Writing a proper normal log-likelihood in R - r

Question

I have a problem regarding the following model,
where I want to make inference on μ and tau, u is a known vector and x is the data vector. The log-likelihood is
I have a problem writing a log-likelihood in R.
x <- c(3.3569,1.9247,3.6156,1.8446,2.2196,6.8194,2.0820,4.1293,0.3609,2.6197)
mu <- seq(0,10,length=1000)
normal.lik1<-function(theta,x){
u <- c(1,3,0.5,0.2,2,1.7,0.4,1.2,1.1,0.7)
mu<-theta[1]
tau<-theta[2]
n<-length(x)
logl <- sapply(c(mu,tau),function(mu,tau){logl<- -0.5*n*log(2*pi) -0.5*n*log(tau^2+u^2)- (1/(2*tau^2+u^2))*sum((x-mu)^2) } )
return(logl)
}
#test if it works for mu=1, tau=2
head(normal.lik1(c(1,2),x))
#Does not work..
I want to be able to plug in the vector for mu and plot it over mu for a fixed value of tau, say 2. I also want to find out the MLE's of tau and mu using the optim function. I tried:
theta.hat<-optim(c(1,1),loglike2,control=list(fnscale=-1),x=x,,method="BFGS")$par
But it does not work.. Any suggestions to how I can write the likelihood?

Answer 1

First, as has been mentioned in the comments to your question, there is no need to use sapply(). You can simply use sum() – just as in the formula of the logLikelihood.
I changed this part in normal.lik1() and multiplied the expression that is assigned to logl by minus 1 such that the function computes the minus logLikelihood. You want to search for the minimum over theta since the function returns positive values.
x < c(3.3569,1.9247,3.6156,1.8446,2.2196,6.8194,2.0820,4.1293,0.3609,2.6197)
u <- c(1,3,0.5,0.2,2,1.7,0.4,1.2,1.1,0.7)
normal.lik1 <- function(theta,x,u){
mu <- theta[1]
tau <- theta[2]
n <- length(x)
logl <- - n/2 * log(2*pi) - 1/2 * sum(log(tau^2+u^2)) - 1/2 * sum((x-mu)^2/(tau^2+u^2))
return(-logl)
}
This can be done using nlm(), for example
nlm(normal.lik1, c(0,1), hessian=TRUE, x=x,u=u)$estimate
where c(0,1) are the starting values for the algorithm.
To plot the logLikelihood for a range of values of mu and some fixed tau you can adjust the function such that mu and tau are separate numeric arguments.
normal.lik2 <- function(mu,tau,x,u){
n <- length(x)
logl <- - n/2 * log(2*pi) - 1/2 * sum(log(tau^2+u^2)) - 1/2 * sum((x-mu)^2/(tau^2+u^2))
return(logl)
}
Then define some range for mu, compute the loglikelihood and use plot().
range.mu <- seq(-10,20,0.1)
loglik <- sapply(range.mu, function(m) normal.lik2(mu=m,tau=2,x=x,u=u))
plot(range.mu, loglik, type = "l")
I'm sure there are more elegant ways to do this but this does the trick.