\}\)</span>. That is, based on observing <span class="math inline">\(Y_{ij}\)</span>, we get probabilities for all <span class="math inline">\(K\)</span> categories. A proper scoring rule is rule that is maximized when the true probabilities are used in its calculation. The best-known proper scoring rules, together with their score ranges are: <span class="math display">\[\begin{array}{rl}
\mbox{logarithmic:} &amp; R(\pi_{ij}, \{Y_{ij} = k\}) = \log(\pi_{ijk}), \quad (-\infty, 0],\\\\
\mbox{quadratic:} &amp