Calculating the Brier or log score from the confusion matrix, or from accuracy, sensitivity, specificity, F1 score etc Suppose I have a confusion matrix, or alternatively any one or more of accuracy, sensitivity, specificity, recall, F1 score or friends for a binary classification problem.
How can I calculate the Brier or log score of my classifications from this information?

A point raised in some related conversations that seem to have inspired the posting of this question is that a confusion matrix can contain probability values if you normalize by the sum of the entries.
$$
\begin{pmatrix}
4 & 2\\
1 & 3
\end{pmatrix}
\overset{\div 10}{\rightarrow}
\begin{pmatrix}
0.4 & 0.2\\
0.1 & 0.3
\end{pmatrix}\\
\begin{pmatrix}
8 & 2\\
3 & 7
\end{pmatrix}
\overset{\div 20}{\rightarrow}
\begin{pmatrix}
0.4 & 0.1\\
0.15 & 0.35
\end{pmatrix}
$$
However, notice that each of these matrices contains four values, while they correspond to ten and twenty classification attempts, respectively. This means there is a mismatch between the number of true $y_i$ and predicted $\hat p_i$ when you go to calculate something like Brier score or log-loss (even c-index or ROCAUC) that sums over the number of observations (ten and twenty in the two respective matrices). Consequently, even this normalization of the confusion matrices does not allow for calculations of the Brier score or log loss, giving further evidence that the confusion matrix is not the whole story. Since the accuracy, sensitivity, specificity, $F_1$, $F_{\beta}$, precision, recall, FPR, FNR, TPR, TNR, PPV, NPV, Matthews correlation coefficient, and all of the other friends mentioned here are derived from the confusion matrix, those would not tell the whole story, either.
