30 Aug 2017 Excercise Solutions

1. Consider the example of the manufactured bouncy balls that are manufactured in four discrete sizes: diameters of 1.0, 2.0, 3.0, and 4.0 (assume units are cm). The manufacturing process is quite imprecise (‘noisy’), however. Assume the noise follows a Normal distribution with mean=0 and variance s2=1. You sample 3 balls of diameter 1.77, 2.23, and 2.7 that were manufactured to be of the same diameter. The prior probability of the balls being from a given diameter are equal. What is the probability of the ball diameter being 1.0, 2.0, 3.0, and 4.0. Show your work. (refer to K2.1.1 pages 20-21 for more details).

size<-c(1,2,3,4)
balls<-c(1.77,2.23,2.7)
L1<-prod(dnorm(balls, mean = size[1], sd = 1))
L2<-prod(dnorm(balls, mean = size[2], sd = 1))
L3<-prod(dnorm(balls, mean = size[3], sd = 1))
L4<-prod(dnorm(balls, mean = size[4], sd = 1))
c(L1,L2,L3,L4)/sum(c(L1,L2,L3,L4))

[1] 0.070566992 0.636868053 0.286163262 0.006401693

2. Explain why the ‘cloud’ of credible regression lines in the left side panel of Figure 2.5 (K2.3, page 26) have the characteristic shape that they do, e.g., narrow range in center of plot but broader towards edges?

1. The slope and intercept covary so that when one is higher then the other is lower and vice versa.

3. Exercise Kruschke 2.1

“[Purpose: To get you actively manipulating mathematical models of probabilities.] Suppose we have a four-sided die from a board game. On a tetrahedral die, each face is an equilateral triangle. When you roll the die, it lands with one face down and the other three faces visible as a three-sided pyramid. The faces are numbered 1–4, with the value of the bottom face printed (as clustered dots) at the bottom edges of all three visible faces. Denote the value of the bottom face as x. Consider the following three mathematical descriptions of the probabilities of x. Model A: p(x) = 1/4. Model B: p(x) = x/10. Model C: p(x) = 12/(25x). For each model, determine the value of p(x) for each value of x. Describe in words what kind of bias (or lack of bias) is expressed by each model.”

Model A: p(x=1) = 0.25 p(x=2) = 0.25 p(x=3) = 0.25 p(x=4) = 0.25

Model B: p(x) = x/10

x<-1:4
p<-x/10
p

[1] 0.1 0.2 0.3 0.4

sum(p)

[1] 1

The bias is towards higher numbers.

Model C: p(x) = 12/(25x)

#x<-1:4
p<-12/(25*x)
p

[1] 0.48 0.24 0.16 0.12

sum(p)

[1] 1

The bias is towards lower numbers.

4. Exercise Kruschke 2.2

“[Purpose: To get you actively thinking about how data cause credibilities to shift.] Suppose we have the tetrahedral die introduced in the previous exercise, along with the three candidate models of the die’s probabilities. Suppose that initially, we are not sure what to believe about the die. On the one hand, the die might be fair, with each face landing with the same probability. On the other hand, the die might be biased, with the faces that have more dots landing down more often (because the dots are created by embedding heavy jewels in the die, so that the sides with more dots are more likely to land on the bottom). On yet another hand, the die might be biased such that more dots on a face make it less likely to land down (because maybe the dots are bouncy rubber or protrude from the surface). So, initially, our beliefs about the three models can be described as p(A) = p(B) = p(C) = 1/3. Now we roll the die 100 times and find these results: #1’s = 25, #2’s = 25, #3’s = 25, #4’s = 25. Do these data change our beliefs about the models? Which[…]”

x<-c(25,25,25,25)
p<-x/sum(x)
p

[1] 0.25 0.25 0.25 0.25

Yes, this supports model A.

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