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A Note on the Chernoff Bound for Random Variables in the Unit Interval

The Chernoff bound is a well-known tool for obtaining a high probability bound on the expectation of a Bernoulli random variable in terms of its sample average. This bound is commonly used in statistical learning theory to upper bound the generalisation risk of a hypothesis in terms of its empirical risk on held-out data, for the case of a binary-valued loss function. However, the extension of this bound to the case of random variables taking values in the unit interval is less well known in the community. In this note we provide a proof of this extension for convenience and future reference.