# d.rel.conn.beta.prior: Estimate the probability distribution of relative... In ConnMatTools: Tools for Working with Connectivity Data

## Description

These functions calculate the probability density function (`d.rel.conn.beta.prior`), the probability distribution function (aka the cumulative distribution function; `p.rel.conn.beta.prior`) and the quantile function (`q.rel.conn.beta.prior`) for the relative (to all settlers at the destination site) connectivity value for larval transport between a source and destination site given a known fraction of marked individuals (i.e., eggs) in the source population. A non-uniform prior is used for the relative connectivity value.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44``` ```d.rel.conn.beta.prior( phi, p, k, n, prior.shape1 = 0.5, prior.shape2 = prior.shape1, prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2), ... ) p.rel.conn.beta.prior( phi, p, k, n, prior.shape1 = 0.5, prior.shape2 = prior.shape1, prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2), ... ) q.rel.conn.beta.prior.func( p, k, n, prior.shape1 = 0.5, prior.shape2 = prior.shape1, prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2), N = 1000, ... ) q.rel.conn.beta.prior( q, p, k, n, prior.shape1 = 0.5, prior.shape2 = prior.shape1, prior.func = function(phi) dbeta(phi, prior.shape1, prior.shape2), N = 1000, ... ) ```

## Arguments

 `phi` Vector of fractions of individuals (i.e., eggs) from the source population settling at the destination population `p` Fraction of individuals (i.e., eggs) marked in the source population `k` Number of marked settlers found in sample `n` Total number of settlers collected `prior.shape1` First shape parameter for Beta distributed prior. Defaults to 0.5. `prior.shape2` Second shape parameter for Beta distributed prior. Defaults to being the same as `prior.shape1`. `prior.func` Function for prior distribution. Should take one parameter, `phi`, and return a probability. Defaults to `function(phi) dbeta(phi,prior.shape1,prior.shape2)`. If this is specified, then inputs `prior.shape1` and `prior.shape2` are ignored. `...` Extra arguments for the `integrate` function used for normalization of probability distributions. `N` Number of points at which to estimate cumulative probability function for reverse approximation of quantile distribution. Defaults to `1000`. `q` Vector of quantiles

## Details

The prior distribution for relative connectivity `phi` defaults to a Beta distribution with both shape parameters equal to 0.5. This is the Reference or Jeffreys prior for a binomial distribution parameter. Both shape parameters equal to 1 corresponds to a uniform prior.

Estimations of the probability distribution are based on numerical integration using the `integrate` function, and therefore are accurate to the level of that function. Some modification of the default arguments to that function may be necessary to acheive good results for certain parameter values.

## Value

Vector of probabilities or quantiles, or a function in the case of `q.rel.conn.beta.prior.func`.

## Functions

• `d.rel.conn.beta.prior`: Returns the probability density for relative connectivity between a pair of sites

• `p.rel.conn.beta.prior`: Returns the cumulative probability distribution for relative connectivity between a paire of sites

• `q.rel.conn.beta.prior.func`: Returns a function to estimate quantiles for the probability distribution function for relative connectivity between a pair of sites.

• `q.rel.conn.beta.prior`: Estimates quantiles for the probability distribution function for relative connectivity between a pair of sites

## Author(s)

David M. Kaplan dmkaplan2000@gmail.com

## References

Kaplan DM, Cuif M, Fauvelot C, Vigliola L, Nguyen-Huu T, Tiavouane J and Lett C (in press) Uncertainty in empirical estimates of marine larval connectivity. ICES Journal of Marine Science. doi:10.1093/icesjms/fsw182.

Other connectivity estimation: `d.rel.conn.dists.func()`, `d.rel.conn.finite.settlement()`, `d.rel.conn.multinomial.unnorm()`, `d.rel.conn.multiple()`, `d.rel.conn.unif.prior()`, `dual.mark.transmission()`, `optim.rel.conn.dists()`, `r.marked.egg.fraction()`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25``` ```library(ConnMatTools) k <- 10 # Number of marked settlers among sample n.obs <- 87 # Number of settlers in sample p <- 0.4 # Fraction of eggs that was marked phi <- seq(0.001,1-0.001,length.out=101) # Values for relative connectivity # Probability distribution assuming infinite settler pool and uniform prior drc <- d.rel.conn.unif.prior(phi,p,k,n.obs) qrc <- q.rel.conn.unif.prior(c(0.025,0.975),p,k,n.obs) # 95% confidence interval # Probability distribution assuming infinite settler pool and using reference/Jeffreys prior drp <- d.rel.conn.beta.prior(phi,p,k,n.obs) prp <- p.rel.conn.beta.prior(phi,p,k,n.obs) qrp <- q.rel.conn.beta.prior(c(0.025,0.975),p,k,n.obs) # 95% confidence interval # Make a plot of different distributions # black = Jeffreys prior; red = uniform prior # Jeffreys prior draws distribution slightly towards zero plot(phi,drp,type="l",main="Probability of relative connectivity values", xlab=expression(phi),ylab="Probability density") lines(phi,drc,col="red") abline(v=qrp,col="black",lty="dashed") abline(v=qrc,col="red",lty="dashed") ```