Notation is confusing me (sufficiency principle)

Question

This example (from a book) illustrates the sufficiency principle and I will write it down so that you get some context :

Let $\boldsymbol{X} = (X_1,X_2..,X_n)$ be a sample of independent Bernoulli variables with probability function (pdf):

$$p(\boldsymbol{x};\theta) = \prod\limits_{i=1}^{i=n}\theta^{x_i}(1-\theta)^{1-x_i} = \theta^{k}(1-\theta)^{n-k}$$ where $k = \sum_{i=1}^{n}x_i\,.$

Consider the statistic$$T = \sum_{i=1}^{n} X_i \sim \mathcal{Bin}(n,\theta)$$with $P^T(k;\theta)=\binom{n}{k}\theta^k(1-\theta)^{n-k} $.

EDIT: The statistic $T$ generates a partition of the sample space $\mathcal{X} = \bigcup_{k=0}^{n} \mathcal{X}_k$ with$$\mathcal{X}_k = \left\{(x_1,x_2...,x_n):k = \sum_{i=1}^{n} x_i \right\}$$

EDIT:The conditional distribution of $\boldsymbol{X}$ given $T=k$ is a distribution with support $\mathcal{X}_k$. Thus for $x \in \mathcal{X}_k = \{x': \sum_i x_i' =k \}$

$$\dfrac{P_{\theta}\{ \{\boldsymbol{x}\} \cap \mathcal{X}_k \}}{P_{\theta}^T(\{k \})}= \dfrac{p(\boldsymbol{x};\theta)}{p^T(k;\theta)} = \dfrac{\theta^{k}(1-\theta)^{n-k}}{\binom{n}{k}\theta^k(1-\theta)^{n-k}}\qquad\boldsymbol{(1)}$$

However, the notation in $\boldsymbol{(1)}$ is a bit confusing. I am trying to compare to what I would have written it as: (is this right?)

$$\dfrac{P_{\theta} \{\boldsymbol{X} = \boldsymbol{x} \cap T (\boldsymbol{X})=k \} }{P\{ T(\boldsymbol{X})= k \} } = \dfrac{p(\boldsymbol{x};\theta)}{p^T(k;\theta)} = \dfrac{\theta^{k}(1-\theta)^{n-k}}{\binom{n}{k}\theta^k(1-\theta)^{n-k}}\qquad\boldsymbol{(2)}$$

Anyhow why does the author use the set notation $\{\boldsymbol{x}\} $ ? I have the same question for $\{k\} $ and also about $\{ \boldsymbol{x} \} \cap \mathcal{X}_k $ and these things are embedded in :

$$\dfrac{P_{\theta}\{ \{\boldsymbol{x}\} \cap \mathcal{X}_k \}}{P_{\theta}^T(\{k \})}$$

Answer 1

Both forms of notation are informal shorthands for more rigorous ones.

A random variable $X$ is a (measurable) function defined on a sample space $\Omega$, with values in a real vector space $V$, endowed with a probability measure $\mathbb{P}$ (on a sigma-algebra $\mathfrak{F}$),

$$X:\Omega\to V.$$

We may ignore all details of the parenthesized conditions about measurability and a sigma-algebra, but they should remind us that probabilities are numbers assigned to events, which are subsets of $\Omega$, not outcomes, which are elements of $\Omega$.

In the context of the question $\Omega$ has not been specified but would, in practice, be the set of all possible outcomes of a sequence of $n$ binary experiments. The process of encoding those outcomes with zeros and ones, with the intention of performing arithmetic operations on those values, is what creates the random variable $X$. Although its values are in the strictest sense only elements of $\{0,1\}^n = \{(x_1,x_2,\ldots, x_n)\,|\, x_i\in\{0,1\}\}$, in order to make sense of sums, averages, and other arithmetic combinations of them we automatically--with no further announcement--consider them to be elements of $\mathbb{R}^n = \{(x_1,x_2,\ldots, x_n)\,|\, x_i\in\mathbb{R}\}$.

Notice the use of conventional mathematical set-builder notation. This abbreviates more formal descriptions of sets, which themselves are permitted by virtue of the Axiom schema of specification. This axiom schema is essentially a set of templates that assert the meaningfulness of defining a set as consisting of elements of a given set that satisfy some additional logical condition. The corresponding notation is of the form

$$\{x\in A\,|\, P(x)\}$$

where $A$ is a set and $P$ is a logical predicate (that is, will be definitely true or false for any element of $A$). Although $P$ can have any logical form, in the instances below it typically is an assertion that something is an element of some set; that is, "$P(x)$" is in the form "$x\in B$", or else it is an assertion that some function attains a specified value; that is, "$P(x)$" is in the form "$f(x) = y$" for a given function $f$ and possible value $y$.

Continuing with the question, we can identify the statistic $T$ as a real-valued function

$$T:\mathbb{R}^n\to\mathbb{R}$$

Consequently, via functional composition and an abuse of notation, "$T$" is also used to represent the random variable indicated in this schematic:

$$T\circ X:\Omega\to\mathbb{R}^n\to\mathbb{R};\\T\circ X(\omega) = T(X(\omega)).$$

We are now in a position to translate the informal language of the question into these more rigorous terms.

1. "$T$ generates a partition of the sample space... ." For each $k\in\mathbb{R}$, define

   $$\mathcal{X}_k = \left(T\circ X\right)^{-1}(k) = \{\omega\in\Omega\,|\, T\circ X(\omega) = k\}.$$

   This is a subset of $\Omega$. It is an event because $T$ is (obviously) measurable, whence $T\circ X$ is measurable.

   In words: $\mathcal{X}_k$ is the set of outcomes (event) for which the statistic $T$ has the value $k$.

2. "$\{\boldsymbol{x}\} \cap \mathcal{X}_k$" clearly is intended as a set intersection. We know what the second set is. The first is a very informal shorthand for

   $$``\{\boldsymbol{x}\}" = X^{-1}(\{\boldsymbol{x}\}) = \{\omega\in\Omega\,|\, X(\omega)\in\{\boldsymbol{x}\} \} = \{\omega\in\Omega\,|\, X(\omega)=\boldsymbol{x}\}.$$

   Notice the two different meanings of "$\{\boldsymbol{x}\}$"! The correct, mathematically conventional meaning used in the middle is that of a (singleton) subset of the vector space $\mathbb{R}^n$. The intended meaning of the shorthand on the left is that of an event in $\Omega$.

   Since $\{\boldsymbol{x}\} \subset \mathbb{R}^n$ is measurable, $\{\boldsymbol{x}\}\subset \Omega$ is an event. (How's that for a confusing statement?!)

   In words: $\{\boldsymbol{x}\}$ is the set of outcomes (event) where $X$ has the particular value $X_1=x_1, X_2=x_2, \ldots, X_n=x_n$.

3. "$\boldsymbol{X} = \boldsymbol{x} \cap T (\boldsymbol{X})=k$" is a little clearer, being a bit more explicit. It would be interpreted as the intersection of

   $$``\boldsymbol{X} = \boldsymbol{x}" = \{\omega\in\Omega\,|\, X(\omega)=\boldsymbol{x}\}$$

   and

   $$``T (\boldsymbol{X})=k" = \{\omega\in\Omega\,|\, T(X(\omega)) = k\}.$$

   In words: These are the same two events described in (2) and (1) above, respectively.

4. "$P_{\theta}^T(\{k \})$" contains a lot of baggage, but the key part is the interpretation of "$\{k\}$" in the context of the superscript "$^T$". This is intended to refer to the event

   $$``\{k\}" = \left(T\circ X\right)^{-1}(k) = \{\omega\in\Omega\,|\,T\circ X(\omega) = k\}.$$

   In words: $P_{\theta}^T(\{k \})$ is the probability (according to some probability measure $\mathbb{P}_\theta$ as defined on $(\Omega, \mathfrak{F})$) that $T$ will have the value $k$.

The informal rule that seems to be operating in (2) and (4) is to abbreviate expressions of the form "$\{\omega\in\Omega\,|\, Z(\omega)= y\}$", where $Z$ is a random variable and $y\in\mathbb{R}$ or $y\in\mathbb{R}^n$, in favor of some notation that mentions only "$Z$" and "$y$".

This is enough of a concordance to decipher the quotations in the question and to see the equivalence in the intended meanings. Although the mathematically more rigorous notation is more explicit, and (hopefully) less subject to misinterpretation, it also grows cumbersome and--taken to an extreme--becomes almost unreadable. Good expository writing is a compromise between being explicit and being readily understandable.

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Reference

Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.