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Currently, I have trained my model through 5-fold cross validation with very small amount of the sample (n=240).

I used whole data set to train and got quite low performance in terms of accuracy, which is bit higher than 70%.

However, if I put my data which was used for training back to trained model to validate, it gives me higher accuracy (80%).

So, my question is it okay to say that I have verified my trained model using training set and got 80 of accuracy? or should I have to stick with 70 % of accuracy that I received from 5-fold cross validation?

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Scoring your model using the training data is not a best practice. The reason is that you have already used this data to develop the "guess" at an accurate model.

This is akin to saying - 'Pick a random number between 1 and 10 (20 times)', here's a list of the last 10 numbers I picked to get you started. You pick 20 new numbers and come up with 5 correct (25%). What you are suggesting is that instead of reporting the 25% accuracy, that you would throw the first 10 back into the mix and report a 50% accuracy because now you've gotten 15 out of 30 right.

So there are two different things here. The MSE of your initial model against the data you gave it - provides you and understanding of how well the model demonstrates history. While the MSE (or other error metric) of the 'test' data shows how well you can predict the future.

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