Orthogonal parametrization In general inference, why orthogonal parameters are useful, and why is it worth trying to find a new parametrization that makes the parameters orthogonal ? 
I have seen some textbook examples, not so many, and would be interested in more concrete examples and/or motivation.
 A: In Maximum Likelihood, the term orthogonal parameters is used when you can achieve a clean factorization of a multi-parameter likelihood function.  Say your data have two parameters $\theta$ and $\lambda$.  If you can rewrite the joint likelihood:
$L(\theta, \lambda) = L_{1}(\theta) L_{2}(\lambda)$
then we call $\theta$ and $\lambda$ orthogonal parameters.  The obvious case is when you have independence, but this is not necessary for the definition as long as factorization can be achieved.  Orthogonal parameters are desirable because, if $\theta$ is of interest, then you can perform inference using $L_{1}$. 
When we don't have orthogonal parameters, we try to find factorizations like
$L(\theta, \lambda) = L_{1}(\theta) L_{2}(\theta, \lambda)$
and perform inference using $L_1$.  In this case, we must argue that the information loss due to excluding $L_{2}$ is low.  This leads to the concept of marginal likelihood.
A: This is a good, if underspecified question.  
Simply put, obtaining an orthogonal parametrization allows for parameters of interest to be conveniently related to other parameters, particularly in establishing needed minimizations. Whether or not this is useful depends on what you are trying to do (in the case of some physics problems, for instance, orthogonal parametrization may obscure the symmetries of interest).
In the case of statistical inference, orthogonal parametrization can allow the use of  statistics by way of minimization (or its dual) on orthogonal parameters.  For instance, Cox and Reid use the orthogonality of nuisance parameters (and their appropriately applied maximum likelihood estimates) to construct a generalization of a liklihood ratio statistic for a parameter of interest. 
To see how orthogonality allows for this requires an understanding of the properties of commonly used mathematical spaces and the construction of estimators, which is essentially an issue of information geometry.  See Information Geometry, Bayesian Inference, Ideal Estimates and Error Decomposition for a lucid, but technical description of orthogonality and its role in statistical inference.
