UNIT II
PROBABILISTIC REASONING
Acting under uncertainty – Bayesian inference – naïve bayes models. Probabilistic reasoning –
Bayesian networks – exact inference in BN – approximate inference in BN – causal networks.
Introduction of Probabilistic Reasoning
A agent working in real world environment almost never has access to whole truth about its
environment. Therefore, agent needs to work under uncertainty.
Earlier agents make the epistemological commitment that either the facts are true, false or else they are
unknown.
When an agent knows enough facts about its environment, the logical approach enables it to derive
plans, which are guaranteed to work.
But when agent works with uncertain knowledge then it might be impossible to construct a complete
and correct description of how its actions will work.
The right thing logical agent , take a rational decision.
The rational decision depends on following things:
• The relative importance of various goals.
• The likelihood and the degree to which, goals will be achieved.
Acting Under Uncertainty
An agent would possess some early basic knowledge of the world (Assume that knowledge is
represented in first order logic sentence).
Using first order logic to handle real word problem domains fails for three main reasons as discussed
below:
1) Laziness:
It is too much work to list the complete set of antecedents or consequents needed to ensure an
exception less rule and too hard to use such rules.
2) Theoretical ignorance:
A particular problem may not have complete theory for the domain.
3) Practical ignorance:
Even if all the rules are known, particular aspects of problem are not checked yet or some details are
not considered at all (missing out the details).
,The agent's knowledge can provide it with a degree of belief with relevant sentences. To this degree of
belief probability theory is applied.
Probability assigns a numerical degree of belief between 0 and 1 to each sentence.
Probability provides a way of summarizing the uncertainty that comes from our laziness and
ignorance.
Assigning probability of 0 to a given sentence corresponds to an unequivocal belief saying that
sentence is false.
Assigning probability of 1 corresponds to an unequivocal belief saying that the sentence is true.
Probabilities between 0 and 1 correspond to intermediate degree of belief in the truth of the sentence.
The beliefs completely depends on percept’s of agent at particular time.
These percept’s constitute the evidence on which probability assertions are based.
When more sentences are added to knowledge base the entailment keeps on changing. Similarly the
probability would also keep on changing with additional knowledge.
All probability statements must therefore, indicate the evidence with respect to which the probability is
being assessed.
As the agent receives new precepts, its probability assessments are updated to reflect the new evidence.
Before the evidence is obtained, we talk about prior or unconditional probability; after the evidence is
obtained, we talk about posterior or conditional probability.
In most cases, an agent will have some evidence from its percepts and will be interested in computing
the posterior probabilities of the outcomes it cares about.
Uncertainty and rational decisions:
The presence of uncertainty drastically changes the way an agent makes decision.
At particular time an agent can have various available decisions, from which it has to make a choice.
To make such choices an agent must have a preferences between the different possible outcomes, of
the various plans.
A particular outcome is completely specified state, along with the expected factors related with the
outcome.
For example: Consider a car driving agent who wants to reach at airport by a specific time say at 7.30
pm.
Here factors like, whether agent arrived at airport on time, what is the length of waiting duration at the
airport are attached with the outcome.
,Utility Theory
Utility theory is used to represent and reason with preferences. The term utility in current context is
used as "quality of being useful".
Utility theory says that every state has a degree of usefulness called as utility. The agent will prefer the
states with higher utility.
The utility of the state is relative to the agent for which utility function is calculated on the basis of
agent's preferences.
For example: The payoff functions for games are utility functions. The utility of a state in which black
has won a game of chess is obviously high for the agent playing black and low for the agent playing
white.
There is no measure that can count test or preferences. Someone loves deep chocolate ice cream and
someone loves chocochip icecream. A utility function can account for altruistic behavior, simply by
including the welfare of other as one of the factors contributing to the agent's own utility.
Decision theory
Preferences as expressed by utilities are combined with probabilities for making rational decisions.
This theory, of rational decision making is called as decision theory.
Decision theory can be summarized as,
Decision theory = Probability theory + Utility theory.
Need of probabilistic reasoning in AI:
o When there are unpredictable outcomes.
o When specifications or possibilities of predicates becomes too large to handle.
o When an unknown error occurs during an experiment.
In probabilistic reasoning, there are two ways to solve problems with uncertain knowledge:
o Bayes' rule
o Bayesian Statistics
As probabilistic reasoning uses probability and related terms, so before understanding probabilistic
reasoning, let's understand some common terms:
Probability: Probability can be defined as a chance that an uncertain event will occur. It is the
numerical measure of the likelihood that an event will occur. The value of probability always
remains between 0 and 1 that represent ideal uncertainties.
, 1. 0 ≤ P(A) ≤ 1, where P(A) is the probability of an event A.
2. P(A) = 0, indicates total uncertainty in an event A.
3. P(A) =1, indicates total certainty in an event A.
We can find the probability of an uncertain event by using the below formula.
o P(¬A) = probability of a not happening event.
o P(¬A) + P(A) = 1.
Event: Each possible outcome of a variable is called an event.
Sample space: The collection of all possible events is called sample space.
Random variables: Random variables are used to represent the events and objects in the real world.
Prior probability: The prior probability of an event is probability computed before observing new
information.
Posterior Probability: The probability that is calculated after all evidence or information has taken
into account. It is a combination of prior probability and new information.
Conditional probability:
Conditional probability is a probability of occurring an event when another event has already
happened.
Let's suppose, we want to calculate the event A when event B has already occurred, "the probability of
A under the conditions of B", it can be written as:
Where P(A⋀B)= Joint probability of a and B
P(B)= Marginal probability of B.
