BDA3 Chapter 1 Exercise 3
Here’s my solution to exercise 3, chapter 1, of Gelman’s Bayesian Data Analysis (BDA), 3rd edition. There are solutions to some of the exercises on the book’s webpage.
Suppose a particular gene for eye colour has two alleles: a dominant X and a recessive x allele. Having \(xx\) gives you blue eyes, otherwise you have brown eyes. Suppose also that the proportion of blueeyed people is \(p^2\), and the proportion of heterozygotes is \(2p(1 – p)\). There are 3 questions to answer:
 What is the probability of a browneyed child of browneyed parents being a heterozygote?
 If such a heterozygote, Judy, has n browneyed children with a random heterozygote, what’s the probability that Judy is a heterozygote?
 Under the conditions of part 2, what is the probability that Judy’s first grandchild has blue eyes?
Simulation
Let’s first set up some data with which we can verify the results via simulation.
Data
We’ll simulate a large population of individuals where the probability of the recessive allele is 0.2.
set.seed(11146)
N < 5000000
p < 0.2
alleles < c('x', 'X')
weights < c(p, 1  p)
df < tibble(id = 1:N %>% as.character()) %>%
mutate(
allele1 = sample(alleles, N, prob = weights, replace = TRUE),
allele2 = sample(alleles, N, prob = weights, replace = TRUE),
genotype = if_else(allele1 == allele2, 'homozygote', 'heterozygote'),
eye_colour = if_else(allele1 == 'x' & allele2 == 'x', 'blue', 'brown')
)
id  allele1  allele2  genotype  eye_colour 

1  X  X  homozygote  brown 
2  x  X  heterozygote  brown 
3  X  x  heterozygote  brown 
4  X  X  homozygote  brown 
5  x  X  heterozygote  brown 
6  X  X  homozygote  brown 
This has the correct distribution of alleles, since \(p^2 \approx\) 0.04 and \((1p)^2\approx\) 0.64.
allele_distribution < df %>%
group_by(allele1, allele2) %>%
tally() %>%
mutate(frac = n / sum(n))
allele1  allele2  n  frac 

x  x  200006  0.2000018 
x  X  800015  0.7999982 
X  x  800802  0.2002016 
X  X  3199177  0.7997984 
This also has the correct distribution of eye colours.
eye_colour_distribution < df %>%
group_by(eye_colour) %>%
tally() %>%
mutate(frac = n / sum(n))
eye_colour  n  frac 

blue  200006  0.0400012 
brown  4799994  0.9599988 
The genotype distribution is also correct, since \(2p(1p) \approx\) 0.32.
genotype_distribution < df %>%
group_by(genotype) %>%
tally() %>%
mutate(frac = n / sum(n))
genotype  n  frac 

heterozygote  1600817  0.3201634 
homozygote  3399183  0.6798366 
Reproduction
Let’s also define a couple of functions to simulate reproduction within our population. The pair
function matches up random individuals from the first table with random individuals from the second.
pair < function(df1, df2) {
inner_join(
df1 %>%
select(matches('\\.(xy)$')) %>%
select(matches('^(idallelegenotypeeye)')) %>%
ungroup() %>%
sample_frac(size = 1) %>%
mutate(row = row_number()),
df2 %>%
select(matches('\\.(xy)$')) %>%
select(matches('^(idallelegenotypeeye)')) %>%
ungroup() %>%
sample_frac(size = 1) %>%
mutate(row = row_number()),
by = 'row'
) %>%
select(row) %>%
return()
}
The reproduce
function then randomly generates a child from the paired individuals.
reproduce < function(pairs, n=1) {
pairs %>%
crossing(child = 1:n) %>%
mutate(
# the variables x and y indicate the allele taken from parent x and y, respectively
x = rbinom(n(), 1, 0.5) + 1,
y = rbinom(n(), 1, 0.5) + 1,
allele1 = if_else(x == 1, allele1.x, allele2.x),
allele2 = if_else(y == 1, allele1.y, allele2.y),
genotype = if_else(allele1 == allele2, 'homozygote', 'heterozygote'),
eye_colour = if_else(allele1 == 'x' & allele2 == 'x', 'blue', 'brown'),
id = paste(id.x, id.y, child, sep = '')
) %>%
return()
}
The kids
table then represents the next generation from random mating within the entire population.
kids < df %>%
pair(df) %>%
reproduce()
allele1.x  allele2.x  allele1.y  allele2.y  allele1  allele2  x  y 

