This set is due 11/4 before class. Any form of readable and intelligible presentation will be accepted. Electronic submissions should go to Fabian with subject BDA: Homework 1. We encourage discussing exercises with fellow students, but only individual solutions will be accepted.

1. Basic notions of probability

Consider the following two-dimensional matrix:

##       blond brown  red black
## blue   0.22  0.21 0.00  0.01
## green  0.00  0.14 0.06  0.01
## brown  0.16  0.15 0.00  0.04
  1. Calculate the marginal probabilities.
  2. Calculate the conditional probability of eye color given blond hair.

2. Intuitions about (conditional) probability

Imagine that there are three cards: one is red on either side, the other is white on either side, and the third is red on one side and white on the other. Suppose a confederate draws a card from this set of three totally at random, and shows a totally random side of that card to you.

  1. What is the probability that what you see is red?
  2. What is the probability that, given that you see red, the other side of the card that the confederate holds is white?
  3. Write down a two-dimensional probability matrix to support your answers.
  4. Now imagine that the confederate draws a card at random but then presents “red” with probability \(\frac{1}{4}\) if there is a red side. What is the conditional probability that the other side of the card is white if you are shown red? (Best use a new two-dimensional probability matrix in support of your answer.)

3. Compare likelihood functions for coin flips

We saw two different likelihood functions for coin flips in the second lecture. The first one is the binomial distribution:

\[P_{\text{binom}}(\langle n_h, n_t \rangle \, | \, \theta) = {{n}\choose{n_h}} \theta^{n_h} \, (1-\theta)^{n_t}\]

The second one was Kruschke’s generalization of the Bernoulli distribution:

\[P_{\text{Bern}}(\langle n_h, n_t \rangle \, | \, \theta) = \theta^{n_h} \, (1-\theta)^{n_t}\]

Prove that no matter what the priors \(P(\theta)\) are and no matter what \(n_h\) and \(n_t\) we observe, the posterior \(P_{\text{binom}}(\theta \, | \, \langle n_h, n_t \rangle)\) derived from the first likelihood function will be identical to the posterior \(P_{\text{Bern}}(\theta \, | \, \langle n_h, n_t \rangle)\) derived from the second.

Please spell out and comment/explain each relevant derivation step. Pay good attention to spelling out and manipulating the normalizing constants.

4. Wagenmakers’ critique of p-value logic

Read Wagenmakers (2007) and answer the following questions:

  1. List the three problems of p-values Wagenmakers discusses. Explain them in your own words. Which one do you think is most severe?
  2. Think of three possible cases – in science or real life – where strong relience on p-value significance testing might be harmful.
  3. What is the definition of a p-value? You can draw from this and this. Note, however, that Gelman misses something crucial in his definition of the p-value (second link)