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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 value: frequency pairs,

and the ProbDist then 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(%s)" % 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(%s)" % 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: %s" % var)

def variable_values(self, var):

"Return the domain of var."

return [True, False]

def __repr__(self):

return 'BayesNet(%r)' % 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 = 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."""

s = []

dist = [0.5, 0.5]

# State Initialization

s = ['A' if probability(dist[0]) else 'B' for i in range(N)]

# Weight Initialization

w = [0 for i 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 i 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(s, w, N)

return s

aima-python/probability.py at master · aesgitrep/aima-python · GitHub

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