Proof of invariance property of MLE I am reading the proof of the invariance property of MLE from Casella and Berger.
In this proof we parametrize :
$\eta  = \tau(\theta)$
There we define the induced likelihood function: 
$ L_{1}^*(\eta|x) = sup_{\theta|\tau(\theta) = \eta} L(\theta|x) \tag{1}$
I have subscripted L*($\eta$|x) by 1 to differentiate between the induced likelihood of $\eta $ and the Likelihood of $\eta$ which are both denoted by $L^*(\eta|x)$
I am not sure why this is being done. (In what follows,L* is the likelihood of $\eta$ ). 
If $\theta_1$ and $\theta_2$ are such that $\tau(\theta_1) = \tau(\theta_2)$ then $L(\theta_1|x)$ = $L^*(\eta  = \tau(\theta_1)|x)$= $L^*(\eta  = \tau(\theta_2)|x)$ = $L(\theta_2|x$) since $\tau(\theta_1)$ =$\tau(\theta_2)$  
Hence there is no need of the supremum in (1).
Where do I misunderstand?
 A: Perhaps the issues here are best understood in the context of an example. Suppose that we are interested in estimating the mean of a normal model with variance 1 i.e. we are considering models of the form $N(\theta,1)$. In this case, the likelihood (for a single data point $x$) is (ignoring the constant) $L(\theta | x)=\exp(-(x-\theta)^2/2)$.
Suppose that we are actually interested in a function of the mean, call it $\eta=\tau(\theta)$. How to define the likelihood $L(\eta|x)$? If $\tau$ is invertible then we just define $L(\eta|x)$ to be $L(\theta=\tau^{-1}(\eta) | x)$ i.e. we set $\theta$ equal to the unique value corresponding to the chosen value of $\eta$. e.g. if $\tau(\theta)=2\theta$ then $L(\eta | x):=L(\theta=\frac{\eta}{2} |x)$.
What if $\tau$ is not invertible? e.g. $\tau(\theta)=\theta^2$. Should $L(\eta|x)$ be $L(\theta=+\sqrt{\eta} | x)$ or should it be defined as $L(\theta=-\sqrt{\eta} | x)$? These two values will usually be different, so the likelihood $L(\eta|x)$ is undefined. Hence Casella and Berger define the induced likelihood. With the chosen definition, it turns out that the invariance property (which is obvious when $\tau$ is invertible) still holds.
A: I believe a lot of misunderstanding arises from notation used. The best explanation of what induced likelihood function is is in the original paper of Zhenna, 1966 (see 1).
Induced likelihood function is one of ways to make $\tau(\theta)$ one-to-one when it is not one-to-one initially. Several methods of achieving this exist, see "On invariance and maximum likelihood estimation" [2]. This 
$L_{1}^*(\eta|x) = sup_{\theta|\tau(\theta) = \eta} L(\theta|x) \tag{1}$
tells to map all values of $\tau(\theta)$ to corresponding maximum value of $\theta$.
An example. Suppose $\tau(\theta)=\sin(\theta) , f(\theta)=2*\theta+5, \theta \in [0,100]$. Obviously, $sup f(\theta) = 205$ The transformation is not one-to-one. Supremum value is attained at $\theta=100$, but many values of $x$ satisfy $sin(x)=sin(100)$. All such values of $x$ are mapped by induced likelihood function to single value, the supremum value, 205 in this example. Thus it is assured that both functions likelihood and induced likelihood are maximized so invariance is preserved.
