This IPy notebook acts as supporting material for topics covered in Chapter 13 Quantifying Uncertainty, Chapter 14 Probabilistic Reasoning, Chapter 15 Probabilistic Reasoning over Time, Chapter 16 Making Simple Decisions and parts of Chapter 25 Robotics of the book* Artificial Intelligence: A Modern Approach*. This notebook makes use of the implementations in probability.py module. Let us import everything from the probability module. It might be helpful to view the source of some of our implementations. Please refer to the Introductory IPy file for more details on how to do so.
from probability import *
from utils import print_table
from notebook import psource, pseudocode, heatmap
Let us begin by specifying discrete probability distributions. The class ProbDist defines a discrete probability distribution. We name our random variable and then assign probabilities to the different values of the random variable. Assigning probabilities to the values works similar to that of using a dictionary with keys being the Value and we assign to it the probability. This is possible because of the magic methods _ getitem _ and _ setitem _ which store the probabilities in the prob dict of the object. You can keep the source window open alongside while playing with the rest of the code to get a better understanding.
psource(ProbDist)
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([('{}: ' + numfmt).format(v, p)
for (v, p) in sorted(self.prob.items())])
def __repr__(self):
return "P({})".format(self.varname)
p = ProbDist('Flip')
p['H'], p['T'] = 0.25, 0.75
p['T']
0.75
The first parameter of the constructor varname has a default value of '?'. So if the name is not passed it defaults to ?. The keyword argument freqs can be a dictionary of values of random variable: probability. These are then normalized such that the probability values sum upto 1 using the normalize method.
p = ProbDist(freqs={'low': 125, 'medium': 375, 'high': 500})
p.varname
'?'
(p['low'], p['medium'], p['high'])
(0.125, 0.375, 0.5)
Besides the prob and varname the object also separately keeps track of all the values of the distribution in a list called values. Every time a new value is assigned a probability it is appended to this list, This is done inside the _ setitem _ method.
p.values
['low', 'medium', 'high']
The distribution by default is not normalized if values are added incrementally. We can still force normalization by invoking the normalize method.
p = ProbDist('Y')
p['Cat'] = 50
p['Dog'] = 114
p['Mice'] = 64
(p['Cat'], p['Dog'], p['Mice'])
(50, 114, 64)
p.normalize()
(p['Cat'], p['Dog'], p['Mice'])
(0.21929824561403508, 0.5, 0.2807017543859649)
It is also possible to display the approximate values upto decimals using the show_approx method.
p.show_approx()
'Cat: 0.219, Dog: 0.5, Mice: 0.281'
The helper function event_values returns a tuple of the values of variables in event. An event is specified by a dict where the keys are the names of variables and the corresponding values are the value of the variable. Variables are specified with a list. The ordering of the returned tuple is same as those of the variables.
Alternatively if the event is specified by a list or tuple of equal length of the variables. Then the events tuple is returned as it is.
event = {'A': 10, 'B': 9, 'C': 8}
variables = ['C', 'A']
event_values(event, variables)
(8, 10)
A probability model is completely determined by the joint distribution for all of the random variables. (Section 13.3) The probability module implements these as the class JointProbDist which inherits from the ProbDist class. This class specifies a discrete probability distribute over a set of variables.
psource(JointProbDist)
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)
Values for a Joint Distribution is a an ordered tuple in which each item corresponds to the value associate with a particular variable. For Joint Distribution of X, Y where X, Y take integer values this can be something like (18, 19).
To specify a Joint distribution we first need an ordered list of variables.
variables = ['X', 'Y']
j = JointProbDist(variables)
j
P(['X', 'Y'])
Like the ProbDist class JointProbDist also employes magic methods to assign probability to different values. The probability can be assigned in either of the two formats for all possible values of the distribution. The event_values call inside _ getitem _ and _ setitem _ does the required processing to make this work.
j[1,1] = 0.2
j[dict(X=0, Y=1)] = 0.5
(j[1,1], j[0,1])
(0.2, 0.5)
It is also possible to list all the values for a particular variable using the values method.
j.values('X')
[1, 0]
In this section we use Full Joint Distributions to calculate the posterior distribution given some evidence. We represent evidence by using a python dictionary with variables as dict keys and dict values representing the values.
This is illustrated in Section 13.3 of the book. The functions enumerate_joint and enumerate_joint_ask implement this functionality. Under the hood they implement Equation 13.9 from the book.
$$\textbf{P}(X | \textbf{e}) = \alpha \textbf{P}(X, \textbf{e}) = \alpha \sum_{y} \textbf{P}(X, \textbf{e}, \textbf{y})$$Here α is the normalizing factor. X is our query variable and e is the evidence. According to the equation we enumerate on the remaining variables y (not in evidence or query variable) i.e. all possible combinations of y
We will be using the same example as the book. Let us create the full joint distribution from Figure 13.3.
full_joint = JointProbDist(['Cavity', 'Toothache', 'Catch'])
full_joint[dict(Cavity=True, Toothache=True, Catch=True)] = 0.108
full_joint[dict(Cavity=True, Toothache=True, Catch=False)] = 0.012
full_joint[dict(Cavity=True, Toothache=False, Catch=True)] = 0.016
full_joint[dict(Cavity=True, Toothache=False, Catch=False)] = 0.064
full_joint[dict(Cavity=False, Toothache=True, Catch=True)] = 0.072
full_joint[dict(Cavity=False, Toothache=False, Catch=True)] = 0.144
full_joint[dict(Cavity=False, Toothache=True, Catch=False)] = 0.008
full_joint[dict(Cavity=False, Toothache=False, Catch=False)] = 0.576
Let us now look at the enumerate_joint function returns the sum of those entries in P consistent with e,provided variables is P's remaining variables (the ones not in e). Here, P refers to the full joint distribution. The function uses a recursive call in its implementation. The first parameter variables refers to remaining variables. The function in each recursive call keeps on variable constant while varying others.
psource(enumerate_joint)
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)])
Let us assume we want to find P(Toothache=True). This can be obtained by marginalization (Equation 13.6). We can use enumerate_joint to solve for this by taking Toothache=True as our evidence. enumerate_joint will return the sum of probabilities consistent with evidence i.e. Marginal Probability.
evidence = dict(Toothache=True)
variables = ['Cavity', 'Catch'] # variables not part of evidence
ans1 = enumerate_joint(variables, evidence, full_joint)
ans1
0.19999999999999998
You can verify the result from our definition of the full joint distribution. We can use the same function to find more complex probabilities like P(Cavity=True and Toothache=True)
evidence = dict(Cavity=True, Toothache=True)
variables = ['Catch'] # variables not part of evidence
ans2 = enumerate_joint(variables, evidence, full_joint)
ans2
0.12
Being able to find sum of probabilities satisfying given evidence allows us to compute conditional probabilities like P(Cavity=True | Toothache=True) as we can rewrite this as $$P(Cavity=True | Toothache = True) = \frac{P(Cavity=True \ and \ Toothache=True)}{P(Toothache=True)}$$
We have already calculated both the numerator and denominator.
ans2/ans1
0.6
We might be interested in the probability distribution of a particular variable conditioned on some evidence. This can involve doing calculations like above for each possible value of the variable. This has been implemented slightly differently using normalization in the function enumerate_joint_ask which returns a probability distribution over the values of the variable X, given the {var:val} observations e, in the JointProbDist P. The implementation of this function calls enumerate_joint for each value of the query variable and passes extended evidence with the new evidence having X = xi. This is followed by normalization of the obtained distribution.
psource(enumerate_joint_ask)
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()
Let us find P(Cavity | Toothache=True) using enumerate_joint_ask.
query_variable = 'Cavity'
evidence = dict(Toothache=True)
ans = enumerate_joint_ask(query_variable, evidence, full_joint)
(ans[True], ans[False])
(0.6, 0.39999999999999997)
You can verify that the first value is the same as we obtained earlier by manual calculation.
A Bayesian network is a representation of the joint probability distribution encoding a collection of conditional independence statements.
A Bayes Network is implemented as the class BayesNet. It consisits of a collection of nodes implemented by the class BayesNode. The implementation in the above mentioned classes focuses only on boolean variables. Each node is associated with a variable and it contains a conditional probabilty table (cpt). The cpt represents the probability distribution of the variable conditioned on its parents P(X | parents).
Let us dive into the BayesNode implementation.
psource(BayesNode)
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)))
The constructor takes in the name of variable, parents and cpt. Here variable is a the name of the variable like 'Earthquake'. parents should a list or space separate string with variable names of parents. The conditional probability table is a dict {(v1, v2, ...): p, ...}, the distribution P(X=true | parent1=v1, parent2=v2, ...) = p. Here the keys are combination of boolean values that the parents take. The length and order of the values in keys should be same as the supplied parent list/string. In all cases the probability of X being false is left implicit, since it follows from P(X=true).
The example below where we implement the network shown in Figure 14.3 of the book will make this more clear.
The alarm node can be made as follows:
alarm_node = BayesNode('Alarm', ['Burglary', 'Earthquake'],
{(True, True): 0.95,(True, False): 0.94, (False, True): 0.29, (False, False): 0.001})
It is possible to avoid using a tuple when there is only a single parent. So an alternative format for the cpt is
john_node = BayesNode('JohnCalls', ['Alarm'], {True: 0.90, False: 0.05})
mary_node = BayesNode('MaryCalls', 'Alarm', {(True, ): 0.70, (False, ): 0.01}) # Using string for parents.
# Equivalant to john_node definition.
The general format used for the alarm node always holds. For nodes with no parents we can also use.
burglary_node = BayesNode('Burglary', '', 0.001)
earthquake_node = BayesNode('Earthquake', '', 0.002)
It is possible to use the node for lookup function using the p method. The method takes in two arguments value and event. Event must be a dict of the type {variable:values, ..} The value corresponds to the value of the variable we are interested in (False or True).The method returns 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.)
john_node.p(False, {'Alarm': True, 'Burglary': True}) # P(JohnCalls=False | Alarm=True)
0.09999999999999998
With all the information about nodes present it is possible to construct a Bayes Network using BayesNet. The BayesNet class does not take in nodes as input but instead takes a list of node_specs. An entry in node_specs is a tuple of the parameters we use to construct a BayesNode namely (X, parents, cpt). node_specs must be ordered with parents before children.
psource(BayesNet)
class BayesNet:
"""Bayesian network containing only boolean-variable nodes."""
def __init__(self, node_specs=None):
"""Nodes must be ordered with parents before children."""
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."""
