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In bias variance trade off graph

Bias is the difference between actual and predicted value in training data set so train error (dotted red curve) and bias(red curve ) looks same

Variance is the difference between actual and predicted value in test data set i was expecting the test error (dotted blue curve) and variance (blue curve) to be same but they are not same ? Please help to understand this part

Analysis done : 1.Model poorly fits data set then we have high bias and high training error this looks reasonable but it also has low variance with high test error

if a model is not able to capture the pattern in the dataset it would not be able to predict correct value on which it was trained on( train set ) so high training error = high bias

so it will also not be able to predict the correct value for test set right then why is that all graph in internet show low variance for low complex model in bias variance trad off graph high test error == low variance ? ? why high test error =! high variance ?

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Variance is the difference between actual and predicted value in test data set

No it's not. It's the sensitivity of the model to the small changes in data. Changes happen in the test data because the model already knows about the training data. So, high variance means high test error.

If the model has high bias, both training and test performance will be poor. For example, a model that always outputs a constant prediction has high bias (in may problems), and low variance, zero variance to be precise. One wouldn't expect low test error in this case.

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  • $\begingroup$ thanks , can you please elaborate you last point about zero variance , "model that always outputs a constant prediction has high bias and and low variance, zero variance to be precise." my query is since there its going to have high test error then it should also have high variance right , You have also mentioned in first part of answered that "high variance means high test error" . $\endgroup$
    – star
    Commented Sep 5, 2021 at 17:48
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    $\begingroup$ High variance leads to high test error, but high test error doesn't mean there is high variance. This is not an equality relation. If a model always outputs constant predictions, then there is no variance in it, it doesn't vary. So, changes in the input will produce no variance in the output. $\endgroup$
    – gunes
    Commented Sep 5, 2021 at 18:53
  • $\begingroup$ @gunes If I may ask an additional question, for the unlikely scenario, that we have a high training error, but a lower test error, that would still be considered high variance? Since the model seems to be sensitive to changes in the data (here a change from training to test data) $\endgroup$
    – kklaw
    Commented Feb 20, 2023 at 18:10

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