes one of these forms:
* A number, the unconditional probability P(X=true). You can
use this form when there are no parents.
* A dict {v: p, ...}, the conditional probability distribution
P(X=true | parent=v) = p. When there's just one parent.
* A dict {(v1, v2, ...): p, ...}, the distribution P(X=true |
parent1=v1, parent2=v2, ...) = p. Each key must have as many
values as there are parents. You can use this form always;
the first two are just conveniences.
In all cases the probability of X being false is left implicit,
since it follows from P(X=true).
>>> X = BayesNode('X', '', 0.2)
>>> Y = BayesNode('Y', 'P', {T: 0.2, F: 0.7})
>>> Z = BayesNode('Z', 'P Q',
... {(T, T): 0.2, (T, F): 0.3, (F, T): 0.5, (F, F): 0.7})
"""
if
isinstance
(
parents
,
str
):
parents
=
parents
.
split
()
# We store the table always in the third form above.
if
isinstance
(
cpt
, (
float
,
int
)):
# no parents, 0-tuple
cpt
=
{():
cpt
}
elif
isinstance
(
cpt
,
dict
):
# one parent, 1-tuple
if
cpt
and
isinstance
(
list
(
cpt
.
keys
())[
0
],
bool
):
cpt
=
{(
v
,):
p
for
v
,
p
in
cpt
.
items
()}
assert
isinstance
(
cpt
,
dict
)
for
vs
,
p
in
cpt
.
items
():
assert
isinstance
(
vs
,
tuple
)
and
len
(
vs
)
==
len
(
parents
)
assert
all
(
isinstance
(
v
,
bool
)
for
v
in
vs
)
assert
0
<=
p
<=
1
self
.
variable
=
X
self
.
parents
=
parents
self
.
cpt
=
cpt
self
.
children
=
[]
def
p
(
self
,
value
,
event
):
"""
Return the conditional probability
P(X=value | parents=parent_values), where parent_values
are the values of parents in event. (event must assign each
parent a value.)
>>> bn = BayesNode('X', 'Burglary', {T: 0.2, F: 0.625})
>>> bn.p(False, {'Burglary': False, 'Earthquake': True})
0.375
"""
assert
isinstance
(
value
,
bool
)
ptrue
=
self
.
cpt
[
event_values
(
event
,
self
.
parents
)]
return
ptrue
if
value
else
1
-
ptrue
def
sample
(
self
,
event
):
"""
Sample from the distribution for this variable conditioned
on event's values for parent_variables. That is, return True/False
at random according with the conditional probability given the
parents.
"""
return
probability
(
self
.
p
(
True
,
event
))
def
__repr__
(
self
):
return
repr
((
self
.
variable
,
' '
.
join
(
self
.
parents
)))
# Burglary example [Figure 13 .2]
T
,
F
=
True
,
False
burglary
=
BayesNet
([
(
'Burglary'
,
''
,
0.001
),
(
'Earthquake'
,
''
,
0.002
),
(
'Alarm'
,
'Burglary Earthquake'
,
{(
T
,
T
):
0.95
, (
T
,
F
):
0.94
, (
F
,
T
):
0.29
, (
F
,
F
):
0.001
}),
(
'JohnCalls'
,
'Alarm'
, {
T
:
0.90
,
F
:
0.05
}),
(
'MaryCalls'
,
'Alarm'
, {
T
:
0.70
,
F
:
0.01
})
])
# ______________________________________________________________________________
# Section 13.2. The Semantics of Bayesian Networks
# Bayesian nets with continuous variables
def
gaussian_probability
(
param
,
event
,
value
):
"""
Gaussian probability of a continuous Bayesian network node on condition of
certain event and the parameters determined by the event
:param param: parameters determined by discrete parent events of current node
:param event: a dict, continuous event of current node, the values are used
as parameters in calculating distribution
:param value: float, the value of current continuous node
:return: float, the calculated probability
>>> param = {'sigma':0.5, 'b':1, 'a':{'h1':0.5, 'h2': 1.5}}
>>> event = {'h1':0.6, 'h2': 0.3}
>>> gaussian_probability(param, event, 1)
0.2590351913317835
"""
assert
isinstance
(
event
,
dict
)
assert
isinstance
(
param
,
dict
)
buff
=
0
for
k
,
v
in
event
.