X  X  X  x  X  X  2  1 
X  x  X  X  x  X  2  1 
X  X  X  x  X  x  1  2 
X  X  X  X  X  X  1  1 
X  x  X  x  X  X  1  1 
x  X  X  X  x  X  1  1 
The parent attributes are contained in the kids
table, with the .x
suffix for one parent and .y
for the other.
Part 1
We’ll use \(A\) to stand for the allele combination, e.g. \(XX\), or \(Xx = xX\), and \(E\) for eye colour. The subscripts \(i = 1, 2\) will be used for each of the two parents, and the absence of subscripts will indicate the variable for the child. We need to calculate the probability that the child is heterogenous given that they are browneyed with browneyed parents:
\[
\mathbb P (A = Xx \mid E, E_1, E_2 = B).
\]
It will be easier to calculate this if we can rewrite it as a probability conditional only on \(A_\bullet\)variables. First note that
\[
\begin{align}
\mathbb P (A, A_1, A_2)
&=
\mathbb P (A \mid A_1, A_2) \mathbb P(A_1, A_2)
\\
&=
\mathbb P (A \mid A_1, A_2) \mathbb P(A_1) \mathbb P (A_2)
\end{align}
\]
using the chain rule and the assumption of random mating. Therefore,
\[
\begin{align}
&
P (A = Xx \mid E_\bullet = B)
\\
&=
\frac{\mathbb P (E_\bullet = B \mid A = Xx) \cdot \mathbb P (A = Xx)}{\mathbb P (E_\bullet = B)}
\\
&=
\frac{
\sum_{a_1, a_2} \mathbb P (E_\bullet = B \mid A = Xx, A_1 = a_1, A_2 = a_2) \cdot \mathbb P (A = Xx \mid A_1 = a_1, A_2 = a_2) \cdot \mathbb P (A_1 = a_1) \cdot \mathbb P (A_2 = a_2)
}{
\sum_{a, a_1, a_2} \mathbb P (E_\bullet = B \mid A = a, A_1 = a_1, A_2 = a_2) \cdot \mathbb P (A = a \mid A_1 = a_1, A_2 = a_2) \cdot \mathbb P (A_1 = a_1) \cdot \mathbb P (A_2 = a_2)
},
\end{align}
\]
where the numerator is marginalised over possible values of \(A_1\) and \(A_2\), and the denominator additionally over \(A\).
The factors involving \(E_\bullet\) are either 1 or 0, depending only on whether the given combination of alleles can give rise to brown eyes or not, respectively. Moreover, \(\mathbb P (A_i = XX) = (1 – p)^2\) and \(\mathbb P (A_i = Xx) = 2p(1 – p)\), where the case \(A_i = xx\) is impossible conditional on everybody having brown eyes. The only nontrivial calculations now involve \(\mathbb P (A = a \mid A_1 = a_1, A_2 = a_2)\):
\[
\begin{align}
\mathbb P (A = Xx \mid A_1 = Xx, A_2 = Xx)
&=
\frac{1}{2}
\\
\mathbb P (A = Xx \mid A_1 = Xx, A_2 = XX)
&=
\frac{1}{2}
\\
\mathbb P (A = Xx \mid A_1 = XX, A_2 = XX)
&=
0
\\
\mathbb P (A = XX \mid A_1 = Xx, A_2 = Xx)
&=
\frac{1}{4}
\\
\mathbb P (A = XX \mid A_1 = Xx, A_2 = XX)
&=
\frac{1}{2}
\\
\mathbb P (A = XX \mid A_1 = XX, A_2 = XX)
&=
1,
\end{align}
\]
as can be verified by inspection.
Now let’s plug in these values into the formula for the desired probability. The numerator is
\[
\begin{align}
&
P (A = Xx \mid E_\bullet = B)
\\
&=
\frac{
\frac{1}{2} \cdot (2p(1 – p))^2
+ \frac{1}{2} \cdot 2 \cdot 2p(1 – p)(1 – p)^2
+ 0 \cdot (1 – p)^4
}{
(\frac{1}{2} + \frac{1}{4}) \cdot 4p^2(1 – p)^2
+ (\frac{1}{2} + \frac{1}{2}) \cdot 4p(1 – p)^3
+ (0 + 1) \cdot (1 – p)^4
}
\\
&=
\frac{(1 – p)^2}{(1 – p)^2}
\frac{
2p^2 + 2p(1 – p)
}{
3p^2 + 4p(1 – p) + (1 – p)^2
}
\\
&=
\frac{
2p^2 + 2p – 2p^2
}{
3p^2 + 4p – 4p^2 + 1 + p^2 – 2p
}
\\
&=
\frac{
2p
}{
1 + 2p
},
\end{align}
\]
as required. This is approximately \(2p\) for small \(p\), and is approximatily \(\frac{1}{2}\) for large \(p\).
Part 1 simulation
To condition on browneyed children from browneyed parents, we can just filter the kids
table. Such a child is called judy
in this exercise.
judy < kids %>%
filter(
eye_colour.x == 'brown',
eye_colour.y == 'brown',
eye_colour == 'brown'
)
judy_genotypes < judy %>%
group_by(genotype) %>%
tally() %>%
mutate(frac = n / sum(n))
genotype  n  frac 

