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"""Probability models. (Chapter 13-15)
"""
from
utils
import
(
product
,
argmax
,
element_wise_product
,
matrix_multiplication
,
vector_to_diagonal
,
vector_add
,
scalar_vector_product
,
inverse_matrix
,
weighted_sample_with_replacement
,
isclose
,
probability
,
normalize
)
from
logic
import
extend
import
random
from
collections
import
defaultdict
from
functools
import
reduce
# ______________________________________________________________________________
def
DTAgentProgram
(
belief_state
):
"""A decision-theoretic agent. [Figure 13.1]"""
def
program
(
percept
):
belief_state
.
observe
(
program
.
action
,
percept
)
program
.
action
=
argmax
(
belief_state
.
actions
(),
key
=
belief_state
.
expected_outcome_utility
)
return
program
.
action
program
.
action
=
None
return
program
# ______________________________________________________________________________
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
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
([(
'%s: '
+
numfmt
)
%
(
v
,
p
)
for
(
v
,
p
)
in
sorted
(
self
.
prob
.
items
())])
def
__repr__
(
self
):
return
"P({})"
.
format
(
self
.
varname
)
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 13.3]
>>> 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
)])
# ______________________________________________________________________________
class
BayesNet
:
"""Bayesian network containing only boolean-variable nodes."""
def
__init__
(
self
,
node_specs
=
[]):
"""Nodes must be ordered with parents before children."""
self
.
nodes
=
[]
self
.
variables
=
[]
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."""
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
):
"""X is a variable name, and parents a sequence of variable
names or a space-separated string. 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 14.2]
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
})
])
# ______________________________________________________________________________
def
enumeration_ask
(
X
,
e
,
bn
):
"""Return the conditional probability distribution of variable X
given evidence e, from BayesNet bn. [Figure 14.9]
>>> 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
))
# ______________________________________________________________________________
def
elimination_ask
(
X
,
e
,
bn
):
"""Compute bn's P(X|e) by variable elimination. [Figure 14.11]
>>> 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
)
# ______________________________________________________________________________
# [Figure 14.12a]: 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
})])
# ______________________________________________________________________________
def
prior_sample
(
bn
):
"""Randomly sample from bn's full joint distribution. The result
is a {variable: value} dict. [Figure 14.13]"""
event
=
{}
for
node
in
bn
.
nodes
:
event
[
node
.
variable
]
=
node
.
sample
(
event
)
return
event
# _________________________________________________________________________
def
rejection_sampling
(
X
,
e
,
bn
,
N
):
"""Estimate the probability distribution of variable X given
evidence e in BayesNet bn, using N samples. [Figure 14.14]
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 14.14]
for
j
in
range
(
N
):
sample
=
prior_sample
(
bn
)
# boldface x in [Figure 14.14]
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
):
"""Estimate the probability distribution of variable X given
evidence e in BayesNet bn. [Figure 14.15]
>>> 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 14.15]
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
# _________________________________________________________________________
def
gibbs_ask
(
X
,
e
,
bn
,
N
):
"""[Figure 14.16]"""
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 14.12:]
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
])
# _________________________________________________________________________
class
HiddenMarkovModel
:
"""A Hidden markov model which takes Transition model and Sensor model as inputs"""
def
__init__
(
self
,
transition_model
,
sensor_model
,
prior
=
[
0.5
,
0.5
]):
self
.
transition_model
=
transition_model
self
.
sensor_model
=
sensor_model
self
.
prior
=
prior
def
sensor_dist
(
self
,
ev
):
if
ev
is
True
:
return
self
.
sensor_model
[
0
]
else
:
return
self
.
sensor_model
[
1
]
def
forward
(
HMM
,
fv
,
ev
):
prediction
=
vector_add
(
scalar_vector_product
(
fv
[
0
],
HMM
.
transition_model
[
0
]),
scalar_vector_product
(
fv
[
1
],
HMM
.
transition_model
[
1
]))
sensor_dist
=
HMM
.
sensor_dist
(
ev
)
return
normalize
(
element_wise_product
(
sensor_dist
,
prediction
))
def
backward
(
HMM
,
b
,
ev
):
sensor_dist
=
HMM
.
sensor_dist
(
ev
)
prediction
=
element_wise_product
(
sensor_dist
,
b
)
return
normalize
(
vector_add
(
scalar_vector_product
(
prediction
[
0
],
HMM
.
