In artificial intelligence, decision-making often involves uncertain knowledge, where information is incomplete, noisy, or probabilistic. To handle uncertainty, AI systems use probability theory, Bayesian reasoning, and graphical models like Bayesian networks to infer and update beliefs based on new evidence.
1.) Random Variables:
A random variable is a variable that can take different values, each with an associated probability. These variables help model uncertainty in real-world problems.
Types of Random Variables:
- Discrete Random Variables: Take a finite number of values (e.g., rolling a die: {1, 2, 3, 4, 5, 6}).
- Continuous Random Variables: Take infinite values within a range (e.g., temperature measurements: {0°C to 100°C}).
Example:
Let X be a random variable representing the weather.
- X=”Sunny” → 0.7 probability
- X=”Rainy” → 0.3 probability
These probabilities help AI systems predict outcomes in uncertain environments.
2.) Prior and Posterior Probability:
Prior Probability (P(A)): The initial probability of an event before considering new evidence.
Posterior Probability (P(A|B)): The updated probability of an event after considering new evidence.
Example:
- Suppose 10% of students have the flu (Prior Probability = 0.1).
- A student has a fever (new evidence).
- The probability that a student actually has the flu given they have a fever is the posterior probability, calculated using Bayes’ Theorem.
3.) Inference Using Full Joint Distribution:
A Full Joint Distribution lists the probabilities of all possible combinations of random variables in a system.
Example:
If we have two variables:
- Weather W: {Sunny, Rainy}
- Traffic T: {Heavy, Light}
The full joint probability distribution:
4.) Bayes’ Rule (Bayes’ Theorem):
Bayes’ Rule is a mathematical formula for updating probabilities when new evidence is observed. It relates prior and posterior probabilities as:
Explanation:
- P(A|B): Probability of event A given B (Posterior)
- P(B|A): Probability of B given A (Likelihood)
- P(A): Probability of A (Prior)
- P(B): Probability of B (Evidence)
Example: (Medical Diagnosis)
- A: Patient has disease D.
- B: Patient tests positive.
Given:
- P(D)=0.01 (1% of people have disease)
- P(Positive∣D)=0.9 (90% of sick patients test positive)
- P(Positive)=0.05 (5% of all patients test positive)
Compute Posterior Probability:
Thus, the probability that a patient actually has the disease after a positive test is 18%, even though the test detects 90% of cases.
5.) Bayesian Networks:
A Bayesian Network is a graphical model that represents probabilistic relationships among a set of variables using nodes and directed edges.
Components:
- Nodes: Represent random variables.
- Edges: Represent probabilistic dependencies.
- Conditional Probability Tables (CPTs): Define relationships between connected nodes.
Example: Medical Diagnosis Bayesian Network
Cold -----> Fever
| |
↓ ↓
Cough Test Result- Fever depends on Cold.
- Test Result depends on Fever.
- Cough depends on Cold.
Using Bayesian Networks, we can infer the probability of a cold given a cough without computing the full joint distribution.
6.) Reasoning in Belief Networks:
Belief networks (Bayesian networks) are used to infer probabilities and make decisions under uncertainty.
Types of Reasoning:
- Diagnostic Reasoning (Effect → Cause): Given symptoms, infer the disease.
- Causal Reasoning (Cause → Effect): Given a disease, predict symptoms.
- Intercausal Reasoning: Adjust beliefs when new evidence is introduced.
Example: Disease Diagnosis
- Known: The patient has cough.
- Infer: What is the probability they have a cold?
- New Evidence: The patient also has fever → Updated Probability using Bayes’ Rule.