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)
The constructor of BayesNet takes each item in node_specs and adds a BayesNode to its nodes object variable by calling the add method. add in turn adds node to the net. Its parents must already be in the net, and its variable must not. Thus add allows us to grow a BayesNet given its parents are already present.
burglary global is an instance of BayesNet corresponding to the above example.
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})
])
burglary
BayesNet([('Burglary', ''), ('Earthquake', ''), ('Alarm', 'Burglary Earthquake'), ('JohnCalls', 'Alarm'), ('MaryCalls', 'Alarm')])
BayesNet method variable_node allows to reach BayesNode instances inside a Bayes Net. It is possible to modify the cpt of the nodes directly using this method.
type(burglary.variable_node('Alarm'))
probability.BayesNode
burglary.variable_node('Alarm').cpt
{(True, True): 0.95, (True, False): 0.94, (False, True): 0.29, (False, False): 0.001}
A Bayes Network is a more compact representation of the full joint distribution and like full joint distributions allows us to do inference i.e. answer questions about probability distributions of random variables given some evidence.
Exact algorithms don't scale well for larger networks. Approximate algorithms are explained in the next section.
We apply techniques similar to those used for enumerate_joint_ask and enumerate_joint to draw inference from Bayesian Networks. enumeration_ask and enumerate_all implement the algorithm described in Figure 14.9 of the book.
psource(enumerate_all)
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))
enumerate_all recursively evaluates a general form of the Equation 14.4 in the book.
$$\textbf{P}(X | \textbf{e}) = α \textbf{P}(X, \textbf{e}) = α \sum_{y} \textbf{P}(X, \textbf{e}, \textbf{y})$$such that P(X, e, y) is written in the form of product of conditional probabilities P(variable | parents(variable)) from the Bayesian Network.
enumeration_ask calls enumerate_all on each value of query variable X and finally normalizes them.
psource(enumeration_ask)
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()
Let us solve the problem of finding out P(Burglary=True | JohnCalls=True, MaryCalls=True) using the burglary network. enumeration_ask takes three arguments X = variable name, e = Evidence (in form a dict like previously explained), bn = The Bayes Net to do inference on.
ans_dist = enumeration_ask('Burglary', {'JohnCalls': True, 'MaryCalls': True}, burglary)
ans_dist[True]
0.2841718353643929
The enumeration algorithm can be improved substantially by eliminating repeated calculations. In enumeration we join the joint of all hidden variables. This is of exponential size for the number of hidden variables. Variable elimination employes interleaving join and marginalization.
Before we look into the implementation of Variable Elimination we must first familiarize ourselves with Factors.
In general we call a multidimensional array of type P(Y1 ... Yn | X1 ... Xm) a factor where some of Xs and Ys maybe assigned values. Factors are implemented in the probability module as the class Factor. They take as input variables and cpt.
There are certain helper functions that help creating the cpt for the Factor given the evidence. Let us explore them one by one.
psource(make_factor)
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)
make_factor is used to create the cpt and variables that will be passed to the constructor of Factor. We use make_factor for each variable. It takes in the arguments var the particular variable, e the evidence we want to do inference on, bn the bayes network.
Here variables for each node refers to a list consisting of the variable itself and the parents minus any variables that are part of the evidence. This is created by finding the node.parents and filtering out those that are not part of the evidence.
The cpt created is the one similar to the original cpt of the node with only rows that agree with the evidence.
psource(all_events)
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)
The all_events function is a recursive generator function which yields a key for the orignal cpt which is part of the node. This works by extending evidence related to the node, thus all the output from all_events only includes events that support the evidence. Given all_events is a generator function one such event is returned on every call.
We can try this out using the example on Page 524 of the book. We will make f5(A) = P(m | A)
f5 = make_factor('MaryCalls', {'JohnCalls': True, 'MaryCalls': True}, burglary)
f5
<probability.Factor at 0x116484e48>
f5.cpt
{(True,): 0.7, (False,): 0.01}
f5.variables
['Alarm']
Here f5.cpt False key gives probability for P(MaryCalls=True | Alarm = False). Due to our representation where we only store probabilities for only in cases where the node variable is True this is the same as the cpt of the BayesNode. Let us try a somewhat different example from the book where evidence is that the Alarm = True
new_factor = make_factor('MaryCalls', {'Alarm': True}, burglary)
new_factor.cpt
{(True,): 0.7, (False,): 0.30000000000000004}
Here the cpt is for P(MaryCalls | Alarm = True). Therefore the probabilities for True and False sum up to one. Note the difference between both the cases. Again the only rows included are those consistent with the evidence.
We are interested in two kinds of operations on factors. Pointwise Product which is used to created joint distributions and Summing Out which is used for marginalization.
psource(Factor.pointwise_product)
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)
Factor.pointwise_product implements a method of creating a joint via combining two factors. We take the union of variables of both the factors and then generate the cpt for the new factor using all_events function. Note that the given we have eliminated rows that are not consistent with the evidence. Pointwise product assigns new probabilities by multiplying rows similar to that in a database join.
psource(pointwise_product)
def pointwise_product(factors, bn):
return reduce(lambda f, g: f.pointwise_product(g, bn), factors)
pointwise_product extends this operation to more than two operands where it is done sequentially in pairs of two.
psource(Factor.sum_out)
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)
Factor.sum_out makes a factor eliminating a variable by summing over its values. Again events_all is used to generate combinations for the rest of the variables.
psource(sum_out)
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
sum_out uses both Factor.sum_out and pointwise_product to finally eliminate a particular variable from all factors by summing over its values.
