# Second-guessing Experts

Second-guessing experts work in the framework Prediction with expert advice. Each expert
announces a continuous function of the Forecaster's predictions.
Therefore, the loss of each expert is determined
by the Forecaster’s prediction as well as by the outcome chosen by Reality.
Second-guessing experts are a generalization of experts in the standard Protocol: a standard
expert can be interpreted as a constant function. The protocol of the game is as follows.
**Protocol**:

{$\quad$}Initialize {$L_0=0, L_0^k=0, k=1,2,\dots$}

{$\quad$}FOR {$t=1,2,\dots$}:

{$\qquad$}Experts announce {$\xi^k\in\Gamma\to\Gamma, k=1,2,\dots$}

{$\qquad$}Forecaster announces {$\gamma\in\Gamma$}

{$\qquad$}Reality announces {$\omega\in\Omega$}

{$\qquad$}Update cumulative loss {$L_t=L_{t-1} + \lambda(\omega,\gamma)$}, {$L_t(k)=L_{t-1}(k) + \lambda(\omega,\xi^k()), k=1,2,\dots$}

{$\quad$}END

As it is proven by Chernov et.al, 2010, it is possible to modify the Aggregating Algorithm or use directly the defensive forecasting technique to prove the same bounds on the learner's loss as in the standard protocol.

### Bibliography

- Alexey Chernov, Yuri Kalnishkan, Fedor Zhdanov and Vladimir Vovk. Supermartingales in Prediction with Expert Advice,
*Theoretical Computer Science*,**411**, pp.2647-2669, 2010.