Probability Frequentist deﬁnition A probability P(A) is the proportion of outcomes where the fact A is true after we repeat the experience or observed the situation a near inﬁnite amount of times.

The Dice Problem “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?”

The Dice Problem “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?” “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?”

The Dice Problem “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?” “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?”

The Dice Problem “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?” H 4 6 8 12 “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?”

The Dice Problem “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?” H 4 6 8 12 Prior P(H) 1/4 1/4 1/4 1/4 “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?”

The Dice Problem “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?” H 4 6 8 12 Prior P(H) 1/4 1/4 1/4 1/4 Likelihood P(E|H) 0 1/6 1/8 1/12 “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?”

The Dice Problem “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?” H 4 6 8 12 Prior P(H) 1/4 1/4 1/4 1/4 Likelihood P(E|H) 0 1/6 1/8 1/12 P(H) * P(E|H) 0 1/24 1/32 1/48 “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?”

The Dice Problem “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?” H 4 6 8 12 Prior P(H) 1/4 1/4 1/4 1/4 Likelihood P(E|H) 0 1/6 1/8 1/12 P(H) * P(E|H) 0 1/24 1/32 1/48 Posterior P(H|E) 0 4/9 3/9 2/9 “Suppose I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die and a 12-sided die. Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?”

The Dice Problem “Now suppose I roll the same die from before again, and this time I get an 8. What are the probabilities now?” H Prior P(H) Likelihood P(E|H) P(H) * P(E|H) Posterior P(H|E) 4 0 0 0 0 6 4/9 0 0 0 8 3/9 1/8 1/24 9/13 12 2/9 1/12 1/54 4/13

The Euro Problem “Lukas was playing with a Belgian one-euro coin. He spun it on its edge 10 times and it ended with the heads side up 8 of those times. Intrigued, he proceed to repeat the experience 240 more times. At the end, the coin came up heads 140 times and tails 110. Is this data evidence enough to state that the coin is biased rather than fair?”

The Euro Problem “Lukas was playing with a Belgian one-euro coin. He spun it on its edge 10 times and it ended with the heads side up 8 of those times. Intrigued, he proceed to repeat the experience 240 more times. At the end, the coin came up heads 140 times and tails 110. Is this data evidence enough to state that the coin is biased rather than fair?” “Lukas was playing with a Belgian one-euro coin. He spun it on its edge 10 times and it ended with the heads side up 8 of those times. Intrigued, he proceed to repeat the experience 240 more times. At the end, the coin came up heads 140 times and tails 110. Is this data evidence enough to state that the coin is biased rather than fair?”

Choosing a Prior So far, we’ve chosen uninformative priors - we started with no knowledge or assumptions about the hypotheses’ probabilities. What if we actually know more or believe in something else about them?

Summary • Bayesian inference provides a rational way to combine prior beliefs with new evidence • Beliefs can be updated iteratively as evidence arrives • With enough data, different initial beliefs tend to converge on the same posterior