heterozygote  1280529  0.2858268 
homozygote  3199559  0.7141732 
This is very close to the theoretical value of \(\frac{2p}{1 + 2p}\approx\) 0.286.
Part 2
Denote by \(E_{C_\bullet} = B\) the condition that all of Judy’s children have brown eyes, and by \(A^p = a\) the condition that Judy’s partner has allele combination \(a\). Then
\[
\begin{align}
&
\mathbb P (A = Xx \mid E_\bullet = B = E_{C_\bullet}, A^p = Xx)
\\
&=
\frac{
\mathbb P (A = Xx \mid E_\bullet = B, A^p = Xx)
\cdot
\mathbb P (E_{C_\bullet} = B \mid E_\bullet = B, A^p = Xx = A)
}{
\mathbb P (E_{C_\bullet} = B \mid E_\bullet = B, A^p = Xx)
}
\\
&=
\frac{
\mathbb P (A = Xx \mid E_\bullet = B)
\cdot
\mathbb P (E_{C_\bullet} = B \mid A^p = Xx = A)
}{
\sum_a
\mathbb P (E_{C_\bullet} = B \mid E_\bullet = B, A^p = Xx, A = a)
\cdot
\mathbb P (A = a \mid E_\bullet = B, A^p = Xx)
}
\\
&=
\frac{
\mathbb P (A = Xx \mid E_\bullet = B)
\cdot
\mathbb P (E_{C_\bullet} = B \mid A^p = Xx = A)
}{
\sum_a
\mathbb P (E_{C_\bullet} = B \mid A^p = Xx, A = a)
\cdot
\mathbb P (A = a \mid E_\bullet = B)
}
\\
&=
\frac{
\frac{2p}{1 + 2p}
\cdot
(\frac{3}{4})^n
}{
\mathbb P (E_{C_\bullet} = B \mid A^p = Xx = A)
\cdot
\mathbb P (A = Xx \mid E_\bullet = B)
+
\mathbb P (E_{C_\bullet} = B \mid A^p = Xx, A = XX)
\cdot
\mathbb P (A = XX \mid E_\bullet = B)
}
\\
&=
\frac{
\frac{2p}{1 + 2p}
\cdot
(\frac{3}{4})^n
}{
\frac{2p}{1 + 2p}
\cdot
(\frac{3}{4})^n
+
\frac{1}{1 + 2p}
}
\\
&=
\frac{2p \cdot (\frac{3}{4})^n}{2p \cdot (\frac{3}{4})^n + 1}
\\
&=
\frac{2p \cdot 3^n}{2p \cdot 3^n + 4^n}
,
\end{align}
\]
where we have used conditional independence several times for the probability of the child’s alleles given the parents’ alleles. As \(n \rightarrow \infty\), this probability shrinks to 0.
Part 2 simulation
To simulate part 2, we need to pair judy
with heterozygotes from the general population, then filter for those children with brown eyes.
judy_kids < df %>%
filter(genotype == 'heterozygote') %>%
pair(judy) %>%
reproduce() %>%
ungroup() %>%
filter(eye_colour == 'brown')
Amongst judy_kids
, Judy’s attributes have the .y
suffix. Given the above conditions, the probability of her possible genotypes are then:
judy_genotypes_posterior < judy_kids %>%
group_by(genotype.y) %>%
tally() %>%
mutate(frac = n / sum(n))
genotype.y  n  frac 

