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# data is MMFM
# Likelihood
thetas <- seq(0,1,0.05)
like <- c()
for (theta in thetas) like <- c(like, (theta^3)*(1-theta)^1)
thetas[which.max(like)]
0.75
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plot(thetas, like)

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chi2 <- 2*(like[which(thetas==0.75)]/like[which(thetas==0.05)])
print(chi2)
1 - pchisq(chi2, df=1)
[1] 1776.316
0
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# Belief 1
belief1 <- 1/thetas
belief1[1] <- NA
belief1 <- belief1/sum(belief1, na.rm=T)
par(mfrow=c(1,3))
plot(thetas,like,type="l")
plot(thetas,belief1,type="l")
plot(thetas,belief1*like,type="l")

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thetas[which.max(like*belief1)]
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# Belief 2
belief2 <- 1/(thetas^2)
belief2[1] <- NA
belief2 <- belief2/sum(belief2, na.rm=T)
par(mfrow=c(1,3))
plot(thetas,like,type="l")
plot(thetas,belief2,type="l")
plot(thetas,belief2*like,type="l")
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thetas[which.max(like*belief2)]
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# Belief 3
belief3 <- 1/(thetas^3)
belief3[1] <- NA
belief3 <- belief3/sum(belief3, na.rm=T)
par(mfrow=c(1,3))
plot(thetas,like,type="l")
plot(thetas,belief3,type="l")
plot(thetas,belief3*like,type="l")
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thetas[which.max(like*belief3)]
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# make it a legitimate probability function?
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prob <- belief2*like
prob <- prob / sum(prob, na.rm=T)
plot(thetas,prob,type="l")
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sum(prob[thetas<=0.20],na.rm=T)
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sum(prob[thetas<=0.05],na.rm=T)
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