The beta distribution

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A convenient choice for rapid Bayesian inference

The beta distribution is a convenient choice for representing Bayesian inference on a probabilistic quantity such as probability of extirpation or probability of sighting. It is a distribution on the interval $[0, 1]$ which allows us to quickly estimate these probabilities and also summarise our confidence of these estimates, represented by how peaked the distribution is around its maximum.

You can explore the different forms the beta distribution takes by adjusting the parameters using the sliders below. The initial parameters reflect the posterior distribution for extirpation that we will derive for Primula pauciflora in all habitat in the upcoming example (with alpha and beta switched since we are considering probability of extirpation rather than presence).

Expand for mathematical details

make_beta <- function (prob, weight) {
  c(prob * weight, (1 - prob) * weight)
}

# Beta distribution statistics from https://en.wikipedia.org/wiki/Beta_distribution
beta_mean <- function (bf) {
  bf$alpha / (bf$alpha + bf$beta)
}

beta_variance <- function (bf) {
  bf$alpha * bf$beta / ((bf$alpha + bf$beta)^2 * (bf$alpha + bf$beta + 1))
}

We model the probability of a species’ extirpation in a particular area. For consistency with mathematical treatments, we present calculations with a random variable $θ$ encoding the probability of presence in the range $[0, 1]$ and convert to $1-θ$, the probability of extirpation when we summarise our results. Observations of presence or absence $y_t$ are governed by a Bernoulli process with parameter $θ$, where $p(y_t |θ)=θ$ if $y_t =1$ represents presence.

Following the notation of Royle and Dorazio (2008), we updated our prior beliefs (prior distribution $π(θ)$) using a probabilistic model (likelihood function $f(y|θ)$) to derive posterior beliefs (posterior distribution $π(θ|y)$). This produces the classical form of Bayes theorem for inference written as: $$π(θ|y) ∝ π(θ)f(y|θ)$$ (Equation 1)

We modeled the probability of presence by the beta distribution on $[0, 1]$, which is described by two shape parameters $α$ and $β$ and takes the form: $$f(y|α,β) ∝ y^{(α-1)} (1-y)^{(β-1)}$$ (Equation 2)

This is a convenient choice because, under the Bayesian framework, we can select a conjugate prior within this family, which leads to a posterior distribution in the same family. Given our observations of target species take the form of binary variables (all representing non-detection), these can be placed within a hierarchical Bayesian modeling context where the shape parameters $α$ and $β$ represent hyperparameters for our modeled distributions.

Inferring extirpation Effective search effort