Is Maximum Likelihood implemented differently in different supervised classification systems? Some functions are identical between systems, for example nearest neighbour analysis. It would be unusual to have nearest neighbour implemented differently as appears in academic papers.
Is it the same with maximum likelihood (supervised classification)? How likely is it that the same input imagery raster and same training polygons result a different classified image in ERDAS, ENVI, ArcGIS and IDRISI?
 A: You should not expect the same results using different software. "Maximum likelihood" is a general term for a common way of estimating parameters for a statistical model: attempt to find the values of the parameters that maximize the likelihood function for the model.
Different software packages may run different models for classification, and may also parameterize them differently. This means they are likely starting with different likelihood functions and will have different results.
Furthermore, in most cases, there is not a closed-form solution to the likelihood maximization, so the problem is a numerical optimization problem. Different algorithms will find different solutions (though hopefully they will be quite close to each other - this will be true if the likelihood function is "well-behaved" which is not always the case). For example, even starting with the same model, one piece of software may use L-BFGS and another Nelder-Mead.
There could be other differences as well, maybe some of the algorithms use bootstrapping for robustness, or cross-validation to improve predictive accuracy. And for classification, your software may be automatically choosing an operating point. All of these, if implemented, are probably implemented differently in different software packages.
