## Description

## 1. Bayes Network.

Incidences of diseases A and B (DA, DB) depend on the exposure (E).

Disease A is additionally influenced by risk factors (R). Both diseases lead to symptoms (S).

Results of the test for disease A (TA) are affected also by disease B. Positive test will be

denoted as TA = 1, negative as TA = 0. The Bayes Network is shown in Figure 1. Needed

conditional probabilities are shown in Table 1.

E R

DB DA

S TA

Figure 1: The DAG of the Bayesian networks

Table 1: The known (or elicited) conditional probabilities

E 0 1

0.8 0.2

R 0 1

0.7 0.3

DA 0 1

E

cRc 0.9 0.1

E

cR 0.4 0.6

ERc 0.5 0.5

ER 0.3 0.7

DB 0 1

E

c 0.8 0.2

E 0.3 0.7

S 0 1

Dc

ADc

B 0.95 0.05

Dc

ADB 0.6 0.4

DADc

B 0.4 0.6

DADB 0.1 0.9

TA 0 1

Dc

ADc

B 0.92 0.08

Dc

ADB 0.8 0.2

DADc

B 0.15 0.85

DADB 0.03 0.97

(a) What is the probability of disease A (DA = 1), if disease B is not present (DB = 0),

but symptoms are present (S = 1).

(b) What is the probability of exposure (E = 1), if symptoms are present (S = 1) and

test is positive (TA = 1).

Hint: You can solve this problem by any of the 3 ways: (i) use of WinBUGS or Open2

BUGS, (ii) direct simulation using Octave/MATLAB, R, or Python, and (iii) exact calculation.

## 2. Times to Failure.

Three devices are monitored until failure. The observed lifetimes

are 0.9, 1.8, and 0.3 years. If the lifetimes ate modeled as exponential distribution with rate

λ,

Ti ∼ Exp(λ), f(t|λ) = λe−λt, t > 0, λ > 0.

Assume exponential prior on λ,

λ ∼ Exp(2), π(λ) = 2e

−2λ

, λ > 0.

(a) Find the posterior distribution of λ.

(b) Find the Bayes estimator for λ.

(c) Find the MAP estimator for λ.

(d) Numerically find 95% equitailed confidence interval for λ.

(e) Find the posterior probability of hypothesis H0 : λ ≤ 1/2.

## 3. Gibbs and High/Low Protein Diet in Rats.

Armitage and Berry (1994, p. 111)

report data on the weight gain of 19 female rats between 28 and 84 days after birth. The

rats were placed on diets with high (12 animals) and low (7 animals) protein content.

High protein Low protein

134 70

146 118

104 101

119 85

124 107

161 132

107 94

83

113

129

97

123

We want to test the hypothesis on dietary effect. Did a low protein diet result in significantly lower weight gain?

The classical t test against one sided alternative will be significant. We will do the test

Bayesian way using Gibbs sampler.

Assume that high-protein diet measurements y1i

, i = 1, . . . , 12 are coming from normal

distribution N (θ1, 1/τ1), where τ1 is precision parameter,

f(y1i

|θ1, τ1) ∝ τ

1/2

1

exp

−

τ1

2

(y1i − θ1)

2

, i = 1, . . . , 12.

Low-protein diet measurements y2i

, i = 1, . . . , 7 are coming from normal distribution

N (θ2, 1/τ2),

f(y2i

|θ2, τ2) ∝ τ

1/2

2

exp

−

τ2

2

(y2i − θ2)

2

, i = 1, . . . , 7.

Assume that θ1 and θ2 have normal priors N (θ10, 1/τ10) and N (θ20, 1/τ20), respectively. Take

prior means as θ10 = θ20 = 110 (apriori no preference) and precisions as τ10 = τ20 = 1/100.

Assume that τ1 and τ2 have the gamma Ga(a1, b2) and Ga(a2, b2) priors with shapes

a1 = a2 = 0.01 and rates b1 = b2 = 4.

(a) Construct Gibbs sampler that will sample θ1, τ1, θ2, and τ2 from their posteriors.

(b) Find sample differences θ1 − θ2 . Proportion of positive differences approximates

posterior probability of hypothesis H0 : θ1 > θ2. What is this proportion?

(c) Using sample quantiles find the 95% equitailed credible set for θ1 − θ2. Does this set

contain 0?