ANSWERS TO ACTIVITIES

TOPIC 15 - Learning About Models Using Bayes' Rule

ÝActivity 15-1

(a)

		DATA
		+	-	TOTAL
MODEL	Have Disease
	Don't Have Disease
	TOTAL			10,000

(d)

		DATA
		+	-	TOTAL
MODEL	Have Disease			10
	Don't Have Disease			9990
	TOTAL			10,000

(k)

		DATA
		+	-	TOTAL
MODEL	Have Disease	9	1	10
	Don't Have Disease	999	8990	9990
	TOTAL	1008	8991	10,000

(m)

		DATA
		+	PROPORTION
MODEL	Have Disease	9	9/1008 = .0089
	Don't Have Disease	999	999/1008 = .9911
	TOTAL	1008	1.0000

Activity 15-2

(a)Ý

		COUNT	PROPORTION
MODEL	fair die	2700	.9
	fixed die	300	.1
	TOTAL	3000	1.0

NUMBER ON DIE
1	2	3	4	5	6	TOTAL
450	450	450	450	450	450	2700

(d) The fixed dice:

NUMBER ON DIE
1	2	3	4	5	6	TOTAL
100	100	100	0	0	0	300

(e)

		NUMBER ON DIE
		1	2	3	4	5	6	TOTAL
MODEL	fair die	450	450	450	450	450	450	2700
	fixed die	100	100	100	0	0	0	300
	TOTAL

(f) 450 + 100 = 550

(g) 450

(h) 450/550 = .818

(i) 150/550 = .182

Activity 15-3

The Bayes' box:

		DATA
		tagged	not tagged	TOTAL
	one fish	250	0	250
MODEL	two fish	125	125	250
	three fish	83	167	250
	four fish	62	188	250
	TOTAL	520	480	1000

If you observed a tagged fish, look only at the "tagged column":

		DATA
		tagged	PROPORTION
	one fish	250	250/520 = .48
MODEL	two fish	125	125/520 = .24
	three fish	83	83/520 = .16
	four fish	62	62/520 = .12
	TOTAL	520	1.00

Activity 15-4

The likelihoods:

data result = "not defective"

MODEL	PRIOR	LIKELIHOOD
0 defective	.7	1
1 defective	.1	.75
2 defective	.1	.50
3 defective	.05	.25
4 defective	.05	0

Computing posterior probabilities:

MODEL	PRIOR	LIKELIHOOD	PRODUCT	POSTERIOR
0 defective	.7	1	.7	.8358
1 defective	.1	.75	.075	.0896
2 defective	.1	.50	.050	.0597
3 defective	.05	.25	.0125	.0149
4 defective	.05	0	0	0
SUM			.8375	1.000

(f) The model "4 defectives" is not possible -- has a probability of 0.

(g) Using PRIOR, Prob(2 or fewer defectives) = Prob(0, 1, 2 defectives)Ý = .7 + .1 + .1 = .9

(h) Using POSTERIOR, Prob(2 or fewer defectives) = Prob(0, 1, 2 defectives) = .8358 + .0896 + .0597 = 0.9851

ÝÝÝ Activity 15-5

Bayes' box after observing the data "D scored first":

MODEL	PRIOR	LIKE	PRODUCT	POSTERIOR
D wins	.6	21/29 = .724	.434	.798
P wins	.4	8/29Ý = .276	.110	.202
SUM			.544	1.000

ÝÝÝ Activity 15-7

(a)Ý

		PRIOR
MODEL	white bag	.5
	mixed bag	.5
	TOTAL	1.0

(b) Bayes' box:

		DATA (ball drawn)
		white	black	TOTAL
MODEL	white bag	150	0	150
	mixed bag	75	75	150
	TOTAL	225	75	300

Observe "white ball drawn"

		NEW PROBS
MODEL	white bag	150/225 = .67
	mixed bag	75/225 = .33
	TOTAL	1.0

New Bayes' box:

		DATA (ball drawn)
		white	black	TOTAL
MODEL	white bag	200	0	200
	mixed bag	50	50	100
	TOTAL	250	50	300

Observe "white ball drawn"

		NEW PROBS
MODEL	white bag	200/250 = ..8
	mixed bag	50/250 = .2
	TOTAL	1.0

(d) New ball drawn -- new prior probs of "white bag" and "mixed bag" are .8 and .2.

New Bayes' box:

		DATA (ball drawn)
		white	black	TOTAL
MODEL	white bag	240	0	240
	mixed bag	30	30	60
	TOTAL	270	30	300

Observe "white ball drawn"

		NEW PROBS
MODEL	white bag	240/270 = ..89
	mixed bag	30/270 = .11
	TOTAL	1.0

Activity 15-9

Example 1:

Likelihoods if Data = "Heads on coin"

MODEL	LIKELIHOOD
fair	1/2
two-headed	1
two-tailed	0

Likelihoods if Data = "Tails on coin"

MODEL	LIKELIHOOD
fair	1/2
two-headed	0
two-tailed	1

Example 2:

Four models are

MODEL

(0 men, 3 women)

(1 man, 2 women)

(2 men, 1 women)

(3 men, 0 woman)

Data result: "person chosen is male"

MODEL	LIKELIHOOD
(0 men, 3 women)	0
(1 man, 2 women)	1/3
(2 men, 1 women)	2/3
(3 men, 0 woman)	1

Data result: "person chosen is female"

MODEL	LIKELIHOOD
(0 men, 3 women)	1
(1 man, 2 women)	2/3
(2 men, 1 women)	1/3
(3 men, 0 woman)	0

Computation of posterior probabilities given data "person chosen is female"

MODEL	PRIOR	LIKELIHOOD	PRODUCT	POSTERIOR
(0 men, 3 women)	.25	1	.25	.5
(1 man, 2 women)	.25	2/3 = .67	.167	.334
(2 men, 1 women)	.25	1/3 = .33	.083	.166
(3 men, 0 woman)	.25	0	0	0
SUM			.5	1

Activity 15-12

		DATA
		HH	HT	TH	HH	TOTAL
MODEL	coin is fair	125	125	125	125	500
	coin is two-headed	500	0	0	0	500
	TOTAL	625	125	125	125	1000

Since we observe datum HH, focus on HH column of table:

		DATA
		HH	PROBABILITY
MODEL	coin is fair	125	125/625 = .2
	coin is two-headed	500	500/625 = .8
	TOTAL	625	1

So probability that coin is two-headed has increased from .5 to .8 after seeing HH.