items
():
# buffer varianle to calculate h1*a_h1 + h2*a_h2
buff
+=
param
[
'a'
][
k
]
*
v
res
=
1
/
(
param
[
'sigma'
]
*
np
.
sqrt
(
2
*
np
.
pi
))
*
np
.
exp
(
-
0.5
*
((
value
-
buff
-
param
[
'b'
])
/
param
[
'sigma'
])
**
2
)
return
res
def
logistic_probability
(
param
,
event
,
value
):
"""
Logistic probability of a discrete node in Bayesian network with continuous parents,
:param param: a dict, parameters determined by discrete parents of current node
:param event: a dict, names and values of continuous parent variables of current node
:param value: boolean, True or False
:return: int, probability
"""
buff
=
1
for
_
,
v
in
event
.
items
():
# buffer variable to calculate (value-mu)/sigma
buff
*=
(
v
-
param
[
'mu'
])
/
param
[
'sigma'
]
p
=
1
-
1
/
(
1
+
np
.
exp
(
-
4
/
np
.
sqrt
(
2
*
np
.
pi
)
*
buff
))
return
p
if
value
else
1
-
p
class
ContinuousBayesNode
:
""" A Bayesian network node with continuous distribution or with continuous distributed parents """
def
__init__
(
self
,
name
,
d_parents
,
c_parents
,
parameters
,
type
):
"""
A continuous Bayesian node has two types of parents: discrete and continuous.
:param d_parents: str, name of discrete parents, value of which determines distribution parameters
:param c_parents: str, name of continuous parents, value of which is used to calculate distribution
:param parameters: a dict, parameters for distribution of current node, keys corresponds to discrete parents
:param type: str, type of current node's value, either 'd' (discrete) or 'c'(continuous)
"""
self
.
parameters
=
parameters
self
.
type
=
type
self
.
d_parents
=
d_parents
.
split
()
self
.
c_parents
=
c_parents
.
split
()
self
.
parents
=
self
.
d_parents
+
self
.
c_parents
self
.
variable
=
name
self
.
children
=
[]
def
continuous_p
(
self
,
value
,
c_event
,
d_event
):
"""
Probability given the value of current node and its parents
:param c_event: event of continuous nodes
:param d_event: event of discrete nodes
"""
assert
isinstance
(
c_event
,
dict
)
assert
isinstance
(
d_event
,
dict
)
d_event_vals
=
event_values
(
d_event
,
self
.
d_parents
)
if
len
(
d_event_vals
)
==
1
:
d_event_vals
=
d_event_vals
[
0
]
param
=
self
.
parameters
[
d_event_vals
]
if
self
.
type
==
"c"
:
p
=
gaussian_probability
(
param
,
c_event
,
value
)
if
self
.
type
==
"d"
:
p
=
logistic_probability
(
param
,
c_event
,
value
)
return
p
# harvest-buy example. Figure 13.5
harvest_buy
=
BayesNet
([
(
'Subsidy'
,
''
,
0.001
),
(
'Harvest'
,
''
,
0.002
),
(
'Cost'
,
'Subsidy'
,
'Harvest'
,
{
True
: {
'sigma'
:
0.5
,
'b'
:
1
,
'a'
: {
'Harvest'
:
0.5
}},
False
: {
'sigma'
:
0.6
,
'b'
:
1
,
'a'
: {
'Harvest'
:
0.5
}}},
'c'
),
(
'Buys'
,
''
,
'Cost'
, {
T
: {
'mu'
:
0.5
,
'sigma'
:
0.5
},
F
: {
'mu'
:
0.6
,
'sigma'
:
0.6
}},
'd'
)])
# ______________________________________________________________________________
# 13.3 Exact Inference in Bayesian Networks
# 13.3.1 Inference by enumeration
def
enumeration_ask
(
X
,
e
,
bn
):
"""
Return the conditional probability distribution of variable X
given evidence e, from BayesNet bn. [Figure 13.10]
>>> enumeration_ask('Burglary', dict(JohnCalls=T, MaryCalls=T), burglary
... ).show_approx()
'False: 0.716, True: 0.284'
"""
assert
X
not
in
e
,
"Query variable must be distinct from evidence"
Q
=
ProbDist
(
X
)
for
xi
in
bn
.
