Sample spaces

Suppose that we want to describe one English personal pronoun token and eventually ask whether it is nominative, accusative, singular, or plural. What, exactly, are the possible results? A recording may contain an utterance of them. A transcript represents that utterance with the string them, while an annotation might represent it with the label third person plural accusative. A particular probability model must choose which of these representations defines its outcomes. This choice is the outcome-representation problem.

Assumed background

We assume familiarity with the basic set theory reviewed in Languages as formal objects, including sets, membership, subsets, power sets, union, intersection, difference, and complement. We use those concepts here to define probability spaces.

Begin with surface form as the represented result. Before introducing notation, the setup is this: one pronoun token is selected, and its surface form must be one of twelve strings. We collect those strings in the set

\Omega \equiv\{\textit{I},\textit{me},\textit{you},\textit{it},\textit{he},\textit{him}, \textit{she},\textit{her},\textit{we},\textit{us},\textit{they},\textit{them}\}.

Basically, an outcome is one represented result, and the sample space is the collection of all results the model allows. More specifically, an outcome is one possible result represented by a probability model. The sample space \Omega is the set of all such outcomes.

Each member of \Omega is one outcome. If the recorded form is her, then her is the realized outcome.

The sample space contains twelve represented surface strings, not an inventory of every English pronoun use. Some strings are syncretic: the surface forms you and it, for instance, do not by themselves determine syntactic case. Thus an analysis of case must add an annotation to the outcome or use a deliberately restricted case-coded sample space.

Membership in the sample space

At this stage, we have not assigned probabilities. We can only ask whether a proposed outcome belongs to the sample space. The notation x\in\Omega says that x is a member of \Omega, while x\notin\Omega says that it is not.

For the space above,

\textit{him}\in\Omega \qquad\text{and}\qquad \textit{herself}\notin\Omega.

The first statement says that the model has an outcome corresponding to him. The second says that the model has no outcome corresponding to herself, not that speakers cannot produce that form.

A minimal base R representation makes the same membership check.

Code

omega <- c("I", "me", "you", "it", "he", "him", "she", "her",
           "we", "us", "they", "them")

"him" %in% omega
"herself" %in% omega

The first expression returns TRUE; the second returns FALSE. The code is only a finite encoding of the set written above. It does not yet define which outcomes are common.

Outcome granularity

Now consider a vowel production study. At least three sample spaces are possible.

One outcome is an entire recording.
One outcome is a segmented vowel token.
One outcome is an acoustic measurement taken at a specified point in a vowel.

Under the first representation, one outcome is an entire recording. Under the second, it is a vowel token. Under the third, it is one acoustic value.

Suppose a speaker produces heed twice and we estimate F1 at the vowel midpoint. Under the second representation, the two vowel tokens are two possible observed units. Under the third representation, values such as 285 Hz and 301 Hz are possible observed results. A question about how midpoint F1 differs across tokens belongs naturally to the measurement representation. The estimate does not by itself establish that a token belongs to /i/ or to any other phonological category; that claim requires a category definition and additional evidence.

What counts as one outcome also determines which identifiers must accompany the observed value. In a judgment experiment, the response value 6 alone may not identify the trial that matters for analysis. We may instead represent an outcome as the pair

\langle\text{participant and sentence trial},\ 6\rangle.

If the analysis will compare responses across participants and sentences, the outcome must retain the participant and sentence identifiers.

Omission and impossibility

An omitted outcome is not an impossible result. A space may appear complete because every listed outcome is assigned a place, even though the list omits results that the observation process can produce.

For instance, a forced choice experiment may offer the responses agent and patient. If participants can also skip the trial, then

\Omega\equiv\{\text{agent},\text{patient}\}

does not represent the recorded process unless skipped trials are removed by an explicit rule. A model using this space cannot distinguish a skipped response from missing data created later in processing.

The sample space need not contain every conceivable linguistic object. It must contain every result that the stated observation process can produce, including a skipped response when skipping is recorded as a possible response.

This is the exact upshot: \Omega records the model’s answer to “what could happen?” It does not yet say how outcomes should be grouped or how probable they are. The next page groups outcomes into events that answer questions such as “was the pronoun plural?”

Check your understanding

A study records a seven point judgment for each participant and sentence. Give one sample space whose outcomes are response values. Then give a different sample space whose outcomes retain trial identity.
In the pronoun space above, what does \textit{herself}\notin\Omega claim about the model? What does it not claim about English?
A corpus annotation distinguishes nouns, verbs, and adjectives but labels every other token OTHER. Is OTHER an outcome? Explain your answer from the stored annotation process.
A vowel study stores one F1 trajectory with eleven time points for each token. What question would make one trajectory a suitable outcome? What different question would make one time point a suitable outcome?

