 Solution

Under the independent-gene version of the model, integrating both component labels and effects givesThis product uses independence of the gene effects and observation errors conditional on the shared parameters. Maximize its logarithm over $0\le q\le 1$, for example with a one-dimensional bounded optimizer or a grid followed by refinement. Its derivatives areThus the log likelihood function is concave. An interior maximum-likelihood estimate solves the score equation; otherwise it is at zero or one, determined by the endpoint score signs. An alternative expectation-maximization algorithm iteratesFor $V=0$, the components coincide and $q$ is not identifiable; for $V>0$ the full observation values carry information about the mixing fraction.