transition_model
[
0
]),
scalar_vector_product
(
prediction
[
1
],
HMM
.
transition_model
[
1
])))
def
forward_backward
(
HMM
,
ev
,
prior
):
"""[Figure 15.4]
Forward-Backward algorithm for smoothing. Computes posterior probabilities
of a sequence of states given a sequence of observations."""
t
=
len
(
ev
)
ev
.
insert
(
0
,
None
)
# to make the code look similar to pseudo code
fv
=
[[
0.0
,
0.0
]
for
i
in
range
(
len
(
ev
))]
b
=
[
1.0
,
1.0
]
bv
=
[
b
]
# we don't need bv; but we will have a list of all backward messages here
sv
=
[[
0
,
0
]
for
i
in
range
(
len
(
ev
))]
fv
[
0
]
=
prior
for
i
in
range
(
1
,
t
+
1
):
fv
[
i
]
=
forward
(
HMM
,
fv
[
i
-
1
],
ev
[
i
])
for
i
in
range
(
t
,
-
1
,
-
1
):
sv
[
i
-
1
]
=
normalize
(
element_wise_product
(
fv
[
i
],
b
))
b
=
backward
(
HMM
,
b
,
ev
[
i
])
bv
.
append
(
b
)
sv
=
sv
[::
-
1
]
return
sv
# _________________________________________________________________________
def
fixed_lag_smoothing
(
e_t
,
HMM
,
d
,
ev
,
t
):
"""[Figure 15.6]
Smoothing algorithm with a fixed time lag of 'd' steps.
Online algorithm that outputs the new smoothed estimate if observation
for new time step is given."""
ev
.
insert
(
0
,
None
)
T_model
=
HMM
.
transition_model
f
=
HMM
.
prior
B
=
[[
1
,
0
], [
0
,
1
]]
evidence
=
[]
evidence
.
append
(
e_t
)
O_t
=
vector_to_diagonal
(
HMM
.
sensor_dist
(
e_t
))
if
t
>
d
:
f
=
forward
(
HMM
,
f
,
e_t
)
O_tmd
=
vector_to_diagonal
(
HMM
.
sensor_dist
(
ev
[
t
-
d
]))
B
=
matrix_multiplication
(
inverse_matrix
(
O_tmd
),
inverse_matrix
(
T_model
),
B
,
T_model
,
O_t
)
else
:
B
=
matrix_multiplication
(
B
,
T_model
,
O_t
)
t
+=
1
if
t
>
d
:
# always returns a 1x2 matrix
return
[
normalize
(
i
)
for
i
in
matrix_multiplication
([
f
],
B
)][
0
]
else
:
return
None
# _________________________________________________________________________
def
particle_filtering
(
e
,
N
,
HMM
):
"""Particle filtering considering two states variables."""
dist
=
[
0.5
,
0.5
]
# Weight Initialization
w
=
[
0
for
_
in
range
(
N
)]
# STEP 1
# Propagate one step using transition model given prior state
dist
=
vector_add
(
scalar_vector_product
(
dist
[
0
],
HMM
.
transition_model
[
0
]),
scalar_vector_product
(
dist
[
1
],
HMM
.
transition_model
[
1
]))
# Assign state according to probability
s
=
[
'A'
if
probability
(
dist
[
0
])
else
'B'
for
_
in
range
(
N
)]
w_tot
=
0
# Calculate importance weight given evidence e
for
i
in
range
(
N
):
if
s
[
i
]
==
'A'
:
# P(U|A)*P(A)
w_i
=
HMM
.
sensor_dist
(
e
)[
0
]
*
dist
[
0
]
if
s
[
i
]
==
'B'
:
# P(U|B)*P(B)
w_i
=
HMM
.
sensor_dist
(
e
)[
1
]
*
dist
[
1
]
w
[
i
]
=
w_i
w_tot
+=
w_i
# Normalize all the weights
for
i
in
range
(
N
):
w
[
i
]
=
w
[
i
]
/
w_tot
# Limit weights to 4 digits
for
i
in
range
(
N
):
w
[
i
]
=
float
(
"{0:.4f}"
.
format
(
w
[
i
]))
# STEP 2
s
=
weighted_sample_with_replacement
(
N
,
s
,
w
)
return
s
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