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Is it correct to say that, in the Bias-Variance trade-off:

the bias error represents the ability of the model to possibly map the trend of the training set ?

and the variance represents how much the model is prone to variations if the dataset changes ?

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I would say that both the definitions you gave above refer to variance.

Variance is an index of model complexity. Thus:

high variance -> high complexity -> many degrees of freedom in model -> it overfits training data -> if dataset changes, accuracy degrades a lot

Bias is an index of the systematic error of the model, as it shows the difference between the expected value of various model's predictions for a given target output and the ground truth.

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Bias are the simplifying assumptions made by a model to make the target function easier to learn.

Variance is the amount that the estimate of the target function will change if different training data was used.

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