ads2-probability-bayes-markov

name	ads2-probability-bayes-markov
description	Use this skill for ADS/ADS2 conditional probability, Bayes theorem, Bayes factors, prior/posterior reasoning, logic-card implication tasks, probability trees, lie-detector/base-rate problems, turtle/beach problems, and Markov-chain simulation. Trigger on Bayes, posterior, prior, conditional probability, Bayes factor, Markov chain, probability tree, lie detector, turtle, or card hypothesis.

ADS2 Probability, Bayes, And Markov Chains

Define events before calculating. Most errors come from confusing P(A|B) with P(B|A).

Conditional Probability

H = hypothesis/event of interest
D = observed data/evidence
P(H|D) = probability of H given D
P(D|H) = probability of observing D if H is true

Bayes Theorem

P(H|D) = P(D|H) P(H) / P(D)

For two exhaustive hypotheses:

P(H1|D) = P(D|H1)P(H1) / (P(D|H1)P(H1) + P(D|H2)P(H2))

State assumptions about priors.

Turtles Template

p_west <- 0.5
p_east <- 0.5
p_logger_given_west <- 0.1
p_logger_given_east <- 0.4

p_east_given_logger <- (p_logger_given_east * p_east) /
  (p_logger_given_west * p_west + p_logger_given_east * p_east)
p_east_given_logger

Report:

Assuming the ecologist was equally likely to reach either beach and there are no other beaches, the probability of being on East Beach after observing a Loggerhead turtle is 0.80.

Bayes Factor

posterior odds = Bayes factor * prior odds
Bayes factor = P(D|H1) / P(D|H2)
prior odds = P(H1) / P(H2)

Use Bayes factor when comparing two hypotheses without needing P(D).

Logic Card Problems

For implication A -> B, only A and not B falsifies the hypothesis.

Example: "Every vowel has an even number on the back."

Turn over:

• the vowel card, to check for an odd number;
• the odd-number card, to check for a vowel.

Do not need to turn over consonant or even-number cards.

Markov Chain Concepts

A Markov chain has:

• states;
• transition probabilities between states;
• outgoing probabilities from each state summing to 1;
• next state depending only on current state.

Exam tasks usually ask to draw states/transitions or simulate hitting time.

Markov Simulation Skeleton

set.seed(123)

one_run <- function(pattern = c(1, 0, 0, 1)) {
  coin <- rbinom(length(pattern), 1, 0.5)
  pointer <- 1
  while (TRUE) {
    window <- coin[pointer:(pointer + length(pattern) - 1)]
    if (all(window == pattern)) break
    pointer <- pointer + 1
    coin <- c(coin, rbinom(1, 1, 0.5))
  }
  length(coin)
}

times <- replicate(5000, one_run())
mean(times)
quantile(times, c(0.025, 0.975))

State your coding, for example 1 = H, 0 = T.