This is an example to illustrate MLE. Actually, it could be easy demonstrated that when the parametric family is the normal density function, then the MLE of $$\mu$$ is the mean of the observations and the MLE of $$\sigma$$ is the uncorrected standard deviation of the observations.

Suppose that we have the following independent observations and we know that they come from the same probability density function

x<-c(29,41,32,28,38) #our observations

library('ggplot2')
dat<-data.frame(x,y=0) #we plotted our observations in the x-axis
p<-ggplot(data=dat,aes(x=x,y=y))+geom_point()
p

We don’t know the exact probability density function, but we know that the shape is the shape of the normal density function $f(x|\theta)=f(x|\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ Let’s plot two arbitrary probability density functions fixing the parameters of the parametric family

xseq<-seq(20,45,.1)
f1<-dnorm(xseq,33,7) #mu=33, sigma=7
f2<-dnorm(xseq,27,4) #mu=27, sigma=4
curve1<-data.frame(xseq,f1)
curve2<-data.frame(xseq,f2)

p<-p+geom_line(data=curve1,aes(x=xseq,y=f1))+geom_line(data=curve2,aes(x=xseq,y=f2))
p

It is clear that the probability density function $$f$$ for $$\mu=33$$ and $$\sigma=7$$ describes the observations better than the other one. We are interested in the $$f$$ that maximizes $$L$$.

The joint density function is $f(x|\mu,\sigma)=f(x_1|\mu,\sigma)f(x_2|\mu,\sigma)...f(x_5|\mu,\sigma)=$ $=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x_1-\mu)^2}{2\sigma^2}}\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x_2-\mu)^2}{2\sigma^2}}...\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x_5-\mu)^2}{2\sigma^2}}.$ that when considered as a function of the parameters is $L(\mu,\sigma|x)=L(\mu,\sigma|29,41,32,28,38)=f(x|\mu,\sigma)=f(29,41,32,28,38|\mu,\sigma)=$ $=f(29|\mu,\sigma)f(41|\mu,\sigma)...f(38|\mu,\sigma)=$ $=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(29-\mu)^2}{2\sigma^2}}\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(41-\mu)^2}{2\sigma^2}}...\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(38-\mu)^2}{2\sigma^2}}$

and $log(L)=log(f(x|\mu,\sigma))=log(f(x_1|\mu,\sigma))+log(f(x_2|\mu,\sigma))+...+log(f(x_5|\mu,\sigma))=$ $=log(f(29|\mu,\sigma))+log(f(41|\mu,\sigma))+...+log(f(38|\mu,\sigma))$

logL<-function(p) sum(log(dnorm(x,p[1],p[2]))) # p=c(p[1],p[2])=c(mu,sigma)
#logL<-function(p) sum(dnorm(x,p[1],p[2],log=T)) #alternative way to write it

We can calculate the $$log(L)$$ for the two previous examples to verify that $$log(L)$$ is larger for $$\mu=33$$ and $$\sigma=7$$

logL(c(33,7))
## [1] -15.66098
logL(c(27,4))
## [1] -22.36991
logL(c(33,7))>logL(c(27,4))
## [1] TRUE

Let’s plot the $$log(L)$$ for some values of $$\mu$$ and $$\sigma$$

library('plyr')
dLogL<-expand.grid(museq=seq(20,45,.1),sigmaseq=seq(4,10,1))
dLogL<-ddply(dLogL,.(museq,sigmaseq),transform,logLike=logL(c(museq,sigmaseq)))
ggplot(data=dLogL,aes(x=museq,y=logLike,color=factor(sigmaseq)))+geom_line()

So it seems that values of $$\mu$$ around 34 and $$\sigma$$ around 5 maximizes log(L). Let’s maximize it properly using optim. Because optim search for a minimum, we need to minimize $$-log(L)$$

negLogL<-function(p) -logL(p)
estPar<-optim(c(20,10),negLogL)
MLEparameters<-estPar\$par
MLEparameters
## [1] 33.598814  5.083355

So, we found that from the parametric family, the probability density function that better characterizes the observations according to MLE is the one described by the parameters $$\mu=$$ 33.5988136 and $$\sigma=$$ 5.0833546. Let’s plot the distribution in red in the previous graph

xseq<-seq(20,45,.1)
fMLE<-dnorm(xseq,MLEparameters[1],MLEparameters[2])
curveMLE<-data.frame(xseq,fMLE)

p+geom_line(data=curveMLE,aes(x=xseq,y=fMLE),color='red')

It could be easy demonstrated that actually when the parametric family is the normal density function, then the MLE of $$\mu$$ is the mean of the observations

mean(x)
## [1] 33.6

and the MLE of $$\sigma$$ is the uncorrected standard deviation of the observations

sqrt(mean((x-mean(x))^2))
## [1] 5.083306

References

Knoblauch, K., & Maloney, L. T. (2012). Modeling Psychophysical Data in R. New York: Springer.

Myung, I. J. (2003). Tutorial on maximum likelihood estimation. Journal of Mathematical Psychology.

Prins, N., & Kingdom, F. A. A. (2010). Psychophysics: a practical introduction. London: Academic Press.