# Infimum of the difference between KL Divergence and Entropy [closed]

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 $$0\times \ln0 = 0$$

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

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