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Suppose $y_1 = 1, y_2 = y_3 = ...=y_n = 0\ (n\geq2)$, and $\sum_{i=1}^np(y_i|x;\theta) = 1$, $0\leq p(y_i|x;\theta)\leq1$. Meanwhile $\alpha > 0$ is a constant.

Let's define a function $$f(\theta)= -\sum_{i=1}^n (y_i - \alpha p(y_i|x;\theta)) \ln p(y_i|x;\theta) $$

Please help to find the infimum of the $f(\theta)$, i.e. $\inf\ f(\theta)$.

Notes: for $0\times \ln0 = 0$

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closed as off-topic by Xi'an, mdewey, kjetil b halvorsen, Jeremy Miles, Peter Flom Jan 13 at 11:40

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