variable_values
(
X
):
Q
[
xi
]
=
enumerate_all
(
bn
.
variables
,
extend
(
e
,
X
,
xi
),
bn
)
return
Q
.
normalize
()
def
enumerate_all
(
variables
,
e
,
bn
):
"""
Return the sum of those entries in P(variables | e{others})
consistent with e, where P is the joint distribution represented
by bn, and e{others} means e restricted to bn's other variables
(the ones other than variables). Parents must precede children in variables.
"""
if
not
variables
:
return
1.0
Y
,
rest
=
variables
[
0
],
variables
[
1
:]
Ynode
=
bn
.
variable_node
(
Y
)
if
Y
in
e
:
return
Ynode
.
p
(
e
[
Y
],
e
)
*
enumerate_all
(
rest
,
e
,
bn
)
else
:
return
sum
(
Ynode
.
p
(
y
,
e
)
*
enumerate_all
(
rest
,
extend
(
e
,
Y
,
y
),
bn
)
for
y
in
bn
.
variable_values
(
Y
))
# ______________________________________________________________________________
# 13.3.2 The variable elimination algorithm
def
elimination_ask
(
X
,
e
,
bn
):
"""
Compute bn's P(X|e) by variable elimination. [Figure 13.12]
>>> elimination_ask('Burglary', dict(JohnCalls=T, MaryCalls=T), burglary
... ).show_approx()
'False: 0.716, True: 0.284'
"""
assert
X
not
in
e
,
"Query variable must be distinct from evidence"
factors
=
[]
for
var
in
reversed
(
bn
.
variables
):
factors
.
append
(
make_factor
(
var
,
e
,
bn
))
if
is_hidden
(
var
,
X
,
e
):
factors
=
sum_out
(
var
,
factors
,
bn
)
return
pointwise_product
(
factors
,
bn
).
normalize
()
def
is_hidden
(
var
,
X
,
e
):
"""Is var a hidden variable when querying P(X|e)?"""
return
var
!=
X
and
var
not
in
e
def
make_factor
(
var
,
e
,
bn
):
"""
Return the factor for var in bn's joint distribution given e.
That is, bn's full joint distribution, projected to accord with e,
is the pointwise product of these factors for bn's variables.
"""
node
=
bn
.
variable_node
(
var
)
variables
=
[
X
for
X
in
[
var
]
+
node
.
parents
if
X
not
in
e
]
cpt
=
{
event_values
(
e1
,
variables
):
node
.
p
(
e1
[
var
],
e1
)
for
e1
in
all_events
(
variables
,
bn
,
e
)}
return
Factor
(
variables
,
cpt
)
def
pointwise_product
(
factors
,
bn
):
return
reduce
(
lambda
f
,
g
:
f
.
pointwise_product
(
g
,
bn
),
factors
)
def
sum_out
(
var
,
factors
,
bn
):
"""Eliminate var from all factors by summing over its values."""
result
,
var_factors
=
[], []
for
f
in
factors
:
(
var_factors
if
var
in
f
.
variables
else
result
).
append
(
f
)
result
.
append
(
pointwise_product
(
var_factors
,
bn
).
sum_out
(
var
,
bn
))
return
result
class
Factor
:
"""A factor in a joint distribution."""
def
__init__
(
self
,
variables
,
cpt
):
self
.
variables
=
variables
self
.
cpt
=
cpt
def
pointwise_product
(
self
,
other
,
bn
):
"""Multiply two factors, combining their variables."""
variables
=
list
(
set
(
self
.
variables
)
|
set
(
other
.
variables
))
cpt
=
{
event_values
(
e
,
variables
):
self
.
