# How to determine if the predicted probabilities from sklearn logistic regresssion are accurate?

I am totally new to machine learning and I'm trying to use scikit-learn to make a simple logistic regression model with 1 input variable (X) and a binary outcome (Y). My data consists of 325 samples, with 39 successes and 286 failures. The data was split into a training and test (30%) set.

My goal is actually to obtain the predicted probabilities of success for any given X based on my data, not for classification prediction per se. That is, I will be taking the predicted probabilities for use in a separate model I'm building and won't be using the logistic regression as a classifier at all. So it's important that the predicted probabilities actually fit the data.

However, I am having some trouble understanding whether or not my model is a good fit to the data, or if the computed probabilities are actually accurate.

I am getting the following metrics:

• Classification accuracy: metrics.accuracy_score(Y_test, predicted) = 0.92. My understanding of this metric is that the model has a high chance of making correct predictions, so it looks to me like the model is a good fit.

• Log loss: cross_val_score(LogisticRegression(), X, Y, scoring='neg_log_loss', cv=10) = -0.26 This is probably the most confusing metric for me, and apparently the most important as it is the accuracy of the predicted probabilities. I know that the closer to zero the score is the better - but how close is close enough?

• AUC: metrics.roc_auc_score(Y_test, probs[:, 1]) = 0.9. Again, this looks good, since the closer the ROC score is to 1 the better.

• Confusion Matrix: metrics.confusion_matrix(Y_test, predicted) =

        [  88,  0]
[8,  2]


My understanding here is that the diagonal gives the numbers of correct predictions in the training set so this looks ok.

• Report: metrics.classification_report(Y_test, predicted) =

            precision    recall  f1-score   support

0.0       0.92      1.00      0.96        88
1.0       1.00      0.20      0.33        10

avg / total       0.93      0.92      0.89        98


According to this classification report, the model has good precision so it is a good fit. I am not sure how to interpret the recall or if this report is bad news for my model- the sklearn documentation states that the recall is a models ability to find all positive samples - so a score of 0.2 for a prediction of 1 would mean that it only finds the positives 20% of the time? That sounds like a really bad fit to the data.

I'd really appreciate if someone could clarify that I am interpeting these metrics the right way - and perhaps shed some light on whether my model is good or bogus. Also, if there are any other tests I could do to determine if the computed probabilities are accurate please let me know.

If these aren't good metric scores, I'd really appreciate some direction on where to go next in terms of improvement.

Thanks!!

## 1 Answer

Sklearn has a module for calibration curves and also Brier score. Both of these address the issue of probability accuracy. A calibration curve is a scatter plot where one axis is predicted probability and the other is the true probability. This is computed by taking, e.g., 100 cases, Q=40 of which are truly positive. The actaul probability of these 100 being positive is 0.4. We put each of these cases through the fit model and produce a probability estimate for each. Ideally, if we averaged these 100 estimates we would get 0.4. With a calibration curve, just repeat this process for different Qs and you should get a diagonal line. Brier score is MSE wrt this line.

You are correct that Log loss is difficult to interpret and also related.

Precision/recall are not so useful for this. They depend on a threshold and are traded off.

• Thanks for that, it is exactly the kind of help I'm looking for! I have another silly question though: According to my above output, the accuracy of predicting the outcome is not all that good (I am presuming because if there is, say, a probability of 0.3 to predict success for a given X the classification would predict a failure every time even in cases when it should be a success). Is it possible that a logistic regression model can assign accurate probabilities but still fail at accurate prediction? – none Sep 25 '17 at 0:30
• The classification depends on a threshold (commonly 0.5) such that if P(success)>=0.1, classify as 1 and classify as 0 otherwise. You can change the threshold--say to 0.0001. In this case, if P(success)>=0.0001 classify as 1 and 0 otherwise. You can imagine that you will get many more predictions of success this way than if you had used a 0.5 threshold. With the threshold at 0.0001, accuracy might not be as high as it is with the threshold at 0.5. So accuracy depends on the threshold and therefore could be bad even while the model is giving accurate probabilities. – user0 Sep 25 '17 at 1:39
• Sometimes the probabilities are also accurate in a certain range, even if the overall accuracy is not great. A model with globally very good predicted probabilities must give good accuracy. So to answer your question I don't think a model that gives accurate probabilities will fail at accurate prediction as long as you choose a good threshold, but the converse doesn't hold. A model with very high accuracy could give very bad probabilities (some models like SVM do this by nature). – user0 Sep 25 '17 at 1:42