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I tried to build in R a function that calculates, for a given vector of features, the value of the quadratic discriminant function for a specific class;

$$\delta_{k}(x)=-\frac{1}{2} \log \left|\Sigma_{k}\right|-\frac{1}{2}\left(x-\mu_{k}\right)^{T} \Sigma_{k}^{-1}\left(x-\mu_{k}\right)+\log \pi_{k}$$

By using the R function qda() i can get, for each observation, the posterior class probability for each class. My question is; how do i get from the quadratic discriminant function to the corresponding posterior class probability?

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    $\begingroup$ In discriminatory classification, you decide for the class $k$ with the highest discriminant $\delta_k(x)$. You can thus always transform the discriminant value into a continuous confidence value $\in[0,1]$ by division with $\sum_k \delta_k(x)$. $\endgroup$
    – cdalitz
    Dec 29, 2021 at 16:17

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When logistic regression was developed by DR Cox in 1957 or 1958 it provided a direct probability model not needing to assume distributions for X such as the multivariate normal assumption needed for discriminant analysis. And if multivariate normality with equal variances across groups holds, Bayes' rule turns the discriminant model into a logistic model. If unequal variances/covariances are allowed for, you get quadratic logistic regression instead of simple linear (in the logit) logistic regression. So discriminant analysis has been obsolete for many years in my view. It requires more assumptions, is less direct, and doesn't buy you anything except a tiny increase in efficiency if multivariate normality happened to hold. And logistic regression has been extended to multinomial and to ordinal Y.

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