Attempts and failures are information

09 Dec, 2025

Note to self on learning and taking actions. Will rewrite it when I understand it deeper, one day.

Attempt process.

\begin{matrix} X_{t} & \in {0, 1}, t = 1, 2, \dots \\ ℙ (X_{t} = 1 ∣ p) & = p, \\ ℙ (X_{t} = 0 ∣ p) & = 1 - p, \\ X_{1}, X_{2}, \dots & i.i.d. \end{matrix}

Aggregates after $n$ attempts.

\begin{matrix} S_{n} & = \sum_{t = 1}^{n} X_{t}, \\ F_{n} & = n - S_{n}, \\ {\hat{p}}_{n} & = \frac{S_{n}}{n} . \end{matrix}

Probability of at least one success in $n$ attempts.

\begin{matrix} P_{n} (p) & = ℙ (S_{n} \geq 1 ∣ p) = 1 - (1 - p)^{n} . \end{matrix}

Bayesian update.

\begin{matrix} p & ~ B e t a (α_{0}, β_{0}), \\ p ∣ X_{1 : n} & ~ B e t a (α_{0} + S_{n}, β_{0} + F_{n}), \\ {\tilde{p}}_{n} & : = 𝔼 [p ∣ X_{1 : n}] = \frac{α_{0} + S_{n}}{α_{0} + β_{0} + n} . \end{matrix}

Target planning.

\begin{matrix} q & \in (0, 1), \\ n^{*} (p, q) & : = min {n \in ℕ : 1 - (1 - p)^{n} \geq q} \\ = ⌈ \frac{\log (1 - q)}{\log (1 - p)} ⌉ . \end{matrix}

Plug-in rule for next block size.

\begin{matrix} {\tilde{p}}_{n} & \in {{\hat{p}}_{n}, 𝔼 [p ∣ X_{1 : n}]}, \\ n_{n + 1}^{plan} & = n^{*} ({\tilde{p}}_{n}, q) . \end{matrix}

Change-of-strategy indicator.

\begin{matrix} p_{min} & \in (0, 1), \\ I_{n} & : = 1 {{\tilde{p}}_{n} < p_{min}} . \end{matrix}