p
(
e
)
*
other
.
p
(
e
)
for
e
in
all_events
(
variables
,
bn
, {})}
return
Factor
(
variables
,
cpt
)
def
sum_out
(
self
,
var
,
bn
):
"""Make a factor eliminating var by summing over its values."""
variables
=
[
X
for
X
in
self
.
variables
if
X
!=
var
]
cpt
=
{
event_values
(
e
,
variables
):
sum
(
self
.
p
(
extend
(
e
,
var
,
val
))
for
val
in
bn
.
variable_values
(
var
))
for
e
in
all_events
(
variables
,
bn
, {})}
return
Factor
(
variables
,
cpt
)
def
normalize
(
self
):
"""Return my probabilities; must be down to one variable."""
assert
len
(
self
.
variables
)
==
1
return
ProbDist
(
self
.
variables
[
0
],
{
k
:
v
for
((
k
,),
v
)
in
self
.
cpt
.
items
()})
def
p
(
self
,
e
):
"""Look up my value tabulated for e."""
return
self
.
cpt
[
event_values
(
e
,
self
.
variables
)]
def
all_events
(
variables
,
bn
,
e
):
"""Yield every way of extending e with values for all variables."""
if
not
variables
:
yield
e
else
:
X
,
rest
=
variables
[
0
],
variables
[
1
:]
for
e1
in
all_events
(
rest
,
bn
,
e
):
for
x
in
bn
.
variable_values
(
X
):
yield
extend
(
e1
,
X
,
x
)
# ______________________________________________________________________________
# 13.3.4 Clustering algorithms
# [Figure 13.14a]: sprinkler network
sprinkler
=
BayesNet
([
(
'Cloudy'
,
''
,
0.5
),
(
'Sprinkler'
,
'Cloudy'
, {
T
:
0.10
,
F
:
0.50
}),
(
'Rain'
,
'Cloudy'
, {
T
:
0.80
,
F
:
0.20
}),
(
'WetGrass'
,
'Sprinkler Rain'
,
{(
T
,
T
):
0.99
, (
T
,
F
):
0.90
, (
F
,
T
):
0.90
, (
F
,
F
):
0.00
})])
# ______________________________________________________________________________
# 13.4 Approximate Inference for Bayesian Networks
# 13.4.1 Direct sampling methods
def
prior_sample
(
bn
):
"""
Randomly sample from bn's full joint distribution. The result
is a {variable: value} dict. [Figure 13.15]
"""
event
=
{}
for
node
in
bn
.
nodes
:
event
[
node
.
variable
]
=
node
.
sample
(
event
)
return
event
# _________________________________________________________________________
def
rejection_sampling
(
X
,
e
,
bn
,
N
=
10000
):
"""
[Figure 13.16]
Estimate the probability distribution of variable X given
evidence e in BayesNet bn, using N samples.
Raises a ZeroDivisionError if all the N samples are rejected,
i.e., inconsistent with e.
>>> random.seed(47)
>>> rejection_sampling('Burglary', dict(JohnCalls=T, MaryCalls=T),
... burglary, 10000).show_approx()
'False: 0.7, True: 0.3'
"""
counts
=
{
x
:
0
for
x
in
bn
.
variable_values
(
X
)}
# bold N in [Figure 13.16]
for
j
in
range
(
N
):
sample
=
prior_sample
(
bn
)
# boldface x in [Figure 13.16]
if
consistent_with
(
sample
,
e
):
counts
[
sample
[
X
]]
+=
1
return
ProbDist
(
X
,
counts
)
def
consistent_with
(
event
,
evidence
):
"""Is event consistent with the given evidence?"""
return
all
(
evidence
.
get
(
k
,
v
)
==
v
for
k
,
v
in
event
.
items
())
# _________________________________________________________________________
def
likelihood_weighting
(
X
,
e
,
bn
,
N
=
10000
):
"""
[Figure 13.17]
Estimate the probability distribution of variable X given
evidence e in BayesNet bn.