The algorithm described in Figure 14.11 of the book is implemented by the function elimination_ask. We use this for inference. The key idea is that we eliminate the hidden variables by interleaving joining and marginalization. It takes in 3 arguments X the query variable, e the evidence variable and bn the Bayes network.
The algorithm creates factors out of Bayes Nodes in reverse order and eliminates hidden variables using sum_out. Finally it takes a point wise product of all factors and normalizes. Let us finally solve the problem of inferring
P(Burglary=True | JohnCalls=True, MaryCalls=True) using variable elimination.
psource(elimination_ask)
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()
elimination_ask('Burglary', dict(JohnCalls=True, MaryCalls=True), burglary).show_approx()
'False: 0.716, True: 0.284'
elimination_ask
has some critical point to consider and some optimizations could be performed:
Operation on factors:
sum_out
and pointwise_product
function used in elimination_ask
is where space and time complexity arise in the variable elimination algorithm (AIMA3e pg. 526).
The only trick is to notice that any factor that does not depend on the variable to be summed out can be moved outside the summation.
Variable ordering:
Elimination ordering is important, every choice of ordering yields a valid algorithm, but different orderings cause different intermediate factors to be generated during the calculation (AIMA3e pg. 527). In this case the algorithm applies a reversed order.
In general, the time and space requirements of variable elimination are dominated by the size of the largest factor constructed during the operation of the algorithm. This in turn is determined by the order of elimination of variables and by the structure of the network. It turns out to be intractable to determine the optimal ordering, but several good heuristics are available. One fairly effective method is a greedy one: eliminate whichever variable minimizes the size of the next factor to be constructed.
Variable relevance
Some variables could be irrelevant to resolve a query (i.e. sums to 1). A variable elimination algorithm can therefore remove all these variables before evaluating the query (AIMA3e pg. 528).
An optimization is to remove 'every variable that is not an ancestor of a query variable or evidence variable is irrelevant to the query'.
Let's see how the runtimes of these two algorithms compare. We expect variable elimination to outperform enumeration by a large margin as we reduce the number of repetitive calculations significantly.
%%timeit
enumeration_ask('Burglary', dict(JohnCalls=True, MaryCalls=True), burglary).show_approx()
105 µs ± 11.9 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
%%timeit
elimination_ask('Burglary', dict(JohnCalls=True, MaryCalls=True), burglary).show_approx()
262 µs ± 54.7 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
In this test case we observe that variable elimination is slower than what we expected. It has something to do with number of threads, how Python tries to optimize things and this happens because the network is very small, with just 5 nodes. The elimination_ask
has some critical point and some optimizations must be perfomed as seen above.
Exact inference fails to scale for very large and complex Bayesian Networks. This section covers implementation of randomized sampling algorithms, also called Monte Carlo algorithms.
psource(BayesNode.sample)
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))
Before we consider the different algorithms in this section let us look at the BayesNode.sample method. It samples from the distribution for this variable conditioned on event's values for parent_variables. That is, return True/False at random according to with the conditional probability given the parents. The probability function is a simple helper from utils module which returns True with the probability passed to it.
The idea of Prior Sampling is to sample from the Bayesian Network in a topological order. We start at the top of the network and sample as per P(Xi | parents(Xi) i.e. the probability distribution from which the value is sampled is conditioned on the values already assigned to the variable's parents. This can be thought of as a simulation.
psource(prior_sample)
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
The function prior_sample implements the algorithm described in Figure 14.13 of the book. Nodes are sampled in the topological order. The old value of the event is passed as evidence for parent values. We will use the Bayesian Network in Figure 14.12 to try out the prior_sample
Traversing the graph in topological order is important. There are two possible topological orderings for this particular directed acyclic graph.
N = 1000
all_observations = [prior_sample(sprinkler) for x in range(N)]
Now we filter to get the observations where Rain = True
rain_true = [observation for observation in all_observations if observation['Rain'] == True]
Finally, we can find P(Rain=True)
answer = len(rain_true) / N
print(answer)
0.503
Sampling this another time might give different results as we have no control over the distribution of the random samples
N = 1000
all_observations = [prior_sample(sprinkler) for x in range(N)]
rain_true = [observation for observation in all_observations if observation['Rain'] == True]
answer = len(rain_true) / N
print(answer)
0.519
To evaluate a conditional distribution. We can use a two-step filtering process. We first separate out the variables that are consistent with the evidence. Then for each value of query variable, we can find probabilities. For example to find P(Cloudy=True | Rain=True). We have already filtered out the values consistent with our evidence in rain_true. Now we apply a second filtering step on rain_true to find P(Rain=True and Cloudy=True)
rain_and_cloudy = [observation for observation in rain_true if observation['Cloudy'] == True]
answer = len(rain_and_cloudy) / len(rain_true)
print(answer)
0.8265895953757225
Rejection Sampling is based on an idea similar to what we did just now. First, it generates samples from the prior distribution specified by the network. Then, it rejects all those that do not match the evidence.
psource(rejection_sampling)
def rejection_sampling(X, e, bn, N=10000):
"""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)
The function keeps counts of each of the possible values of the Query variable and increases the count when we see an observation consistent with the evidence. It takes in input parameters X - The Query Variable, e - evidence, bn - Bayes net and N - number of prior samples to generate.
consistent_with is used to check consistency.
psource(consistent_with)
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())
To answer P(Cloudy=True | Rain=True)
p = rejection_sampling('Cloudy', dict(Rain=True), sprinkler, 1000)
p[True]
0.8035019455252919
Rejection sampling takes a long time to run when the probability of finding consistent evidence is low. It is also slow for larger networks and more evidence variables. Rejection sampling tends to reject a lot of samples if our evidence consists of a large number of variables. Likelihood Weighting solves this by fixing the evidence (i.e. not sampling it) and then using weights to make sure that our overall sampling is still consistent.
