Custom threshold in logistic regression and interpretation of probabilities

Let's say I choose threshold in logistic regression equals to $$0.8$$. If it is lower then class is $$0$$ else is $$1$$. Then, how do I interpret the outcome on test point $$x$$, $$h(x)=0.6$$, where $$h$$ is my logistic regression model. I think I cannot say that with probability equals $$0.6$$ class of $$x$$ is $$1$$? Is there any way to interpret this? Should I just scale appropriate intervals so in that case the outcome $$0.6$$ I can interpret as probability of $$0.375$$ that point is in class $$1$$?

• Since logistic regression does not use any threshold by itself & using threshold for making hard classifications has no impact on the probabilities predicted by logistic regression, what exactly do you mean? – Tim Jun 15 at 16:09
• Yes. I understand that if I use threshold $0.5$ (usually it is default) then everything can be interpreted nicely. But how do I interpret things when I do not use $0.5$? – amad Jun 15 at 16:28

Logistic regression results can be expressed in terms of the probability of class membership. I take your terminology $$h(x) = 0.6$$ to mean that your model $$h$$ predicts that case $$x$$ has probability of 0.6 of belonging to Class 1. That's the probability estimate from the model. It has nothing to do directly with a probability cutoff.
So if a case has $$h(x) = 0.6$$ that still means it has a 60% chance of actually being in Class 1. You just will have made a decision not to assign it to Class 1 for your purposes because you don't want to risk that it really is in Class 0.
• Thanks. If I choose threshold $0.6$ because it minimizes error and gives the best predictions then it means that my model is not correct? Am I right that on the training set error is minimized for the threshold $0.5$ and changing threshold may minimize error only on the test set? – amad Jun 15 at 16:33