# Classification vs. Prediction

Recently there has been discussion about the distinction between classification and prediction. In particular it concerns the case where outcome $$Y\sim Bern(p)$$. Usually $$p$$ depends on covariates but we can ignore that here. The claim is that a prediction must be an estimate $$\hat{p}$$ of the probability $$P(Y=1)$$, and a classification, $$\delta(\hat{p})$$, is not a prediction, but instead a decision (where $$\delta(x)=1$$ if $$x>t$$ and zero otherwise).

But because $$Y$$ is binary, shouldn’t a prediction for $$Y$$ be binary? Or, another way, an estimate of $$p$$ is an estimate of a parameter - in this case it is an estimate of the mean of the Bernoulli distribution. An estimate of $$p$$ is not an estimate of the random variable $$Y.$$ By definition prediction is a guess at random variable while estimation is a guess at a parameter. So $$\hat{p}$$ is not a prediction of $$Y,$$ it is an estimate of $$p$$, and $$\delta(\hat{p})$$ is not a classification/decision, it is a prediction.

• Your question might become clearer if accompanied by a definition of what you mean by "prediction," because this term has various meanings in statistics. – whuber Jan 20 at 18:38
• @whuber added, it is the definition you gave in the linked answer. – user0 Jan 20 at 23:49
• You don't use it in the sense I gave, because in general a parameter is not a probability. – whuber Jan 21 at 0:25
• @whuber updated - here I am referring to the Bernoulli case – user0 Feb 15 at 0:58
• Thank you. I am reminded of the colloquial sense of prediction, such as what the weatherman announces. When they say "there's a 60% chance of rain tomorrow," isn't that generally understood as a prediction? Of course if someone says "it will rain tomorrow," that's a prediction, too, but it is no longer probabilistic. Accordingly, in statistics a "prediction" is generally thought to be a characterization of an unobserved random variable based on inferred relationships with observed random variables, in contradistinction to a definite statement about a value of the unobserved variable. – whuber Feb 15 at 15:23