The beta distribution

The uniform distribution placed constant density on a bounded interval. The beta family allows that density to change across the unit interval. Suppose \Theta represents a speaker’s latent probability of choosing the double object construction in a declared discourse context.

The beta distribution is a family of absolutely continuous distributions with topological support [0,1]:

\Theta\sim\operatorname{Beta}(\alpha,\beta), \qquad \alpha>0,\ \beta>0.

Its density has the form

f_\Theta(\theta) \equiv \begin{cases} \dfrac{\theta^{\alpha-1}(1-\theta)^{\beta-1}} {B(\alpha,\beta)},&0<\theta<1,\\ 0,&\text{otherwise}. \end{cases}

The quantity B(\alpha,\beta) is a normalizing constant that makes the total density area one. The parameters \alpha and \beta control how that area is arranged across the interval.

Beginning from constant density

Set \alpha=1 and \beta=1. Then

\theta^{\alpha-1}(1-\theta)^{\beta-1} =\theta^0(1-\theta)^0 =1.

The density is constant across (0,1). This case matches a uniform density on the unit interval.

Changing the relative parameter sizes

When \alpha>\beta, more density tends to lie toward one. When \beta>\alpha, more density tends to lie toward zero.

For instance, \operatorname{Beta}(5,2) places more density on high construction preference values, while \operatorname{Beta}(2,5) places more on low values. Interchanging the parameters reflects the density around .5.

When both parameters exceed one, the density may have an interior peak. When both lie below one, density may rise near both boundaries. The family can thus represent several shapes with the same bounded support.

Deriving the mean and variance

The mean is

\mathbb{E}[\Theta] =\frac{\alpha}{\alpha+\beta}.

For \operatorname{Beta}(2,5),

\mathbb{E}[\Theta] =\frac{2}{7} \approx.286.

The variance is

\operatorname{Var}(\Theta) =\frac{\alpha\beta} { (\alpha+\beta)^2(\alpha+\beta+1)}.

For \operatorname{Beta}(2,5),

\operatorname{Var}(\Theta) =\frac{10}{49(8)} \approx.0255.

The parameters jointly determine center and concentration. Neither parameter alone is a mean or variance.

Holding the mean fixed while changing concentration

Both \operatorname{Beta}(2,2) and \operatorname{Beta}(20,20) have mean .5. Their variances differ:

\operatorname{Var}(\operatorname{Beta}(2,2))=.05

and

\operatorname{Var}(\operatorname{Beta}(20,20))\approx.0061.

The larger equal parameters concentrate more density near .5. Equal means do not imply equal variances.

Inspecting the shapes in base R

Code

theta <- seq(.001, .999, length.out = 300)

plot(theta, dbeta(theta, 2, 5), type = "l",
     xlab = "double object preference", ylab = "density")
lines(theta, dbeta(theta, 5, 2), lty = 2)
lines(theta, dbeta(theta, 2, 2), lty = 3)
lines(theta, dbeta(theta, 20, 20), lty = 4)

The plot separates changes in location from changes in concentration.

Keeping the modeled level explicit

The variable \Theta is a latent probability for a speaker and context. It is not one binary construction choice. A single token remains a zero and one outcome, while \Theta describes the chance of the target outcome across comparable tokens.

A proportion calculated from finitely many observed tokens is also not automatically a direct beta observation. Its discreteness and denominator remain part of the measurement process.

Checking the response support

A beta distribution is not appropriate merely because a response is written between zero and one. The family is continuous on the open interval and assigns no probability to exact endpoints. A response process that produces structural zeros, structural ones, or a finite grid needs those features represented explicitly.

Support is necessary but not sufficient. The modeled level and observation process must also match.

Check your understanding

Compute the mean of \operatorname{Beta}(3,7).
Why do \operatorname{Beta}(2,2) and \operatorname{Beta}(20,20) have the same mean but different concentration?
Explain the difference between a latent construction probability and one observed construction choice.
Why can exact endpoint responses cause a support mismatch for a direct beta model?

The beta family changes shape within a bounded interval. The next page introduces a symmetric location and scale family over the full real line.

