Chernoff Bounds using importance sampling identity

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WMDhamnekar
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How to use importance sampling identity to obtain the Chernoff bounds as given below?

Let X have moment generating function ##\phi(t)= E[e^{tX}]##. Then, for any c > 0 ,

##P[X\geq c ]\leq e^{-tc} \phi(t), \text{if t > 0}##

##P[X \leq c]\leq e^{-tc}\phi(t), \text{if t<0} ##

Solution:
Here's an example of using the importance sampling identity to obtain Chernoff bounds for a binomial random variable.

Suppose we have a binomial random variable ##X \sim \text{Bin}(n,p)##, where n is the number of trials and p is the probability of success on each trial. We want to bound the probability that X deviates from its mean by more than a certain amount, i.e., we want to bound ##P(X \geq (1+\delta)np)## for some ##\delta > 0##.

Let ##X = \sum_{i=1}^n X_i##, where the ##X_i## are independent Bernoulli random variables with parameter p. Let ##Q_i## be a Bernoulli distribution with parameter q, and let ##Q = Q_1 \times \cdots \times Q_n##.

Then, by the importance sampling identity, we have ##P(X \geq (1+\delta)np) = E_Q\left[\prod_{i=1}^n \frac{dP_i}{dQ_i}I_{X \geq (1+\delta)np}\right]##

Now, let ##\phi_i(t) = E[e^{tX_i}] = 1 + p(e^t - 1)## and ##\psi_i(t) = E_Q[e^{tX_i}] = 1 + q(e^t - 1)##.

By choosing ##Q_i## such that ##\frac{dP_i}{dQ_i} = e^{tX_i - \log \phi_i(t)}##, we obtain ##P(X \geq (1+\delta)np) = E_Q\left[e^{tX - n\log \phi(t)}I_{X \geq (1+\delta)np}\right]##

Applying Markov's inequality, we get ##P(X \geq (1+\delta)np) \leq e^{-t(1+\delta)np}E_Q\left[e^{tX - n\log \phi(t)}\right] = e^{-t(1+\delta)np}\prod_{i=1}^n\psi_i(t)e^{-\log \phi_i(t)} = e^{-t(1+\delta)np}\left(\frac{1+q(e^t-1)}{1+p(e^t-1)}\right)^n##

This is the Chernoff bound for a binomial random variable. By choosing an appropriate value of t, we can obtain a tight bound on the tail probability.

In my opinion, this solution seems to look correct. Do you have any doubts? Let me know, so that I shall clear them.
 
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FAQ: Chernoff Bounds using importance sampling identity

What are Chernoff Bounds?

Chernoff Bounds are a probabilistic technique used to bound the tail distributions of random variables. They provide exponentially decreasing bounds on the probability that the sum of random variables deviates significantly from its expected value.

What is Importance Sampling?

Importance Sampling is a statistical technique used to estimate properties of a particular distribution while only having samples from a different distribution. It involves re-weighting the samples to account for the difference between the target and the sampling distributions.

How does the Importance Sampling Identity relate to Chernoff Bounds?

The Importance Sampling Identity allows for the re-weighting of probabilities in Chernoff Bounds to make the analysis more tractable. By choosing an appropriate alternative distribution, the Chernoff Bound can be tightened, leading to more accurate probability estimates.

Why use Importance Sampling with Chernoff Bounds?

Using Importance Sampling with Chernoff Bounds can lead to more efficient and tighter bounds. This is particularly useful in high-dimensional problems or when dealing with rare events, where direct computation of probabilities would be infeasible or inefficient.

Can you provide an example of Chernoff Bounds using Importance Sampling?

Consider a scenario where we want to bound the probability that the sum of n independent Bernoulli random variables deviates from its mean. Using Chernoff Bounds, we can re-weight the probabilities using an alternative distribution (e.g., a tilted Bernoulli distribution) to derive a tighter bound on the tail probabilities, often resulting in more efficient and accurate estimates.

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