The pseudocode in Figure 14.15 is implemented as likelihood_weighting and weighted_sample.
psource(weighted_sample)
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
weighted_sample samples an event from Bayesian Network that's consistent with the evidence e and returns the event and its weight, the likelihood that the event accords to the evidence. It takes in two parameters bn the Bayesian Network and e the evidence.
The weight is obtained by multiplying P(xi | parents(xi)) for each node in evidence. We set the values of event = evidence at the start of the function.
weighted_sample(sprinkler, dict(Rain=True))
({'Rain': True, 'Cloudy': False, 'Sprinkler': True, 'WetGrass': True}, 0.2)
psource(likelihood_weighting)
def likelihood_weighting(X, e, bn, N=10000):
"""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)
likelihood_weighting implements the algorithm to solve our inference problem. The code is similar to rejection_sampling but instead of adding one for each sample we add the weight obtained from weighted_sampling.
likelihood_weighting('Cloudy', dict(Rain=True), sprinkler, 200).show_approx()
'False: 0.2, True: 0.8'
In likelihood sampling, it is possible to obtain low weights in cases where the evidence variables reside at the bottom of the Bayesian Network. This can happen because influence only propagates downwards in likelihood sampling.
Gibbs Sampling solves this. The implementation of Figure 14.16 is provided in the function gibbs_ask
psource(gibbs_ask)
def gibbs_ask(X, e, bn, N=1000):
"""[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)
In gibbs_ask we initialize the non-evidence variables to random values. And then select non-evidence variables and sample it from P(Variable | value in the current state of all remaining vars) repeatedly sample. In practice, we speed this up by using markov_blanket_sample instead. This works because terms not involving the variable get canceled in the calculation. The arguments for gibbs_ask are similar to likelihood_weighting
gibbs_ask('Cloudy', dict(Rain=True), sprinkler, 200).show_approx()
'False: 0.215, True: 0.785'
Let's take a look at how much time each algorithm takes.
%%timeit
all_observations = [prior_sample(sprinkler) for x in range(1000)]
rain_true = [observation for observation in all_observations if observation['Rain'] == True]
len([observation for observation in rain_true if observation['Cloudy'] == True]) / len(rain_true)
13.2 ms ± 3.45 ms per loop (mean ± std. dev. of 7 runs, 100 loops each)
%%timeit
rejection_sampling('Cloudy', dict(Rain=True), sprinkler, 1000)
11 ms ± 687 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)
%%timeit
likelihood_weighting('Cloudy', dict(Rain=True), sprinkler, 200)
2.12 ms ± 554 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
%%timeit
gibbs_ask('Cloudy', dict(Rain=True), sprinkler, 200)
14.4 ms ± 2.16 ms per loop (mean ± std. dev. of 7 runs, 100 loops each)
As expected, all algorithms have a very similar runtime. However, rejection sampling would be a lot faster and more accurate when the probabiliy of finding data-points consistent with the required evidence is small.
Often, we need to carry out probabilistic inference on temporal data or a sequence of observations where the order of observations matter.
We require a model similar to a Bayesian Network, but one that grows over time to keep up with the latest evidences.
If you are familiar with the mdp
module or Markov models in general, you can probably guess that a Markov model might come close to representing our problem accurately.
Before we start, it will be helpful to understand the structure of a temporal model. We will use the example of the book with the guard and the umbrella. In this example, the state $\textbf{X}$ is whether it is a rainy day (X = True
) or not (X = False
) at Day $\textbf{t}$. In the sensor or observation model, the observation or evidence $\textbf{U}$ is whether the professor holds an umbrella (U = True
) or not (U = False
) on Day $\textbf{t}$. Based on that, the transition model is
$X_{t-1}$ | $X_{t}$ | P$(X_{t} |
---|---|---|
*${False}$* | *${False}$* | 0.7 |
*${False}$* | *${True}$* | 0.3 |
*${True}$* | *${False}$* | 0.3 |
*${True}$* | *${True}$* | 0.7 |
And the the sensor model will be,
$X_{t}$ | $U_{t}$ | P$(U_{t} |
---|---|---|
*${False}$* | *${True}$* | 0.2 |
*${False}$* | *${False}$* | 0.8 |
*${True}$* | *${True}$* | 0.9 |
*${True}$* | *${False}$* | 0.1 |
HMMs are implemented in the HiddenMarkovModel
class.
