Hi, could you elaborate on the use of $\approx$ in your answer?

Statistics is a mathematical subject which deals with randomness. It can be applied in an algorithm (to program a machine) to infer, for example, parameter values. That is what "machine learning" means, at least as how I interpret it. The discussion becomes unnecessarily hairy otherwise.

You can hardly call it machine learning if you don't use a machine. It is the machine that learns, after all. And I have actually deployed models that "learned" their parameters by a random (Monte Carlo) process. However, I must admit that there was a validation step involved afterwards.

@Alex, all points in the right hand side plot "$\textrm{pmf}_{0}(\lambda)$" correspond to a different pmf (with different intensity $\lambda$). But they are all evaluated at an event count of $0$. They are my estimate of probability that the underlying generating process actually has that specific intensity $\lambda$, since my observation of the event count was $0$.

@Glen_b: Done, thanks. How do you think a frequentist would approach this?

@RubenvanBergen: I understand what you say and it is good and useful to point that out to user10882. But I still argue that it is ultimately a technicality. Say you use a gradient descent algorithm that uses the training data to infer the step direction (including the polynomial degree $n$) together with a validation procedure that adds the validation loss to the training loss in each step of the gradient descent algorithm (similar to early stopping). Now the difference between "normal" or "hyper" is not relevant any more: it depends on the procedure.