Manuscript Psy01-A@1 · Psy01-A.tex · section · 94477fa3

An exactly solved selectivity–fragility comparison model

Stress certificates and a conditional trade-off › An exactly solved selectivity–fragility comparison model · lines 911–952

Psy01-A@1#eq:comparisonparameters, Psy01-A@1#eq:comparisonmargin, Psy01-A@1#eq:comparisonobservables

The following model is a deliberately declared affine-Gaussian comparison. It is not a differentiable rewriting of the actual finite magnitude mask.

Let $e \in [0, 1]$ , $f \in [0, F]$ , $0 < F < 1$ , and set

b (e, f) h (f) = b_{0} + k f (e - e_{0}) - λ f^{2}, = h_{0} - r f, v (f) = \frac{v _{0}}{1 - f},

where $0 < e_{0} < 1$ and $k, λ, r, h_{0}, v_{0} > 0$ . For two conflicting rules, take the visual-versus-voice logit difference to be

L_{e, f} (α) = b (e, f) + h (f) (2 α - 1) + τ Z + σ v (f) G,

where $Z, G$ are independent standard Gaussians and $τ > 0$ . The visual label is selected when $L > 0$ .

At zero stress,

A_{lin} (e, f) = Φ (\frac{b ( e , f )}{τ}) .

For the correct visual cue,

S_{lin} (e, f, σ) = Φ (\frac{b ( e , f ) + h ( f )}{τ ^{2} + σ ^{2} v ( f )}) .

Displayed formulas in this passage

Psy01-A@1#eq:comparisonparameters · line 918
$b (e, f) h (f) = b_{0} + k f (e - e_{0}) - λ f^{2}, = h_{0} - r f, v (f) = \frac{v _{0}}{1 - f},$
Psy01-A@1#eq:comparisonmargin · line 928
$L_{e, f} (α) = b (e, f) + h (f) (2 α - 1) + τ Z + σ v (f) G,$
unlabeled · line 938
$A_{lin} (e, f) = Φ (\frac{b ( e , f )}{τ}) .$
Psy01-A@1#eq:comparisonobservables · line 943
$S_{lin} (e, f, σ) = Φ (\frac{b ( e , f ) + h ( f )}{τ ^{2} + σ ^{2} v ( f )}) .$