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I'm looking for a lower-bound on $E\left[\log \left(B + \sum_i a_i X_i\right)\right]$ where $X_i$ are Bernoulli random variables with $p(X_i = 1) = q_i$ and $a_i > 0, B > 0$.

Because $X_i$ is 0 or 1, it's hard to use Jensen inequality as it results in $\log(X_i)$ which maybe $-\infty$ and the lower-bound is meaningless.

Thank you very much for any help.

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  • $\begingroup$ Anyway you must assume at least one of the $X_i$ is not zero $\endgroup$ – rlartiga Apr 7 '14 at 18:41
  • $\begingroup$ I add a positive constant B so that the sum is always positive. $\endgroup$ – user3363540 Apr 7 '14 at 18:48
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If the number of $X_i$ and $a_i$ is some large $n$ then you can use a large deviation inequality to upper bound the probability $P$ that the number of $X_i = 1$ is below a certain threshold $k$ which is less than the expected count which is $n/2$. Then the expected value of your function is at least equal to $(1-P)Y$ where $Y$ is the average value of $\log(B + \sum_{i \in S} a_i)$ where $S$ ranges over all subsets of size $k$.

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Since your function is concave instead of convex, you can only use Jensen's inequality to get an upper bound instead of a lower bound. The upper bound you get is $\log(E(B + \sum_i a_i X_i)) = \log(B + \sum_i a_i / 2)$. I'm not sure how to derive a lower bound when you have a concave function.

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  • $\begingroup$ If $X_i$ is not 0,1. I could apply Jensen on the log sum first, before taking expectation. $\log$ is concave so Jensen gives a lower-bound on the log sum. $\endgroup$ – user3363540 Apr 7 '14 at 18:53

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