Reverse engineering of metacognition

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Tables

Appendix 2—table 1

	Noisy-readout	Noisy-report
Censored Normal	${\displaystyle f_{\mathrm{m}}\left(z\right)=\{\begin{array}{ll}\mathrm{\Phi }\left(-\frac{z^{\ast }}{{\sigma }_{\mathrm{m}}}\right)if& z=0\\ \frac{1}{{\sigma }_{\mathrm{m}}}\varphi \left(-\frac{z-z^{\ast }}{{\sigma }_{\mathrm{m}}}\right)if& z>0\end{array}}$	${\displaystyle f_{\mathrm{m}}\left(c\right)=\{\begin{array}{ll}\mathrm{\Phi }\left(-\frac{c^{\ast }}{{\sigma }_{\mathrm{m}}}\right)if& c=0\\ \frac{1}{{\sigma }_{\mathrm{m}}}\varphi \left(-\frac{c-c^{\ast }}{{\sigma }_{\mathrm{m}}}\right)if& 0<c<1\\ \mathrm{\Phi }\left(\frac{1-c^{\ast }}{{\sigma }_{\mathrm{m}}}\right)if& c=1\end{array}}$
Censored Gumbel	${\displaystyle f_{\mathrm{m}}\left(z\right)=\{\begin{array}{ll}1-exp\left(-e^{{\eta }_{\mathrm{m}}{z}^{\ast }}\right)if& z=0\\ {\eta }_{\mathrm{m}}exp\left({\eta }_{\mathrm{m}}\left(z-z^{\ast }\right)-e^{{\eta }_{\mathrm{m}}\left(z-z^{\ast }\right)}\right)if& z>0\end{array}}$	${\displaystyle f_{\mathrm{m}}\left(c\right)=\{\begin{array}{ll}1-exp\left(-e^{{\eta }_{\mathrm{m}}{c}^{\ast }}\right)if& c=0\\ {\eta }_{\mathrm{m}}exp\left({\eta }_{\mathrm{m}}\left(c-c^{\ast }\right)-e^{{\eta }_{\mathrm{m}}\left(c-c^{\ast }\right)}\right)if& 0<c<1\\ exp\left(-e^{{\eta }_{\mathrm{m}}\left(c^{\ast }-1\right)}\right)if& c=1\end{array}}$
Truncated Normal	${\displaystyle f_{\mathrm{m}}\left(z\right)=\frac{1}{{\sigma }_{\mathrm{m}}}\frac{\varphi \left(-\frac{z-z^{\ast }}{{\sigma }_{\mathrm{m}}}\right)}{1-\mathrm{\Phi }\left(-\frac{z^{\ast }}{{\sigma }_{\mathrm{m}}}\right)}}$	${\displaystyle f_{\mathrm{m}}\left(c\right)=\frac{1}{{\sigma }_{\mathrm{m}}}\frac{\varphi \left(-\frac{c-c^{\ast }}{{\sigma }_{\mathrm{m}}}\right)}{\mathrm{\Phi }\left(\frac{1-c^{\ast }}{{\sigma }_{\mathrm{m}}}\right)-\mathrm{\Phi }\left(-\frac{c^{\ast }}{{\sigma }_{\mathrm{m}}}\right)}}$
Truncated Gumbel	${\displaystyle f_{\mathrm{m}}\left(z\right)={\eta }_{\mathrm{m}}\frac{\mathrm{e}\mathrm{x}\mathrm{p}\left({\eta }_{\mathrm{m}}\left(z-z^{\ast }\right)-e^{{\eta }_{\mathrm{m}}\left(z-z^{\ast }\right)}\right)}{\mathrm{e}\mathrm{x}\mathrm{p}\left(-e^{{\eta }_{\mathrm{m}}{z}^{\ast }}\right)}}$	${\displaystyle f_{\mathrm{m}}\left(c\right)={\eta }_{\mathrm{m}}\frac{\mathrm{e}\mathrm{x}\mathrm{p}\left({\eta }_{\mathrm{m}}\left(c-c^{\ast }\right)-e^{{\eta }_{\mathrm{m}}\left(c-c^{\ast }\right)}\right)}{\mathrm{e}\mathrm{x}\mathrm{p}\left(-e^{{\eta }_{\mathrm{m}}{c}^{\ast }}\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(-e^{{\eta }_{\mathrm{m}}\left(c^{\ast }-1\right)}\right)}}$
Gamma/ Beta	${\displaystyle f_{\mathrm{m}}\left(z\right)=\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}{z}^{\alpha -1}{e}^{-\beta z}}$ Parameterization: $\alpha ={z^{*}}^2+2{\sigma }_{\mathrm{m}}^2+z^{*}\sqrt{z^{*}+4{\sigma }_{\mathrm{m}}^2}$ ${\displaystyle \beta =\frac{1}{2{\sigma }_{\mathrm{m}}^2}\left(z^{\ast }+\sqrt{z^{\ast }+4{\sigma }_{\mathrm{m}}^2}\right)}$	${\displaystyle f_{\mathrm{m}}\left(c\right)=\frac{c^{\alpha -1}{\left(1-c\right)}^{\beta -1}}{\mathrm{B}\left(\alpha ,\beta \right)}}$ Parameterization: ${\displaystyle \alpha =c^{\ast }\left(\frac{1}{{\sigma }_{\mathrm{m}}}-2\right)+1}$ ${\displaystyle \beta =\left(1-c^{\ast }\right)\left(\frac{1}{{\sigma }_{\mathrm{m}}}-2\right)+1}$
Lognormal	${\displaystyle f_{\mathrm{m}}\left(z\right)=\frac{1}{{\hat{\sigma }}_m^{\ast }\sqrt{\pi }}exp{\left(-\frac{Inz-{\hat{z}}^{\ast }}{{\hat{\sigma }}_m^{\ast }}\right)}^2}$ Note: ${\displaystyle {\hat{z}}^{\ast }}$ and ${\displaystyle {\hat{\sigma }}_m^{\ast }}$ represent an analytic parameterization such that the lognormal distribution has mode z* and standard deviation σm. See the published code for details.

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