>>> random.seed(1017)
>>> likelihood_weighting('Burglary', dict(JohnCalls=T, MaryCalls=T),
... burglary, 10000).show_approx()
'False: 0.702, True: 0.298'
"""
W
=
{
x
:
0
for
x
in
bn
.
variable_values
(
X
)}
for
j
in
range
(
N
):
sample
,
weight
=
weighted_sample
(
bn
,
e
)
# boldface x, w in [Figure 14.15]
W
[
sample
[
X
]]
+=
weight
return
ProbDist
(
X
,
W
)
def
weighted_sample
(
bn
,
e
):
"""
Sample an event from bn that's consistent with the evidence e;
return the event and its weight, the likelihood that the event
accords to the evidence.
"""
w
=
1
event
=
dict
(
e
)
# boldface x in [Figure 13.17]
for
node
in
bn
.
nodes
:
Xi
=
node
.
variable
if
Xi
in
e
:
w
*=
node
.
p
(
e
[
Xi
],
event
)
else
:
event
[
Xi
]
=
node
.
sample
(
event
)
return
event
,
w
# _________________________________________________________________________
# 13.4.2 Inference by Markov chain simulation
def
gibbs_ask
(
X
,
e
,
bn
,
N
=
1000
):
"""[Figure 13.19]"""
assert
X
not
in
e
,
"Query variable must be distinct from evidence"
counts
=
{
x
:
0
for
x
in
bn
.
variable_values
(
X
)}
# bold N in [Figure 14.16]
Z
=
[
var
for
var
in
bn
.
variables
if
var
not
in
e
]
state
=
dict
(
e
)
# boldface x in [Figure 14.16]
for
Zi
in
Z
:
state
[
Zi
]
=
random
.
choice
(
bn
.
variable_values
(
Zi
))
for
j
in
range
(
N
):
for
Zi
in
Z
:
state
[
Zi
]
=
markov_blanket_sample
(
Zi
,
state
,
bn
)
counts
[
state
[
X
]]
+=
1
return
ProbDist
(
X
,
counts
)
def
markov_blanket_sample
(
X
,
e
,
bn
):
"""
Return a sample from P(X | mb) where mb denotes that the
variables in the Markov blanket of X take their values from event
e (which must assign a value to each). The Markov blanket of X is
X's parents, children, and children's parents.
"""
Xnode
=
bn
.
variable_node
(
X
)
Q
=
ProbDist
(
X
)
for
xi
in
bn
.
variable_values
(
X
):
ei
=
extend
(
e
,
X
,
xi
)
# [Equation 13.12:]
Q
[
xi
]
=
Xnode
.
p
(
xi
,
e
)
*
product
(
Yj
.
p
(
ei
[
Yj
.
variable
],
ei
)
for
Yj
in
Xnode
.
children
)
# (assuming a Boolean variable here)
return
probability
(
Q
.
normalize
()[
True
])
# _________________________________________________________________________
# 13.4.3 Compiling approximate inference
class
complied_burglary
:
"""compiled version of burglary network"""
def
Burglary
(
self
,
sample
):
if
sample
[
'Alarm'
]:
if
sample
[
'Earthquake'
]:
return
probability
(
0.00327
)
else
:
return
probability
(
0.485
)
else
:
if
sample
[
'Earthquake'
]:
return
probability
(
7.05e-05
)
else
:
return
probability
(
6.01e-05
)
def
Earthquake
(
self
,
sample
):
if
sample
[
'Alarm'
]:
if
sample
[
'Burglary'
]:
return
probability
(
0.0020212
)
else
:
return
probability
(
0.36755
)
else
:
if
sample
[
'Burglary'
]:
return
probability
(
0.0016672
)
else
:
return
probability
(
0.0014222
)
def
MaryCalls
(
self
,
sample
):
if
sample
[
'Alarm'
]:
return
probability
(
0.7
)
else
:
return
probability
(
0.01
)
def
JongCalls
(
self
,
sample
):
if
sample
[
'Alarm'
]:
return
probability
(
0.9
)
else
:
return
probability
(
0.05
)
def
Alarm
(
self
,
sample
):
raise
NotImplementedError
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