heterozygote  343962  0.2313911 
homozygote  1142534  0.7686089 
This is close to the theoretical value of \(\frac{6p}{6p + 4}\approx\) 23.1%.
Part 3
Let’s introduce some notation. Let \(A_g\) be the alleles of Judy’s first grandchild, the child of \(c\) with alleles \(A_c\) whose partner has alleles \(A_c^p\). We wish to calculate \(\mathbb P (A_g = xx \mid E_\bullet = B = E_{C_\bullet}, A^p = Xx)\).
First note that
$$
\[\begin{align}
&
\mathbb P (A_c = Xx \mid E_\bullet = E_{C_\bullet} = B, A^p = Xx, A = a)
\\
&=
\frac{
\mathbb P (E_{C_\bullet} = B \mid E_\bullet = B, A_c = Xx = A^p, A = a)
\cdot
\mathbb P (A_c = Xx \mid E_\bullet = B, A^p = Xx, A = a, A)
}{
\mathbb P (E_{C_\bullet} = B \mid E_\bullet = B, A^p = Xx, A = a)
}
\\
&=
\frac{
\mathbb P (E_{C_\bullet} = B \mid E_\bullet = B, A_c = Xx = A^p, A = a)
\cdot
0.5
}{
\mathbb P (E_{C_\bullet} = B \mid E_\bullet = B, A^p = Xx, A = a)
}
\\
&=
\begin{cases}
\frac{\left(\frac{3}{4}\right)^{n1} \cdot \frac{1}{2}}{\left(\frac{3}{4}\right)^n}
&\text{if } A = Xx
\\
1 \cdot \frac{1}{2} / 1
&\text{othewrise}
\end{cases}
\\
&=
\begin{cases}
\frac{2}{3}
&\text{if } A = Xx
\\
\frac{1}{2}
&\text{othewrise}
\end{cases}
.
\end{align}\]
$$
Thus,
$$
\[\begin{align}
&
\mathbb P (A_c = Xx \mid E_\bullet = B = E_{C_\bullet}, A^p = Xx)
\\
&=
\sum_a
\mathbb P (A_c = Xx \mid E_\bullet = B = E_{C_\bullet}, A^p = Xx, A = a)
\cdot
\mathbb P (A = a \mid E_\bullet = B = E_{C_\bullet}, A^p = Xx)
\\
&=
\frac{2}{3} \cdot
\frac{2p \cdot (\frac{3}{4})^n}{2p \cdot (\frac{3}{4})^n + 1}
+
\frac{1}{2} \cdot
\frac{1}{2p \cdot (\frac{3}{4})^n + 1}
\\
&=
\frac{p\left( \frac{3}{4} \right)^{n1} + 0.5}{2p \cdot \left(\frac{3}{4}\right)^n + 1}
,
\end{align}\]
$$
which converges to \(\frac{1}{2}\) as \(n \rightarrow \infty\).
The probability that Judy’s first grandchild is a homozygote can then be calculated by marginalising over the allele combinations of the child and their partner:
$$
\[\begin{align}
&
\mathbb P (A_g = xx \mid E_\bullet = B = E_{C_\bullet}, A^p = Xx)
\\
&=
\sum_{a_c, a_c^p}
