Solution

Under the independent-gene version of the model, integrating both component labels and effects gives

p (y ∣ q, V) = i = 1 \prod N {(1 - q) f_{0} (y_{i}) + q f_{1} (y_{i})} .

This product uses independence of the gene effects and observation errors conditional on the shared parameters. Maximize its logarithm over

0 \leq q \leq 1

, for example with a one-dimensional bounded optimizer or a grid followed by refinement. Its derivatives are

ℓ^{'} (q) = \sum_{i} \frac{f _{1} ( y _{i} ) - f _{0} ( y _{i} )}{( 1 - q ) f _{0} ( y _{i} ) + q f _{1} ( y _{i} )}, ℓ^{''} (q) = - \sum_{i} \frac{[ f _{1} ( y _{i} ) - f _{0} ( y _{i} ) ] ^{2}}{[( 1 - q ) f _{0} ( y _{i} ) + q f _{1} ( y _{i} ) ] ^{2}} \leq 0.

Thus 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 iterates

w_{i} = \frac{q f _{1} ( y _{i} )}{( 1 - q ) f _{0} ( y _{i} ) + q f _{1} ( y _{i} )}, q_{new} = \frac{1}{N} \sum_{i} w_{i} .

For

V = 0

, the components coincide and

q

is not identifiable; for

V > 0

the full observation values carry information about the mixing fraction.

 Ancestors (11)

There are no discussions about this article yet.

There are currently no matching articles.