[next] [prev] [prev-tail] [tail] [up]

4.3. Lower Bounds

It is important to study lower bounds in order to discover how much room for improvement exists in our upper bounds.

There are two forms of lower bounds: a lower bounds on the true error rate and lower upper bounds which lower bound the value our upper bound could return (given available information). For lower bounds essentially the upper bound theorem applies with the bound reversed. This form of lower bound allow us to construct a two-sided confidence interval on the location of the true error rate. Typically, such lower bounds can be formed by simply changing the sign in upper bound arguments.

THEOREM 4.3.1.

(Discrete Hypothesis Lower Bound) For all hypothesis spaces, H, for all δ ∈ (0,1] PrDm ∃h ∈ H : e(h) ≤ē m,ê(h), δ ∣H∣≤ δ where ē m, k m,δ ≡ minp{p : Bin(m,k,p) = δ}

PROOF. For every individual hypothesis, we know that: PrDm(e(h) ≤ē(m,ê(h),δ)) ≤ δ Applying the union bound for every hypothesis gives us the result. ▫

[next] [prev] [prev-tail] [front] [up]