Optimal classification rule given data, model and loss function

Question

Setup
Suppose I have a data set with a categorical variable $Y$ (with possible values $j=1,\dots,J$) and another variable $X$. I wish to classify $Y$ based on the information in $X$.

For simplicity, suppose I also know the shape of the model that generated the data, though I do not know the parameter values; I will estimate them. E.g., this could be a (multinomial) logistic regression.

I am facing a loss function $l(\hat y,y)$ where $\hat y$ is the predicted class and $y$ is the actual/true class. $l$ can be represented by a matrix $\mathbf{L}$ where rows correspond to actual classes and columns to predicted ones. Each off-diagonal cell $l_{ij}$ ($i\neq j$) of $\mathbf{L}$ contains the loss associated with the specific misclassification.

Steps taken
I fit a (multinomial) logistic regression to the data. Given the fitted coefficients and a new data point with a known $X$ value $x_0$ but an unknown $Y$ value $y_0$, I obtain the fitted class probabilities $\hat p_0$ (a vector). I wish to classify $y_0$, i.e. obtain $\hat y_0$ so as to minimize the expected loss.

Questions
What is the optimal classification rule in this setting?
Could you also recommend a textbook chapter on the topic?

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My problem is somewhat similar to Classification optimal decisions considering a loss function but my setting is frequentist and I do not have the prior distribution of classes (or class prevalence) available.

Answer 1

I have considered the following decision rule:

1. Calculate the vector of "estimated expected loss" (denote it by $\widehat{el}(\mathbf{L},\hat p_0)$) consisting of elements corresponding to classification decisions $j=1,\dots,J$ (assigning class $j$ to $\hat y$) as $\widehat{el}(\mathbf{L},\hat p_0):=\mathbf{L}^\top \hat p_0$.
2. Find the minimal element of the vector and select the corresponding predicted class.

I wonder if this is optimal.