Suppose I calculate a likelihood as a function of M parameters from a chi-square using the incomplete gamma function. I wish to plot likelihood contours in 2 dimensions, i.e. "suppress" the other parameters. Suppose M is 3. If I calculate 3-D contours and project them onto the 2-D plot, then obviously M is 3. If I hold the third parameter constant, then obviously M is 2. What about the case when, for a given point in the 2-D plot, I take the maximum of the likelihood from the set of values described by the third parameter? When I take the average? For these 2 cases, would M be 2 or 3?
Obviously, one has to know M before calculating the likelihood.