Lecture 9 Flashcards
What is a sample space in probability?
The set of all possible outcomes of an experiment.
What is an atomic event?
A single possible outcome within the sample space.
What is an event in probability?
A subset of the sample space that includes one or more atomic events.
What is the probability of an event?
The sum of the probabilities of all atomic events that make up the event.
What is a probability space?
A sample space combined with a probability distribution over the events.
How do you calculate the probability of an event occurring?
Divide the number of favorable outcomes by the total number of possible outcomes.
What is a random variable?
A function that assigns numerical values to outcomes in a probability space.
What is the difference between discrete and continuous random variables?
Discrete variables take on countable values, while continuous variables can take on any value within a range.
What is a probability distribution?
A function that assigns probabilities to all possible values of a random variable.
What is a joint probability distribution?
A probability distribution that considers two or more random variables together.
What does P(A|B) represent in probability?
The probability of event A occurring given that event B has already occurred (conditional probability).
How is conditional probability calculated?
P(A|B) = P(A,B) / P(B), where P(A,B) is the probability of both A and B occurring.
What does Bayes’ theorem allow us to do?
Calculate the probability of a cause given an observed effect.
Why is Bayes’ theorem useful?
It helps update probabilities based on new evidence.
What is the inversion problem in probability?
The problem where we often have data about P(effect|cause) but need to determine P(cause|effect).
What is the difference between frequentist and Bayesian probability?
Frequentist probability is based on observed frequencies, while Bayesian probability represents degrees of belief.
What is a Bayesian network?
A probabilistic graphical model that represents conditional dependencies between random variables.
What does the product rule state?
P(A,B) = P(A|B) * P(B), which allows probabilities of multiple events to be decomposed.
What is the chain rule of probability?
A formula that expresses the joint probability of multiple events using conditional probabilities.
What is the advantage of Bayesian networks?
They simplify probability calculations by assuming conditional independence between variables.
What does it mean for two variables to be independent?
If P(A|B) = P(A), then A and B are independent, meaning knowing B provides no information about A.
What is conditional independence?
Two variables are conditionally independent given a third variable if their probabilities do not depend on each other when the third variable is known.
How does conditional independence reduce computation in Bayesian networks?
It reduces the number of required probability calculations by limiting dependencies to direct parent nodes.
What is the role of parent nodes in Bayesian networks?
Each variable depends only on its direct parent nodes, simplifying the probability distribution.
What is the likelihood of an event in probability?
The probability of observing given data assuming a particular hypothesis is true.
