Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 216 / 4 / d

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- Solution d

Solution

Positive definiteness makes the quadratic term uniquely maximal at

β^{(t + 1)} = d .

For each

j

, maximize the concave function

r_{j} lo g π_{j} + q_{j} lo g (1 - π_{j})

. Its interior critical point is

π_{j}^{(t + 1)} = \frac{r _{j}}{r _{j} + q _{j}} = \frac{r _{j}}{n} .

The same formula gives the appropriate boundary value when

r_{j} = 0

or

q_{j} = 0

.

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