Manuscript Psy03-D@1 · Psy03-D-v1.tex · section · 3e18951d

Causally ordered feedback

Observable task designs are new protocols › Causally ordered feedback · lines 723–748

Psy03-D@1#eq:feedback

Let $π_{t}^{-}$ be the rule belief immediately before prediction at time $t$ . After predicting, reveal the target $Y_{t}$ for use on later trials. For noiseless full-label feedback,

π_{t}^{+} (r) = \frac{π _{t}^{-} ( r ) 1 { f _{r} ( X _{t} ) = Y _{t} }}{\sum _{j} π _{t}^{-} ( j ) 1 { f _{j} ( X _{t} ) = Y _{t} }},

whenever the denominator is positive. For a known transition matrix $P_{t}$ ,

π_{t + 1}^{-} = P_{t}^{T} π_{t}^{+} .

Current prediction uses

c arg max r \sum π_{t}^{-} (r) 1 {f_{r} (X_{t}) = c},

not $π_{t}^{+}$ . A known noisy feedback channel replaces the indicator likelihood by its declared likelihood.

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Psy03-D@1#eq:feedback · line 730
$π_{t}^{+} (r) = \frac{π _{t}^{-} ( r ) 1 { f _{r} ( X _{t} ) = Y _{t} }}{\sum _{j} π _{t}^{-} ( j ) 1 { f _{j} ( X _{t} ) = Y _{t} }},$
unlabeled · line 738
$π_{t + 1}^{-} = P_{t}^{T} π_{t}^{+} .$
unlabeled · line 742
$c arg max r \sum π_{t}^{-} (r) 1 {f_{r} (X_{t}) = c},$

Causally ordered feedback

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