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Let us assume that we have naturally ordered data that we want to classify. Then we can use ordinal regression/classification methods. Yet we can treat those as unordered and use multiclass clasiffication. It seems from first glance like the ordinal case modelling is less studied. Could someone lay down the gains that one gets from going from unordered to ordinal?

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As far as I know there is no standard representation for discrete ordinal variables for ML. To my knowledge the choice is between numerical (real numbers) and categorical (e.g. one hot encoding) representation, so usually if the variable is ordinal then it would be represented as a number because this is the only way to preserve order.

If the target variable is ordinal, then it would usually be represented as a number and therefore the task is not classification (only for categorical target) but regression.

The advantage of doing regression over classification is that the model can use the order information. For example if the target values are 1, 2, 3, 4, 5, the model is less likely to predict 1 instead of 5 (large difference) than 4 instead of 5 (small difference). In classification, all the classes are at the same level: an error between 1 and 5 is the same as an error between 4 and 5.

Note that a regression model predict real values, for example 3.79. If discrete values are required, it makes sense to apply a filter on the predicted value, e.g. if the value is between 3.5 and 4.5 then predict as 4.

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It is best practice to represent data with the highest level of measurement possible. Data that are measured on an ordinal scale and stay on an ordinal scale retain the most information. If the data are measured on an ordinal scale and then transformed to an unordinal scale, information is lost. Often that information can not be regained. If ordinal regression/classification is appropriate, then a more precise model can be fit to better model patterns in the data.

The increased precision extends from the model to evaluation metrics. In unordered predictions, all errors are often treated as equivalent. If the prediction is ordered, off-by-n misprediction can be calculated. Off-by-n misprediction is a common evaluation metric in learning-to-rank.

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