If the probability of A is given and we need to find the probability of B, then it will be given as:
PROBABILISTIC REASONING
Acting under uncertainty – Bayesian inference – naïve bayes models. Probabilistic reasoning –
Bayesian networks – exact inference in BN – approximate inference in BN – causal networks.
Introduction of Probabilistic Reasoning
A agent working in real world environment almost never has access to whole truth about its
environment. Therefore, agent needs to work under uncertainty.
Earlier agents make the epistemological commitment that either the facts are true, false or else they are
unknown.
When an agent knows enough facts about its environment, the logical approach enables it to derive
plans, which are guaranteed to work.
But when agent works with uncertain knowledge then it might be impossible to construct a complete
and correct description of how its actions will work.
The right thing logical agent , take a rational decision.
The rational decision depends on following things:
• The relative importance of various goals.
• The likelihood and the degree to which, goals will be achieved.
Acting Under Uncertainty
An agent would possess some early basic knowledge of the world (Assume that knowledge is
represented in first order logic sentence).
Using first order logic to handle real word problem domains fails for three main reasons as discussed
below:
1) Laziness:
It is too much work to list the complete set of antecedents or consequents needed to ensure an
exception less rule and too hard to use such rules.
2) Theoretical ignorance:
A particular problem may not have complete theory for the domain.
3) Practical ignorance:
Even if all the rules are known, particular aspects of problem are not checked yet or some details are
not considered at all (missing out the details).
,The agent's knowledge can provide it with a degree of belief with relevant sentences. To this degree of
belief probability theory is applied.
Probability assigns a numerical degree of belief between 0 and 1 to each sentence.
Probability provides a way of summarizing the uncertainty that comes from our laziness and
ignorance.
Assigning probability of 0 to a given sentence corresponds to an unequivocal belief saying that
sentence is false.
Assigning probability of 1 corresponds to an unequivocal belief saying that the sentence is true.
Probabilities between 0 and 1 correspond to intermediate degree of belief in the truth of the sentence.
The beliefs completely depends on percept’s of agent at particular time.
These percept’s constitute the evidence on which probability assertions are based.
When more sentences are added to knowledge base the entailment keeps on changing. Similarly the
probability would also keep on changing with additional knowledge.
All probability statements must therefore, indicate the evidence with respect to which the probability is
being assessed.
As the agent receives new precepts, its probability assessments are updated to reflect the new evidence.
Before the evidence is obtained, we talk about prior or unconditional probability; after the evidence is
obtained, we talk about posterior or conditional probability.
In most cases, an agent will have some evidence from its percepts and will be interested in computing
the posterior probabilities of the outcomes it cares about.
Uncertainty and rational decisions:
The presence of uncertainty drastically changes the way an agent makes decision.
At particular time an agent can have various available decisions, from which it has to make a choice.
To make such choices an agent must have a preferences between the different possible outcomes, of
the various plans.
A particular outcome is completely specified state, along with the expected factors related with the
outcome.
For example: Consider a car driving agent who wants to reach at airport by a specific time say at 7.30
pm.
Here factors like, whether agent arrived at airport on time, what is the length of waiting duration at the
airport are attached with the outcome.
,Utility Theory
Utility theory is used to represent and reason with preferences. The term utility in current context is
used as "quality of being useful".
Utility theory says that every state has a degree of usefulness called as utility. The agent will prefer the
states with higher utility.
The utility of the state is relative to the agent for which utility function is calculated on the basis of
agent's preferences.
For example: The payoff functions for games are utility functions. The utility of a state in which black
has won a game of chess is obviously high for the agent playing black and low for the agent playing
white.
There is no measure that can count test or preferences. Someone loves deep chocolate ice cream and
someone loves chocochip icecream. A utility function can account for altruistic behavior, simply by
including the welfare of other as one of the factors contributing to the agent's own utility.
Decision theory
Preferences as expressed by utilities are combined with probabilities for making rational decisions.
This theory, of rational decision making is called as decision theory.
Decision theory can be summarized as,
Decision theory = Probability theory + Utility theory.
Need of probabilistic reasoning in AI:
o When there are unpredictable outcomes.
o When specifications or possibilities of predicates becomes too large to handle.
o When an unknown error occurs during an experiment.
In probabilistic reasoning, there are two ways to solve problems with uncertain knowledge:
o Bayes' rule
o Bayesian Statistics
As probabilistic reasoning uses probability and related terms, so before understanding probabilistic
reasoning, let's understand some common terms:
Probability: Probability can be defined as a chance that an uncertain event will occur. It is the
numerical measure of the likelihood that an event will occur. The value of probability always
remains between 0 and 1 that represent ideal uncertainties.
, 1. 0 ≤ P(A) ≤ 1, where P(A) is the probability of an event A.
2. P(A) = 0, indicates total uncertainty in an event A.
3. P(A) =1, indicates total certainty in an event A.
We can find the probability of an uncertain event by using the below formula.
o P(¬A) = probability of a not happening event.
o P(¬A) + P(A) = 1.
Event: Each possible outcome of a variable is called an event.
Sample space: The collection of all possible events is called sample space.
Random variables: Random variables are used to represent the events and objects in the real world.
Prior probability: The prior probability of an event is probability computed before observing new
information.
Posterior Probability: The probability that is calculated after all evidence or information has taken
into account. It is a combination of prior probability and new information.
Conditional probability:
Conditional probability is a probability of occurring an event when another event has already
happened.
Let's suppose, we want to calculate the event A when event B has already occurred, "the probability of
A under the conditions of B", it can be written as:
Where P(A⋀B)= Joint probability of a and B
P(B)= Marginal probability of B.
If the probability of A is given and we need to find the probability of B, then it will be given as:
