Notation for estimator in Murphy's Machine Learning (Chapter 6)

Question

I'm confused with the notation for estimators in Chapter 6 of Murphy's "Machine Learning: A Probabilistic Perspective":

In section 6.2, he initially refers to an estimator as $\delta$ in the beginning of the section.

...is computed by applying an estimator $\delta$ to some data D, so $\hat{\theta} = \delta(D)$.

Later in that paragraph, he refers to the estimator as the $\hat{\theta}(\cdot)$ function.

Now apply the estimator $\hat{\theta}(\cdot)$ to each D...

Then in section 6.2.1, he says the estimator is equal to a function f().

We could then compute our estimator from each sample, $\hat{\theta^s} = f(x_{1:N}^s)$ and...

My question is why are there three different notations for the estimator? Are they all the same?

Answer 1

No, there is no general difference between those notations. As whuber notices, we could speculate why the author used different notation in different places for apparently the same thing, but this would be a guessing game since it was not stated in the text.

The main distinction is between estimator and estimate and is described in this thread:

an estimator is a definite mathematical procedure that comes up with a number (the estimate) for any possible set of data that a particular problem could produce

So estimator is a function that takes data as input and returns an estimate. Initially Kevin Murphy uses $\hat \theta$ (and later $\hat{\theta^s}$) to denote the estimate and $\delta(\cdot)$ to denote estimator, but later on he uses also $\hat\theta(\cdot)$ or $f(\cdot)$, but he does not give any distinction between the functions, so it seems that in all the cases he simply means some function. This is apparent if you check the section 6.2.1 where once he writes $\hat{\theta^s} = f(x_{1:N}^s)$ so few lines later uses $\hat{\theta^s} = \hat\theta(x_{1:N}^s)$, while in both cases referring to estimates based on bootstrap samples $x_i^s$.

Notice that similar ambiguity may appear in general mathematical notation where some authors write $y(x) = \text{{function body}}$ interchangeably with $y = \text{{function body}}$ dropping the $x$ in brackets to simplify the notation. Yes, if used like this it may be confusing.