TITLE: Is there an entropy proof for bounding a weighted sum of binomial coefficients?
QUESTION [6 upvotes]: Given a probability $p \in (0,1)$ and parameter $\alpha \in (0,1)$, is there an entropy-based proof which yields a good upper bound for the sum
$$\sum_{\ell = 0}^{\alpha n} \binom{n}{\ell}p^\ell(1-p)^{n-\ell}$$
when $n$ is large?
When $p = 1/2$, there is very simple proof (for example, see section 3.1 of this paper) which upper bounds the above quantity by
$$2^{(H(\alpha) - 1)n}$$
when $H(\cdot)$ denotes the binary entropy function.
Is there a proof using similar techniques which gives a bound for the more general sum above (which can be interpreted as the CDF of a binomial distribution with parameter $p$)?
I'd also be interested in other proofs for bounds on the above sum. The appropriate bound has already noted in this answer, but doesn't sketch out a proof establishing this result.

REPLY [4 votes]: As you said, the sum is $\Pr[X \leq \alpha n]$ where $X$ is drawn from a Binomial distribution with $n$ trials having $p$ probability of success. Bounds on this sum (for $\alpha < p$) are called "tail bounds", "concentration inequalities", etc. These bounds are proven for many settings, especially sums of independent random variables, of which Binomials are the nicest special case.
A typical proof approach, which some of us call the Chernoff method, looks like this: if $X = \sum_{i=1}^n Y_i$ where each $Y_i$ is an i.i.d. Bernoulli$(p)$, then
\begin{align}
  \Pr[X \leq k] &= \Pr[e^X \leq e^k]  \\
                &\leq e^{-k} \mathbb{E} e^X  & \text{Markov's inequality}  \\
                &= e^{-k} \prod_{i=1}^n \mathbb{E} e^{Y_i}  & \text{Indpendence}  \\
                &= e^{-k} \left(pe + (1-p)\right)^n  \\
                &\dots
\end{align}
etc. I omitted a detail -- we scale both $X$ and $k$ by some factor $\lambda$, which we choose later -- and stopped the analysis early, but that's how many proofs start.
Starting points:

https://en.wikipedia.org/wiki/Binomial_distribution#Tail_bounds
https://en.wikipedia.org/wiki/Hoeffding%27s_inequality
https://en.wikipedia.org/wiki/Chernoff_bound
https://en.wikipedia.org/wiki/Concentration_inequality
Concentration Inequalities: A Nonasymptotic Theory of Independence by Stephane Boucheron, Gabor Lugosi, Pascal Massart.