Let's have a look.
psource(HiddenMarkovModel)
class HiddenMarkovModel:
"""A Hidden markov model which takes Transition model and Sensor model as inputs"""
def __init__(self, transition_model, sensor_model, prior=None):
self.transition_model = transition_model
self.sensor_model = sensor_model
self.prior = prior or [0.5, 0.5]
def sensor_dist(self, ev):
if ev is True:
return self.sensor_model[0]
else:
return self.sensor_model[1]
We instantiate the object hmm
of the class using a list of lists for both the transition and the sensor model.
umbrella_transition_model = [[0.7, 0.3], [0.3, 0.7]]
umbrella_sensor_model = [[0.9, 0.2], [0.1, 0.8]]
hmm = HiddenMarkovModel(umbrella_transition_model, umbrella_sensor_model)
The sensor_dist()
method returns a list with the conditional probabilities of the sensor model.
hmm.sensor_dist(ev=True)
[0.9, 0.2]
Now that we have defined an HMM object, our task here is to compute the belief $B_{t}(x)= P(X_{t}|U_{1:t})$ given evidence U at each time step t.
There are three primary methods to carry out inference in Hidden Markov Models:
Let's have a look at how we can carry out inference and answer queries based on our umbrella HMM using these algorithms.
This is a general algorithm that works for all Markov models, not just HMMs. In the filtering task (inference) we are given evidence U in each time t and we want to compute the belief $B_{t}(x)= P(X_{t}|U_{1:t})$. We can think of it as a three step process:
The forward algorithm performs the step 2 and 3 at once. It updates, or better say reweights, the initial belief using the transition and the sensor model. Let's see the umbrella example. On Day 0 no observation is available, and for that reason we will assume that we have equal possibilities to rain or not. In the HiddenMarkovModel
class, the prior probabilities for Day 0 are by default [0.5, 0.5].
The observation update is calculated with the forward()
function. Basically, we update our belief using the observation model. The function returns a list with the probabilities of raining or not on Day 1.
psource(forward)
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))
umbrella_prior = [0.5, 0.5]
belief_day_1 = forward(hmm, umbrella_prior, ev=True)
print ('The probability of raining on day 1 is {:.2f}'.format(belief_day_1[0]))
The probability of raining on day 1 is 0.82
In Day 2 our initial belief is the updated belief of Day 1.
Again using the forward()
function we can compute the probability of raining in Day 2
belief_day_2 = forward(hmm, belief_day_1, ev=True)
print ('The probability of raining in day 2 is {:.2f}'.format(belief_day_2[0]))
The probability of raining in day 2 is 0.88
In the smoothing part we are interested in computing the distribution over past states given evidence up to the present. Assume that we want to compute the distribution for the time k, for $0\leq k<t $, the computation can be divided in two parts:
Rather than starting at time 1, the algorithm starts at time t. In the umbrella example, we can compute the backward message from Day 2 to Day 1 by using the backward
function. The backward
function has as parameters the object created by the HiddenMarkovModel
class, the evidence in Day 2 (in our case is True), and the initial probabilities of being in state in time t+1. Since no observation is available then it will be [1, 1]. The backward
function will return a list with the conditional probabilities.
psource(backward)
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])))
b = [1, 1]
backward(hmm, b, ev=True)
[0.6272727272727272, 0.37272727272727274]
Some may notice that the result is not the same as in the book. The main reason is that in the book the normalization step is not used. If we want to normalize the result, one can use the normalize()
helper function.
In order to find the smoothed estimate for raining in Day k, we will use the forward_backward()
function. As in the example in the book, the umbrella is observed in both days and the prior distribution is [0.5, 0.5]
pseudocode('Forward-Backward')
function FORWARD-BACKWARD(ev, prior) returns a vector of probability distributions
inputs: ev, a vector of evidence values for steps 1,…,t
prior, the prior distribution on the initial state, P(X0)
local variables: fv, a vector of forward messages for steps 0,…,t
b, a representation of the backward message, initially all 1s
sv, a vector of smoothed estimates for steps 1,…,t
fv[0] ← prior
for i = 1 to t do
fv[i] ← FORWARD(fv[i − 1], ev[i])
for i = t downto 1 do
sv[i] ← NORMALIZE(fv[i] × b)
b ← BACKWARD(b, ev[i])
return sv
Figure ?? The forward-backward algorithm for smoothing: computing posterior probabilities of a sequence of states given a sequence of observations. The FORWARD and BACKWARD operators are defined by Equations (??) and (??), respectively.
umbrella_prior = [0.5, 0.5]
prob = forward_backward(hmm, ev=[T, T], prior=umbrella_prior)
print ('The probability of raining in Day 0 is {:.2f} and in Day 1 is {:.2f}'.format(prob[0][0], prob[1][0]))
The probability of raining in Day 0 is 0.65 and in Day 1 is 0.88
Since HMMs are represented as single variable systems, we can represent the transition model and sensor model as matrices.
The forward_backward
algorithm can be easily carried out on this representation (as we have done here) with a time complexity of $O({S}^{2} t)$ where t is the length of the sequence and each step multiplies a vector of size $S$ with a matrix of dimensions $SxS$.
The matrix formulation allows to optimize online smoothing with a fixed lag.
psource(fixed_lag_smoothing)
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
This algorithm applies forward
as usual and optimizes the smoothing step by using the equations above.
This optimization could be achieved only because HMM properties can be represented as matrices.