--- title: "The beta distribution" --- The [uniform distribution](uniform-distribution.qmd) placed constant density on a bounded interval. The beta family allows that density to change across the unit interval. Suppose $\Theta$ represents a speaker's latent probability of choosing the double object construction in a declared discourse context. The [**beta distribution**](https://bruno.nicenboim.me/bayescogsci/ch-introBDA.html) is a family of absolutely continuous distributions with topological support $[0,1]$: $$ \Theta\sim\operatorname{Beta}(\alpha,\beta), \qquad \alpha>0,\ \beta>0. $$ Its density has the form $$ f_\Theta(\theta) \equiv \begin{cases} \dfrac{\theta^{\alpha-1}(1-\theta)^{\beta-1}} {B(\alpha,\beta)},&0<\theta<1,\\ 0,&\text{otherwise}. \end{cases} $$ The quantity $B(\alpha,\beta)$ is a normalizing constant that makes the total density area one. The parameters $\alpha$ and $\beta$ control how that area is arranged across the interval. ## Beginning from constant density Set $\alpha=1$ and $\beta=1$. Then $$ \theta^{\alpha-1}(1-\theta)^{\beta-1} =\theta^0(1-\theta)^0 =1. $$ The density is constant across $(0,1)$. This case matches a uniform density on the unit interval. ## Changing the relative parameter sizes When $\alpha>\beta$, more density tends to lie toward one. When $\beta>\alpha$, more density tends to lie toward zero. For instance, $\operatorname{Beta}(5,2)$ places more density on high construction preference values, while $\operatorname{Beta}(2,5)$ places more on low values. Interchanging the parameters reflects the density around $.5$. When both parameters exceed one, the density may have an interior peak. When both lie below one, density may rise near both boundaries. The family can thus represent several shapes with the same bounded support. ## Deriving the mean and variance The mean is $$ \mathbb{E}[\Theta] =\frac{\alpha}{\alpha+\beta}. $$ For $\operatorname{Beta}(2,5)$, $$ \mathbb{E}[\Theta] =\frac{2}{7} \approx.286. $$ The variance is $$ \operatorname{Var}(\Theta) =\frac{\alpha\beta} { (\alpha+\beta)^2(\alpha+\beta+1)}. $$ For $\operatorname{Beta}(2,5)$, $$ \operatorname{Var}(\Theta) =\frac{10}{49(8)} \approx.0255. $$ The parameters jointly determine center and concentration. Neither parameter alone is a mean or variance. ## Holding the mean fixed while changing concentration Both $\operatorname{Beta}(2,2)$ and $\operatorname{Beta}(20,20)$ have mean $.5$. Their variances differ: $$ \operatorname{Var}(\operatorname{Beta}(2,2))=.05 $$ and $$ \operatorname{Var}(\operatorname{Beta}(20,20))\approx.0061. $$ The larger equal parameters concentrate more density near $.5$. Equal means do not imply equal variances. ## Inspecting the shapes in base R ```{r} #| eval: false theta <- seq(.001, .999, length.out = 300) plot(theta, dbeta(theta, 2, 5), type = "l", xlab = "double object preference", ylab = "density") lines(theta, dbeta(theta, 5, 2), lty = 2) lines(theta, dbeta(theta, 2, 2), lty = 3) lines(theta, dbeta(theta, 20, 20), lty = 4) ``` The plot separates changes in location from changes in concentration. ## Keeping the modeled level explicit The variable $\Theta$ is a latent probability for a speaker and context. It is not one binary construction choice. A single token remains a zero and one outcome, while $\Theta$ describes the chance of the target outcome across comparable tokens. A proportion calculated from finitely many observed tokens is also not automatically a direct beta observation. Its discreteness and denominator remain part of the measurement process. ## Checking the response support A beta distribution is not appropriate merely because a response is written between zero and one. The family is continuous on the open interval and assigns no probability to exact endpoints. A response process that produces structural zeros, structural ones, or a finite grid needs those features represented explicitly. Support is necessary but not sufficient. The modeled level and observation process must also match. ## Check your understanding 1. Compute the mean of $\operatorname{Beta}(3,7)$. 2. Why do $\operatorname{Beta}(2,2)$ and $\operatorname{Beta}(20,20)$ have the same mean but different concentration? 3. Explain the difference between a latent construction probability and one observed construction choice. 4. Why can exact endpoint responses cause a support mismatch for a direct beta model? The beta family changes shape within a bounded interval. The [next page](normal-distribution.qmd) introduces a symmetric location and scale family over the full real line.