Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, October 02, 2012, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Jan Hazla

Concentration Bounds using Bayes' Rule

Concentration bounds limit the probability of a random variable deviating too much from a certain value (usually its expectation).

Let us have a look at a simple version of the so called Chernoff bound:
Let A1, A2, ... be independent events with Pr[A_i] = 1/2. Let k be a number between 0 and n/2 and let S(n, k) be the event that at least n/2 + k of the events A_1, ..., A_n occur. We would like to show that Pr[S(n,k)] \leq exp(-Omega(k^2/n)).

The standard proof optimizes certain exponential function of the number of events occurring and applies Markov's inequality.

We propose a different approach: one can use Bayes' rule to bound Pr[S(n,k)] with exp(-k/n) * Pr[S(n-1,k-1)] and apply the induction to finish the proof.

We will present some generalisations and applications of this technique. In particular, we will show how the independence assumption can be relaxed while still obtaining a good bound. We will also sketch how to apply the technique to the problem of counting occurences of a certain subgraph (e.g., a triangle) in a random graph.

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