Here's how we can use fixed_lag_smoothing
for inference on our umbrella HMM.
umbrella_transition_model = [[0.7, 0.3], [0.3, 0.7]]
umbrella_sensor_model = [[0.9, 0.2], [0.1, 0.8]]
hmm = HiddenMarkovModel(umbrella_transition_model, umbrella_sensor_model)
Given evidence T, F, T, F and T, we want to calculate the probability distribution for the fourth day with a fixed lag of 2 days.
e_t = F
evidence = [T, F, T, F, T]
fixed_lag_smoothing(e_t, hmm, d=2, ev=evidence, t=4)
[0.1111111111111111, 0.8888888888888888]
e_t = T
evidence = [T, T, F, T, T]
fixed_lag_smoothing(e_t, hmm, d=1, ev=evidence, t=4)
[0.9938650306748466, 0.006134969325153394]
We cannot calculate probability distributions when $t$ is less than $d$
fixed_lag_smoothing(e_t, hmm, d=5, ev=evidence, t=4)
As expected, the output is None
The filtering problem is too expensive to solve using the previous methods for problems with large or continuous state spaces. Particle filtering is a method that can solve the same problem but when the state space is a lot larger, where we wouldn't be able to do these computations in a reasonable amount of time as fast, as time goes by, and we want to keep track of things as they happen.
Particle filtering can be divided into four steps:
If we have some idea about the prior probability distribution, we drop the initial particles accordingly, or else we just drop them uniformly over the state space.
As time goes by and measurements come in, we are going to move the selected particles into the grid squares that makes the most sense in terms of representing the distribution that we are trying to track. When time goes by, we just loop through all our particles and try to simulate what could happen to each one of them by sampling its next position from the transition model. This is like prior sampling - samples' frequencies reflect the transition probabilities. If we have enough samples we are pretty close to exact values. We work through the list of particles, one particle at a time, all we do is stochastically simulate what the outcome might be. If we had no dimension of time, and we had no new measurements come in, this would be exactly the same as what we did in prior sampling.
As observations come in, don't sample the observations, fix them and downweight the samples based on the evidence just like in likelihood weighting. $$w(x) = P(e/x)$$ $$B(X) \propto P(e/X)B'(X)$$
Rather than tracking weighted samples, we resample. We choose from our weighted sample distribution as many times as the number of particles we initially had and we replace these particles too, so that we have a constant number of particles. This is equivalent to renormalizing the distribution. The samples with low weight are rarely chosen in the new distribution after resampling. This newer set of particles after resampling is in some sense more representative of the actual distribution and so we are better allocating our computational cycles. Now the update is complete for this time step, continue with the next one.
psource(particle_filtering)
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
Here, scalar_vector_product
and vector_add
are helper functions to help with vector math and weighted_sample_with_replacement
resamples from a weighted sample and replaces the original sample, as is obvious from the name.
Here's how we can use particle_filtering
on our umbrella HMM, though it doesn't make much sense using particle filtering on a problem with such a small state space.
It is just to get familiar with the syntax.
umbrella_transition_model = [[0.7, 0.3], [0.3, 0.7]]
umbrella_sensor_model = [[0.9, 0.2], [0.1, 0.8]]
hmm = HiddenMarkovModel(umbrella_transition_model, umbrella_sensor_model)
particle_filtering(T, 10, hmm)
['A', 'A', 'A', 'A', 'A', 'A', 'A', 'A', 'A', 'A']
We got 5 samples from state A
and 5 samples from state B
particle_filtering([F, T, F, F, T], 10, hmm)
['A', 'B', 'A', 'B', 'B', 'B', 'B', 'B', 'B', 'B']
This time we got 2 samples from state A
and 8 samples from state B
Comparing runtimes for these algorithms will not be useful, as each solves the filtering task efficiently for a different scenario.
In the domain of robotics, particle filtering is used for robot localization. Localization is the problem of finding out where things are, in this case, we want to find the position of a robot in a continuous state space.
psource(monte_carlo_localization)
def monte_carlo_localization(a, z, N, P_motion_sample, P_sensor, m, S=None):
"""Monte Carlo localization algorithm from Fig 25.9"""
def ray_cast(sensor_num, kin_state, m):
return m.ray_cast(sensor_num, kin_state)
M = len(z)
W = [0]*N
S_ = [0]*N
W_ = [0]*N
v = a['v']
w = a['w']
if S is None:
S = [m.sample() for _ in range(N)]
for i in range(N):
S_[i] = P_motion_sample(S[i], v, w)
W_[i] = 1
for j in range(M):
z_ = ray_cast(j, S_[i], m)
W_[i] = W_[i] * P_sensor(z[j], z_)
S = weighted_sample_with_replacement(N, S_, W_)
return S
Our implementation of Monte Carlo Localization uses the range scan method.
The ray_cast
helper function casts rays in different directions and stores the range values.
We'll now define a simple 2D map to run Monte Carlo Localization on.
m = MCLmap([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0],
[1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0],
[0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0],
[0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0],
[0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0],
[0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0]])
heatmap(m.m, cmap='binary')
Let's define the motion model as a function P_motion_sample
.
def P_motion_sample(kin_state, v, w):
"""Sample from possible kinematic states.
Returns from a single element distribution (no uncertainity in motion)"""
pos = kin_state[:2]
orient = kin_state[2]
# for simplicity the robot first rotates and then moves
orient = (orient + w)%4
for _ in range(orient):
v = (v[1], -v[0])
pos = vector_add(pos, v)
return pos + (orient,)
Define the sensor model as a function P_sensor
.
def P_sensor(x, y):
"""Conditional probability for sensor reading"""
# Need not be exact probability. Can use a scaled value.
if x == y:
return 0.8
elif abs(x - y) <= 2:
return 0.05
else:
return 0
Initializing variables.
a = {'v': (0, 0), 'w': 0}
z = (2, 4, 1, 6)
Let's run monte_carlo_localization
with these parameters to find a sample distribution S.
