I point out that the $\TPR$, the $\FNR$, the $\TNR$ and the $\FPR$, which are commonly used metrics to gauge the performance of binary classifiers, are retrodictive. I then contrast them with their predictive counterparts and name these according to a coherent, self-explanatory scheme. I also provide equations to convert from one set of rates to the other, and interactive visuals that enable exploring this interdependence.

- From retrodictive to predictive
- From predictive to retrodictive

- the True Positive Rate ($\TPR$) and its complement, the False Negative Rate ($\FNR$); and
- the True Negative Rate ($\TNR$) and its complement, the False Positive Rate ($\FPR$).

$\TPR$ | : | Proportion of objects predicted to be in the positive class among objects observed to be in the positive class |

$\FNR$ | : | Proportion of objects predicted to be in the negative class among objects observed to be in the positive class |

$\TNR$ | : | Proportion of objects predicted to be in the negative class among objects observed to be in the negative class |

$\FPR$ | : | Proportion of objects predicted to be in the positive class among objects observed to be in the negative class |

And they also go by other names:

$\TPR$ | : | sensitivity, recall, hit rate |

$\FNR$ | : | miss rate |

$\TNR$ | : | specificity, selectivity |

$\FPR$ | : | fall-out, false alarm rate |

These rates actually are conditional probabilities that a prediction is correct or incorrect given an observation. As such, they provide a

$A$ | : | An object is predicted to be in the positive class. |

$B$ | : | An object is observed to be in the positive class. |

Using common notations of probability theory, the following holds. \begin{align*} \TPR &= p(A|B) \\ \FNR &= p(\bar{A}|B) \\ \TNR &= p(\bar{A}|\bar{B}) \\ \FPR &= p(A|\bar{B}) \end{align*} The following complement equations hold. \begin{equation*} \TPR + \FNR = 1 \qquad \TNR + \FPR = 1 \end{equation*}

$p(B|A)$ | : | Proportion of objects observed to be in the positive class among objects predicted to be in the positive class |

$p(\bar{B}|A)$ | : | Proportion of objects observed to be in the negative class among objects predicted to be in the positive class |

$p(\bar{B}|\bar{A})$ | : | Proportion of objects observed to be in the negative class among objects predicted to be in the negative class |

$p(B|\bar{A})$ | : | Proportion of objects observed to be in the positive class among objects predicted to be in the negative class |

I think of these probabilities as

$p(B|A)$ | $=$ | $\pTPR$ | : | Predictive True Positive Rate |

$p(\bar{B}|A)$ | $=$ | $\pFPR$ | : | Predictive False Positive Rate |

$p(\bar{B}|\bar{A})$ | $=$ | $\pTNR$ | : | Predictive True Negative Rate |

$p(B|\bar{A})$ | $=$ | $\pFNR$ | : | Predictive False Negative Rate |

The following complement equations hold. \begin{equation*} \pTPR + \pFPR = 1 \qquad \pTNR + \pFNR = 1 \end{equation*} These predictive rates already have names:

$\pTPR$ | : | Positive Predictive Value ($\PPV$); Precision |

$\pFPR$ | : | False Discovery Rate ($\FDR$) |

$\pTNR$ | : | Negative Predictive Value ($\NPV$) |

$\pFNR$ | : | False Omission Rate ($\FOR$) |

The naming scheme herein is I think more coherent and self-explanatory. The $\PPV$-$\NPV$ naming scheme is adequate but incomplete.

I denote $\alpha$ and $\beta$ are the odds of the predicted negative class and the observed negative class, respectively. \begin{equation*} \alpha = \dfrac{ 1 - p(A) }{ p(A) } \qquad \beta = \dfrac{ 1 - p(B) }{ p(B) } \end{equation*} I denote $f$ and $g$ the bivariate and trivariate functions defined as follows. \begin{equation*} f(t,r) = \dfrac{ 1 }{ 1 + t r } \qquad g(t,u,v) = \dfrac{ 1 }{ 1 + t v/u } = f(t,v/u) \end{equation*}

Equations \eqref{eq:TPR} and \eqref{eq:FPR} consolidate into equation \eqref{eq:pTPR-pFNR-to-TPR-FPR}. \begin{align} \label{eq:TPR-FPR-to-pTPR-pFNR} ( \pTPR , \pFNR ) &= h( \beta , \TPR , \FPR ) \\[2mm] \label{eq:pTPR-pFNR-to-TPR-FPR} ( \TPR , \FPR ) &= h( \alpha , \pTPR , \pFNR ) \end{align} Equations \eqref{eq:TPR-FPR-to-pTPR-pFNR} and \eqref{eq:pTPR-pFNR-to-TPR-FPR} further consolidate into the following diagram.

Equations \eqref{eq:TPR} and \eqref{eq:TNR} consolidate into equation \eqref{eq:PPV-NPV-to-TPR-TNR}. \begin{align} \label{eq:TPR-TNR-to-PPV-NPV} ( \PPV , \NPV ) &= k( \beta , \TPR , \TNR ) \\[2mm] \label{eq:PPV-NPV-to-TPR-TNR} ( \TPR , \TNR ) &= k( \alpha , \PPV , \NPV ) \end{align} Equations \eqref{eq:TPR-TNR-to-PPV-NPV} and \eqref{eq:PPV-NPV-to-TPR-TNR} further consolidate into the following diagram.

Equations \eqref{eq:ratios3} and \eqref{eq:ratios4} can be obtained from equations \eqref{eq:TPR} and \eqref{eq:FPR}, respectively. \begin{align} \label{eq:ratios1} \beta \times \dfrac{\FPR}{\TPR} &= \dfrac{\pFPR}{\pTPR} \\[2mm] \label{eq:ratios2} \beta \times \dfrac{\TNR}{\FNR} &= \dfrac{\pTNR}{\pFNR} \\[2mm] \label{eq:ratios3} \alpha \times \dfrac{\pFNR}{\pTPR} &= \dfrac{\FNR}{\TPR} \\[2mm] \label{eq:ratios4} \alpha \times \dfrac{\pTNR}{\pFPR} &= \dfrac{\TNR}{\FPR} \end{align}