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Probability Theory
Lecture 4 Terms
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Terms in this set (11)
Complex Events
Composites of simple events in the same space
e.g., probability (heads) OR probability (tails)
Shared Events
Simultaneous occurrence of multiple simple events in the same space
e.g., probability (heads) AND probability (tails)
Axiom #2
Probability of a complex event equals the sum of outcomes that make up that event
Axiom #3
If two events are independent, the probability of a shared event (both events occur) equals the product of the individual probabilities.
Sampling with Replacement
Choosing an ace from a deck of cards, replacing it, AND then choosing an ace as the second card.
Sampling without Replacement
Choosing an ace from a deck of cards, AND then choosing an ace as the second card.
Conditional Probability
When calculating the probability of a complex event, information on known outcomes can be used to revise the probability calculations..
These updated estimates are termed "conditional probabilities" P (A l B).
Probability of event A, given that event B occurred.
Bayes Theorem
Concerned with conditional probability.
It quantifies the probability that a hypothesis is true if some event has happened.
Goal: to calculate the "Posterior Probability", incorporating the new observation (event)
Bayesian Theory
Approach: Need to consider three quantities:
Probability of observation, if hypothesis is true.
Probability of observation, if hypothesis is not true/
The Prior: Probability of the hypothesis, before the (new) observation was made
Frequentist Statistics
Approach: Outcome is binary: accept / reject
Compare hypothesis and decision about hypothesis
Bayesian vs Frequentist Statistics
To oversimplify:
"Bayesian probability" is the interpretation of probability as the degree of belief in a hypothesis.
Yields probabilities to hypotheses, which vary as additional observations are collected.
"Frequentist probability" is the interpretation of probability as the frequency of a particular outcome in a large number of experimental trials.
Yields the acceptance / rejection of the null hypothesis.
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