Target encoding is a technique used in machine learning and statistics to encode categorical variables as numerical values. The idea behind target encoding is to replace each categorical value with the average value of the target variable (or some other related statistic) for that category. For example, if we have a categorical variable "color" with possible values "red", "green", and "blue", and a target variable "price", we can replace each value of "color" with the average "price" for that color.

The mathematical reason for target encoding effectiveness lies in capturing information about the target variable that is not available in the original categorical variable. When we replace each category with the average value of the target variable, we are essentially using the target variable as a guide to estimate the importance of each category. If a particular category has a high average target value, it will likely be an important predictor of the target variable.

Target encoding can be especially effective when we have a large number of categories, and it is difficult to estimate the probability of each category directly from the data. By using the target variable to estimate the importance of each category, we can effectively reduce the dimensionality of the data and improve the performance of our models.

However, it is important to note that target encoding can also lead to overfitting if not done carefully. One common approach to avoid overfitting is to use some form of regularization, such as adding noise to the encoded values or using cross-validation to estimate the encoding for each fold.

Target encoding is a technique used in machine learning to encode categorical variables as numerical values based on the target variable. The idea is to use the target variable to create a new feature for each unique category in the categorical variable, where the value of the feature is the average value of the target variable for that category.

From a mathematical point of view, target encoding can be effective because it can help capture the relationship between the categorical variable and the target variable. By encoding the categorical variable based on the target variable, the resulting numerical values can provide a more informative representation of the categorical variable, which can help improve the performance of the machine learning model.

One way to think about this is in terms of information theory. The target variable provides information about the relationship between the categorical variable and the target variable. By encoding the categorical variable based on the target variable, we are effectively incorporating this information into the feature representation. This can help improve the model's ability to learn patterns and make accurate predictions.

However, target encoding can also be prone to target leakage, where the encoded feature incorporates information from the target variable that would not be available at prediction time. This can lead to overfitting and poor generalization performance. To mitigate this issue, it is important to use proper cross-validation techniques and to ensure that the encoding is done using only information that would be available at prediction time.

Here are a few resources that you may find helpful:

• "A Preprocessing Scheme for High-Cardinality Categorical Attributes in Classification and Prediction Problems" by Daniele Micci-Barreca (https://dl.acm.org/doi/10.5555/507538.507542)
• "Target Encoding for Categorical Variables: A Comprehensive Guide" by Kamil Mysiak (https://towardsdatascience.com/target-encoding-for-categorical-variables-a-comprehensive-guide-1b138fb5c57f)
• "Target Encoding: What it is and How to Do it (Right)" by Will Koehrsen (https://towardsdatascience.com/target-encoding-22b919b4dcf6)