S = monte_carlo_localization(a, z, 1000, P_motion_sample, P_sensor, m)
Let's plot the values in the sample distribution S
.
grid = [[0]*17 for _ in range(11)]
for x, y, _ in S:
if 0 <= x < 11 and 0 <= y < 17:
grid[x][y] += 1
print("GRID:")
print_table(grid)
heatmap(grid, cmap='Oranges')
GRID: 0 0 12 0 143 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 52 201 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 5 19 9 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 166 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 11 75 0 0 0 0 0 0 0 0 0 0 0 73 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 124 0 0 0 0 0 0 1 0 3 0 0 0 0 0 0 0 0 0 0 14 4 15 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
The distribution is highly concentrated at (5, 3)
, but the robot is not very confident about its position as some other cells also have high probability values.
Let's look at another scenario.
a = {'v': (0, 1), 'w': 0}
z = (2, 3, 5, 7)
S = monte_carlo_localization(a, z, 1000, P_motion_sample, P_sensor, m, S)
grid = [[0]*17 for _ in range(11)]
for x, y, _ in S:
if 0 <= x < 11 and 0 <= y < 17:
grid[x][y] += 1
print("GRID:")
print_table(grid)
heatmap(grid, cmap='Oranges')
GRID: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
In this case, the robot is 99.9% certain that it is at position (6, 7)
.
We now move into the domain of probabilistic decision making.
Here we'll see how a decision-theoretic agent is implemented in the module.
psource(DTAgentProgram)
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
The DTAgentProgram
function is pretty self-explanatory.
Before we discuss what an information gathering agent is, we'll need to know what decision networks are. For an agent in an environment, a decision network represents information about the agent's current state, its possible actions, the state that will result from the agent's action, and the utility of that state. Decision networks have three primary kinds of nodes which are:
A description of the agent's utility as a function is associated with a utility node.
psource(DecisionNetwork)
class DecisionNetwork(BayesNet):
"""An abstract class for a decision network as a wrapper for a BayesNet.
Represents an agent's current state, its possible actions, reachable states
and utilities of those states."""
def __init__(self, action, infer):
"""action: a single action node
infer: the preferred method to carry out inference on the given BayesNet"""
super(DecisionNetwork, self).__init__()
self.action = action
self.infer = infer
def best_action(self):
"""Return the best action in the network"""
return self.action
def get_utility(self, action, state):
"""Return the utility for a particular action and state in the network"""
raise NotImplementedError
def get_expected_utility(self, action, evidence):
"""Compute the expected utility given an action and evidence"""
u = 0.0
prob_dist = self.infer(action, evidence, self).prob
for item, _ in prob_dist.items():
u += prob_dist[item] * self.get_utility(action, item)
return u
The DecisionNetwork
class inherits from BayesNet
and has a few extra helper methods.
Before we proceed, we need to know a few more terms.
An information gathering agent is an agent with certain properties that explores decision networks as and when required with heuristics driven by VPI calculations of nodes. A sensible agent should ask questions in a reasonable order, should avoid asking irrelevant questions, should take into account the importance of each piece of information in relation to its cost and should stop asking questions when that is appropriate. VPI is used as the primary heuristic to consider all these points in an information gathering agent as the agent ultimately wants to maximize the utility and needs to find the optimal cost and extent of finding the required information.
psource(InformationGatheringAgent)
class InformationGatheringAgent(Agent):
"""A simple information gathering agent. The agent works by repeatedly selecting
the observation with the highest information value, until the cost of the next
observation is greater than its expected benefit. [Figure 16.9]"""
def __init__(self, decnet, infer, initial_evidence=None):
"""decnet: a decision network
infer: the preferred method to carry out inference on the given decision network
initial_evidence: initial evidence"""
self.decnet = decnet
self.infer = infer
self.observation = initial_evidence or []
self.variables = self.decnet.nodes
def integrate_percept(self, percept):
"""Integrate the given percept into the decision network"""
raise NotImplementedError
def execute(self, percept):
"""Execute the information gathering algorithm"""
self.observation = self.integrate_percept(percept)
vpis = self.vpi_cost_ratio(self.variables)
j = argmax(vpis)
variable = self.variables[j]
if self.vpi(variable) > self.cost(variable):
return self.request(variable)
return self.decnet.best_action()
def request(self, variable):
"""Return the value of the given random variable as the next percept"""
raise NotImplementedError
def cost(self, var):
"""Return the cost of obtaining evidence through tests, consultants or questions"""
raise NotImplementedError
def vpi_cost_ratio(self, variables):
"""Return the VPI to cost ratio for the given variables"""
v_by_c = []
for var in variables:
v_by_c.append(self.vpi(var) / self.cost(var))
return v_by_c
def vpi(self, variable):
"""Return VPI for a given variable"""
vpi = 0.0
prob_dist = self.infer(variable, self.observation, self.decnet).prob
for item, _ in prob_dist.items():
post_prob = prob_dist[item]
new_observation = list(self.observation)
new_observation.append(item)
expected_utility = self.decnet.get_expected_utility(variable, new_observation)
vpi += post_prob * expected_utility
vpi -= self.decnet.get_expected_utility(variable, self.observation)
return vpi
The cost
method is an abstract method that returns the cost of obtaining the evidence through tests, consultants, questions or any other means.
With this we conclude this notebook.