--- title: "Sample spaces" --- Suppose that we want to describe one English personal pronoun token and eventually ask whether it is nominative, accusative, singular, or plural. What, exactly, are the possible results? A recording may contain an utterance of *them*. A transcript represents that utterance with the string `them`, while an annotation might represent it with the label `third person plural accusative`. A particular probability model must choose which of these representations defines its outcomes. This choice is the **outcome-representation problem**. ::: {.callout-note title="Assumed background"} We assume familiarity with the basic set theory reviewed in [*Languages as formal objects*](https://aaronstevenwhite.io/intro-to-cl/languages-as-formal-objects/), including sets, membership, subsets, power sets, union, intersection, difference, and complement. We use those concepts here to define probability spaces. ::: Begin with surface form as the represented result. Before introducing notation, the setup is this: one pronoun token is selected, and its surface form must be one of twelve strings. We collect those strings in the set $$ \Omega \equiv\{\textit{I},\textit{me},\textit{you},\textit{it},\textit{he},\textit{him}, \textit{she},\textit{her},\textit{we},\textit{us},\textit{they},\textit{them}\}. $$ Basically, an outcome is one represented result, and the sample space is the collection of all results the model allows. More specifically, an [**outcome**](https://bruno.nicenboim.me/bayescogsci/ch-intro.html) is one possible result represented by a probability model. The [**sample space**](https://bruno.nicenboim.me/bayescogsci/ch-intro.html) $\Omega$ is the set of all such outcomes. Each member of $\Omega$ is one outcome. If the recorded form is *her*, then *her* is the realized outcome. The sample space contains twelve represented surface strings, not an inventory of every English pronoun use. Some strings are syncretic: the surface forms *you* and *it*, for instance, do not by themselves determine syntactic case. Thus an analysis of case must add an annotation to the outcome or use a deliberately restricted case-coded sample space. ## Membership in the sample space At this stage, we have not assigned probabilities. We can only ask whether a proposed outcome belongs to the sample space. The notation $x\in\Omega$ says that $x$ is a member of $\Omega$, while $x\notin\Omega$ says that it is not. For the space above, $$ \textit{him}\in\Omega \qquad\text{and}\qquad \textit{herself}\notin\Omega. $$ The first statement says that the model has an outcome corresponding to *him*. The second says that the model has no outcome corresponding to *herself*, not that speakers cannot produce that form. A minimal base R representation makes the same membership check. ```{r} #| eval: false omega <- c("I", "me", "you", "it", "he", "him", "she", "her", "we", "us", "they", "them") "him" %in% omega "herself" %in% omega ``` The first expression returns `TRUE`; the second returns `FALSE`. The code is only a finite encoding of the set written above. It does not yet define which outcomes are common. ## Outcome granularity Now consider a vowel production study. At least three sample spaces are possible. 1. One outcome is an entire recording. 2. One outcome is a segmented vowel token. 3. One outcome is an acoustic measurement taken at a specified point in a vowel. Under the first representation, one outcome is an entire recording. Under the second, it is a vowel token. Under the third, it is one acoustic value. Suppose a speaker produces *heed* twice and we estimate F1 at the vowel midpoint. Under the second representation, the two vowel tokens are two possible observed units. Under the third representation, values such as 285 Hz and 301 Hz are possible observed results. A question about how midpoint F1 differs across tokens belongs naturally to the measurement representation. The estimate does not by itself establish that a token belongs to /i/ or to any other phonological category; that claim requires a category definition and additional evidence. What counts as one outcome also determines which identifiers must accompany the observed value. In a judgment experiment, the response value `6` alone may not identify the trial that matters for analysis. We may instead represent an outcome as the pair $$ \langle\text{participant and sentence trial},\ 6\rangle. $$ If the analysis will compare responses across participants and sentences, the outcome must retain the participant and sentence identifiers. ## Omission and impossibility An omitted outcome is not an impossible result. A space may appear complete because every listed outcome is assigned a place, even though the list omits results that the observation process can produce. For instance, a forced choice experiment may offer the responses *agent* and *patient*. If participants can also skip the trial, then $$ \Omega\equiv\{\text{agent},\text{patient}\} $$ does not represent the recorded process unless skipped trials are removed by an explicit rule. A model using this space cannot distinguish a skipped response from missing data created later in processing. The sample space need not contain every conceivable linguistic object. It must contain every result that the stated observation process can produce, including a skipped response when skipping is recorded as a possible response. This is the exact upshot: $\Omega$ records the model's answer to “what could happen?” It does not yet say how outcomes should be grouped or how probable they are. The [next page](events.qmd) groups outcomes into events that answer questions such as “was the pronoun plural?” ## Check your understanding 1. A study records a seven point judgment for each participant and sentence. Give one sample space whose outcomes are response values. Then give a different sample space whose outcomes retain trial identity. 2. In the pronoun space above, what does $\textit{herself}\notin\Omega$ claim about the model? What does it not claim about English? 3. A corpus annotation distinguishes nouns, verbs, and adjectives but labels every other token `OTHER`. Is `OTHER` an outcome? Explain your answer from the stored annotation process. 4. A vowel study stores one F1 trajectory with eleven time points for each token. What question would make one trajectory a suitable outcome? What different question would make one time point a suitable outcome?