If gene frequencies are in Hardy-Weinberg equilibrium, the genotypes AA, Aa and aa occur with probabilities (1–0)², 20(1-0) and 0². Suppose that on haptoglobin type in a sample of n people, n₁ individuals have genotypes AA, n₂ individuals have genotypes Aa and n3 individuals have genotypes aa: AA n1 Haptoglobin Type Aa n2 aa n3 Thus, (n₁, n2, n3) is multinomial with parameters n and (1 - 0)², 20(1-0) and 0². (a) Write down the likelihood function L(0) of 0. (b) Assume that the prior for is uniform between 0 and 1. Find the posterior
If gene frequencies are in Hardy-Weinberg equilibrium, the genotypes AA, Aa and aa occur with probabilities (1–0)², 20(1-0) and 0². Suppose that on haptoglobin type in a sample of n people, n₁ individuals have genotypes AA, n₂ individuals have genotypes Aa and n3 individuals have genotypes aa: AA n1 Haptoglobin Type Aa n2 aa n3 Thus, (n₁, n2, n3) is multinomial with parameters n and (1 - 0)², 20(1-0) and 0². (a) Write down the likelihood function L(0) of 0. (b) Assume that the prior for is uniform between 0 and 1. Find the posterior
MATLAB: An Introduction with Applications
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
Transcribed Image Text:a)
If gene frequencies are in Hardy-Weinberg equilibrium, the genotypes AA, Aa and aa
occur with probabilities (1-0)², 20(1-0) and 0². Suppose that on haptoglobin type in
a sample of n people, n₁ individuals have genotypes AA, n₂ individuals have genotypes
Aa and n3 individuals have genotypes aa:
b)
AA
n1
Haptoglobin Type
Aa
n2
Solution:
Thus, (n₁, n2, n3) is multinomial with parameters n and (1 - 0)², 20(1-0) and 0².
(a) Write down the likelihood function L(0) of 0.
(b) Assume that the prior for is uniform between 0 and 1. Find the posterior
distribution of given data n₁, n2 and n3. Is this a known distribution?
(c) Find the Jeffrey's prior for 0.
f(e\n₁, n₂, n3) 0 x
L(0)=P(N₁ = n₁, №₂ = n₂, N3 = 13)
n!
n₁!· n₂! n3!
x
x
aa
n3
(1
π₁ (0)=√(1-0)²
2n
0)²″ ¹(20(1 – 0))^²(0)²n ³3
n!
n₁!·n₂! n3!
n!
n₁!·n₂! -n₂!
22xn!
n₁!·n₂! n3!
2n
2n
(1 – 0)²″ ¹(20(1 – 0))^²(0)²n 3
(1-0)
(1 -0
3
¹(2″ ²0″²(1 – 0)^²) (0)²¹ ³
2n₁+n₂
n₂+2n3
The Jeffreys prior for the parameter of the continuous probability density function is
2n₁ n₂(-20²+20-1) n3
0²(1-0)² 0²
+

Transcribed Image Text:Continue with Question 3 and answer the following:
(a) Use the data on the haptoglobin type and compute the MLE Ô for 0 and I(Ô) the
Fisher information evaluated at the MLE.
(b) Use the normal approximation for posterior distribution to find a 95% credible
interval for 0.
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VIEWStep 3: Determine the posterior distribution of θ
VIEWStep 4: Determine the Jeffrey's prior for θ
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VIEWStep 6: Find the Fisher information at the MLE
VIEWStep 7: Find a 95% credible interval for θ
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