# Inferring a prior belief after observing a behavior?

#### nottolina

##### New Member
In my experiment, a participant goes through a maze made of 32 T intersections. At each intersection he must choose whether to go either to the left or to the right: one option will lead to another T intersection, while the other option will lead to a blind alley.

If I code as 1 the times the correct turn is to the right and as 0 the times the correct turn is to the left, this is my maze:

Code:
    turn_right <- c(1,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,1,1,1,0,1,1,0,1,1,1,1,0,1,0,0,1)
At each intersection, a sign points either to the left or to the right. A storm has messed up the signs, so that now only 50% of them are correct. The participant knows that the storm has made some damage to the signage system, but he does not what kind of damage.

These are my signs, where a 1 means that the sign points to the correct direction, and a 0 means that the sign points to the wrong direction.

Code:
    sign <- c(1,0,1,0,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0,0,1,1,0,0,1)
Now I observe the behavior of my participant. Sometimes he follows the sign (1), sometimes he does not (0):

Code:
trust_sign <- c(0,0,0,0,0,0,0,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0)
My question: can I infer what is the prior belief of the participant before entering the maze? That is, how much he trusts the signage system?

Since we have binary choices, I thought I could model the participant's choices (trust_sign) with a beta distribution:

Code:
 maze <- data.frame(turn_right, sign, trust_sign)
sum32 <- sum(maze\$trust_sign[1:32])
curve(dbeta(x, sum32, 32- sum32),add=TRUE,lty="solid",ylim=c(0,6),ylab="Probability Density",las=1)

I can also calculate the likelihood of a sign being correct given the actual maze:

Code:
 k = 16 # number of times a sign is correct
n = 32 # total number of intersections

numSteps = 200 ## x-axis for plotting
x = seq(0, 1, 1 / numSteps)

L = x^k * (1 - x)^(n - k) ## Likelihood function

L = L / sum(L) * numSteps ## Just normalize likelihood

plot(x, L, type = 'l', lwd = 3, ylim = c(0,6),
main = "Bernoulli Likelihood",
xlab = expression(theta), ylab = "pdf")

Given that likelihood and the behavior seen before, what is the belief of my participant prior to entering the maze? Is this the right framework for this question or am I missing something?