Class conducted on: 22-08-2024 by Yash Shah

Uncertainty

Uncertainty in probability refers to the lack of complete knowledge or predictability about an outcome or event. It's a fundamental concept in probability theory that deals with the inherent randomness and variability in real-world phenomena.

Key Concepts

• Probability: A measure of the likelihood of an event occurring, expressed as a number between 0 and 1.
• Random Variables: Variables whose possible values are outcomes of a random phenomenon.
• Conditional Probability: The probability of an event occurring given that another event has already occurred.
• Independence: When the occurrence of one event does not affect the probability of another event.
• Bayes' Rule: A method for updating probabilities based on new evidence or information.
• Joint Probability: The probability of two or more events occurring together.
• Bayesian Networks: Graphical models that represent probabilistic relationships among a set of variables.

Probability Rules

• Negation: P(¬a) = 1 - P(a). This stems from the fact that the sum of the probabilities of all the possible worlds is 1, and the complementary literals a and ¬a include all the possible worlds.

• Inclusion-Exclusion: P(a ∨ b) = P(a) + P(b) - P(a ∧ b). This can interpreted in the following way: the worlds in which a or b are true are equal to all the worlds where a is true, plus the worlds where b is true. However, in this case, some worlds are counted twice (the worlds where both a and b are true)). To get rid of this overlap, we subtract once the worlds where both a and b are true (since they were counted twice).

  Here is an example: Suppose I eat ice cream 80% of days and cookies 70% of days. If we’re calculating the probability that today I eat ice cream or cookies P(ice cream ∨ cookies) without subtracting P(ice cream ∧ cookies), we erroneously end up with 0.7 + 0.8 = 1.5. This contradicts the axiom that probability ranges between 0 and 1. To correct for counting twice the days when I ate both ice cream and cookies, we need to subtract P(ice cream ∧ cookies) once.

• Marginalisation: P(a) = P(a, b) + P(a, ¬b). The idea here is that b and ¬b are disjoint probabilities. That is, the probability of b and ¬b occurring at the same time is 0. We also know b and ¬b sum up to 1. Thus, when a happens, b can either happen or not. When we take the probability of both a and b happening in addition to the probability of a and ¬b, we end up with simply the probability of a.

• Conditioning: P(a) = P(a | b)P(b) + P(a | ¬b)P(¬b). This is a similar idea to marginalisation. The probability of event a occurring is equal to the probability of a given b times the probability of b, plus the probability of a given ¬b time the probability of ¬b.

Bayes Probability

Bayes' Probability, also known as Bayes' Theorem or Bayes' Rule, is a fundamental concept in probability theory and statistics. It provides a way to update the probability of a hypothesis based on new evidence or data.

The formula for Bayes' Theorem is:

$$ P(A|B) = \frac{P(B|A) * P(A)}{P(B)} $$