The bridgedist Basics
Author: Bruce J. Swihart
Date: 2024-10-18

# The `bridgedist` Basics

#### Bruce J. Swihart

#### 2024-10-18

    library(bridgedist)

    ## The argument `scale` has been replaced with `phi` in this version.

    ## The top of page 77o in Wang Louis 2003 states phi is not a scale.

    ## Please start using the `phi` argument now.

Recreate the plot from Wang and Louis (2003) where the Bridge, Normal, and Logistic all have unit variance and mean 0 with ggplot2:

    library(reshape2)
    library(ggplot2)
    xaxis = seq(-4,4,.01)
    df = data.frame( xaxis,
                     Bridge = dbridge(xaxis, phi=1/sqrt(1+3/pi^2)),
                     Normal = dnorm(xaxis),
                     Logistic = dlogis(xaxis, scale=sqrt(3/pi^2)))
    melt.df <- melt(df, id.vars = "xaxis")
    colnames(melt.df) <- c("x", "Distribution", "value")
    ggplot(melt.df, aes(x, value, color=Distribution)) +
      geom_line(size=1.05) +
      ylab("Probability density function")

    ## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
    ## ℹ Please use `linewidth` instead.
    ## This warning is displayed once every 8 hours.
    ## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
    ## generated.

Fig. 1. Probability density functions of the Gaussian, logistic and bridge, for logistic, distributions each with zero mean and unit variance.

The implication is that a random variable from a Bridge distribution plus random variable from a standard logistic distribution is a logistic random variable with a scale greater than one (1/phi).

    phi <- 0.5
    df = data.frame(
                     Bridge = rbridge(1e5, phi=phi),
                     Std_Logistic = rlogis(1e5),
                     BridgePlusStd_Logistic = rbridge(1e5, phi=phi) +  rlogis(1e5),
                     Logistic = rlogis(1e5, scale=1/phi)
    )
    melt.df <- melt(df)
    colnames(melt.df) <- c("Distribution", "value")
    ggplot(melt.df, aes(value)) +
      facet_grid(.~Distribution) +
      geom_histogram()

Fig. 2. 10000 random variates in each panel. From left to right: the bridge distribution, the logistic with scale=1, the sum of the previous two, and the logistic with scale=1/phi. Note how similar the third and fourth panel, the application supporting the theory.