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I am reading John Langford's paper on practical prediction theory (link), and I have the following doubt with definition of Binomial Tail inversion.

The paper says that binomial tail inversion is the largest true error such that the probability of having k or more errors is at least $\delta$.

But the expression used in the paper reads "largest true error such that probability of having k or less errors is at least $\delta$"

Can anyone help?

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I think this is a typo.

In the Definition 2.1 and Definition 2.2, the estimation of $c_D$ is done by empirical summation of $\mathbf{1}(c(x_i) = y_i)$.

Later in the Definition 3.1 it used the following notation : $$ ... = \Pr_{...} \left( \sum_i Z_i < k \right) = ... $$ In this notation $Z_i=1$ corresponds to correct decisions or $c(x_i) = y_i$, and $Z_i=0$ corresponds to the mistakes or $c(x_i) \neq y_i$. I suppose that in this binomial representation $Z_i = 1$ (correct decision) corresponds to heads" (And similarly $Z_i = 0$ (wrong decision) corresponds totail").

That said, the right sentence seems to be "... having k or less errors is at least $\delta$"

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