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I have to methods of projecting random samples in $\mathbb{R}^n$ onto a manifold defined by $C(q)=0$, which is a lower-dimensional subset. Now, samples in $\mathbb{R}^n$ are uniformly distributed. ...

This is actually a variant of a well-known problem of how the parameters of a multinomial distribution can be estimated by maximum likelihood, and this arises from a final year project I undertook ...