How to calculate the likelihood?

Question

We will define the following probabilities for a PCR test:

p - The probability that a pcr test of a **sick** person, will show us a positive result.

q - The probability that a pcr test of a **healthy** person, will show us a positive result.

s - The general percentage of patients in the population.

Let q = 10%, p = 95% and s = 20%

We want to generate data of 1000 people how came to check,

I did it in this way:

s = 0.2

p = 0.95

q = 0.1

true_result = c()

test_result = c()

for (i in 1:1000){

x = runif(1,0,1)

if (x<=s){

true_result = append(true_result,1)

r = runif(1,0,1)

if (r<=p){

test_result = append(test_result,1)

}else{

test_result = append(test_result,0)

}

}else{

true_result = append(true_result,0)

r = runif(1,0,1)

if (r<=q){

test_result = append(test_result,1)

}

else{

test_result = append(test_result,0)

}

}

}

Now, we want to generate 10,000 sample from the posterior.

And for that we need the prior and likelihood.

Let the priors to be:

**f_q(q) = 2q if x in [0,1] and 0 otw.**

**f_p(p)= 2(1-p) if x in [0,1] and 0 otw.**

**s ~ u(0,1)**

I calculated the joint prior in this way:

JointPrior <- function(p,s,q){

if(p<=0 | p>=1){

return (0)

}

if (q<= 0 | q>=1){

return(0)

}

return(s*2*q*2*(1-p))

}

The part where I get stuck is in the Likelihood calculation..

How can I calculate the likelihood? can someone help me with that?

Answer 1

Likelihood <- function(p, s, q, test_result, true_result){

n11 <- sum(test_result == 1 & true_result == 1)

n10 <- sum(test_result == 0 & true_result == 1)

n01 <- sum(test_result == 1 & true_result == 0)

n00 <- sum(test_result == 0 & true_result == 0)

return ((p * s)^n11 * ((1 - p) * s)^n10 * (q * (1 - s))^n01 * ((1 - q) * (1 - s))^n00)

}