\mathbb P (A_g = xx \mid E_\bullet = B = E_{C_\bullet}, A^p = Xx, A_c = a_c, A_c^p = a_c^p)
\cdot
\mathbb P (A_c = a_c, A_c^p = a_c^p \mid E_\bullet = B = E_{C_\bullet}, A^p = Xx)
\\
&=
\sum_{a_c, a_c^p}
\mathbb P (A_g = xx \mid A_c = a_c, A_c^p = a_c^p)
\cdot
\mathbb P (A_c^p = a_c^p )
\cdot
\mathbb P (A_c = a_c \mid E_\bullet = B = E_{C_\bullet}, A^p = Xx)
\\
&=
\sum_{a_c^p}
\mathbb P (A_g = xx \mid A_c = Xx, A_c^p = a_c^p)
\cdot
\mathbb P (A_c^p = a_c^p )
\cdot
\mathbb P (A_c = Xx \mid E_\bullet = B = E_{C_\bullet}, A^p = Xx)
\\
&=
\frac{p\left( \frac{3}{4} \right)^{n1} + 0.5}{2p \cdot \left(\frac{3}{4}\right)^n + 1}
\cdot
\sum_{a_c^p}
\mathbb P (A_g = xx \mid A_c = Xx, A_c^p = a_c^p)
\cdot
\mathbb P (A_c^p = a_c^p )
\\
&=
\frac{p\left( \frac{3}{4} \right)^{n1} + 0.5}{2p \cdot \left(\frac{3}{4}\right)^n + 1}
\cdot
\left(
\mathbb P (A_g = xx \mid A_c = Xx, A_c^p = Xx)
\cdot
\mathbb P (A_c^p = Xx )
+
\mathbb P (A_g = xx \mid A_c = Xx, A_c^p = xx)
\cdot
\mathbb P (A_c^p = xx )
\right)
\\
&=
\frac{p\left( \frac{3}{4} \right)^{n1} + 0.5}{2p \cdot \left(\frac{3}{4}\right)^n + 1}
\cdot
\left(
\frac{1}{4}
\cdot
2p(1 – p)
+
\frac{1}{2}
\cdot
p^2
\right)
\\
&=
\frac{p\left( \frac{3}{4} \right)^{n1} + 0.5}{2p \cdot \left(\frac{3}{4}\right)^n + 1}
\cdot
\frac{p}{2}
,
\end{align}\]
$$
since

the grandchild can only be blueeyed if the (browneyed) child has at least one xallele, i.e. the child is \(Xx\);

\(A\) and \(A^p\) are independent by the random mating assumption; and

\(A_c\) and \(A_c^p\) are independent by the random mating assumption.
As \(n \rightarrow \infty\), this probability converges to \(\frac{p}{4}\).
Part 3 simulation
To simulate Judy’s grandkids, we pair up judy_kids
with members of the general population.
judy_grandkids < judy_kids %>%
pair(df) %>%
reproduce()
judy_grandkids %>%
summarise(mean(eye_colour == 'blue')) %>%
pull() %>%
signif(3)
[1] 0.054
The above fraction of grandkids with blue eyes is consistent with the theoretical value of \(\frac{4p + 0.5}{6p + 4}\frac{p}{2} \approx\) 0.0538.
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