# d.rel.conn.multinomial.unnorm: Calculates unnormalized probability density for relative... In ConnMatTools: Tools for Working with Connectivity Data

## Description

This functions calculates the unnormalized probability density function for the relative (to all settlers at the destination site) connectivity value for larval transport between multiple source sites to a destination site. An arbitrary number of source sites can be evaluated.

## Usage

 ```1 2 3 4 5 6 7 8``` ```d.rel.conn.multinomial.unnorm( phis, ps, ks, n.sample, log = FALSE, dirichlet.prior.alphas = 1/(length(phis) + 1) ) ```

## Arguments

 `phis` Vector of fractions of individuals (i.e., eggs) from the source populations settling at the destination population `ps` Vector of fractions of individuals (i.e., eggs) marked in each of the source populations `ks` Vector of numbers of marked settlers from each source population found in the sample `n.sample` Vector of total numbers of settlers collected `log` Boolean indicating whether or not to return the log probability density. Defaults to `FALSE`. `dirichlet.prior.alphas` Parameter value for a Dirichlet prior distribution for the `phis`. Can be a single value for a Dirichlet prior with uniform parameters, or a vector of length = `length(phis)+1`. Defaults to `1/(length(phis)+1)`, the value for the "reference distance" non-informative prior of Berger et al. 2015.

## Details

As this function returns the unnormalized probability density, it must be normalized somehow to be produce a true probability density. This can be acheived using a variety of approaches, including brute force integration of the unnormalized probability density and MCMC algorithms.

## Value

The unnormalized probability density value. If `log=TRUE`, then the logarithm of the probability density value will be returned.

## 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.

Berger JO, Bernardo JM, Sun D (2015) Overall Objective Priors. Bayesian Analysis 10:189-221. doi:10.1214/14-BA915

Other connectivity estimation: `d.rel.conn.beta.prior()`, `d.rel.conn.dists.func()`, `d.rel.conn.finite.settlement()`, `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 26 27 28 29 30 31 32 33 34 35 36 37``` ```library(ConnMatTools) ps <- c(0.7,0.5) # Fraction of eggs "marked" at each source site ks <- c(4,5) # Number of marked settlers among sample from each source site n.sample <- 20 # Total sample size. Must be >= sum(ks) phis0 = runif(3,min=0.05) phis0 = phis0 / sum(phis0) phis0 = phis0[1:2] # Don't include relative connectivity of unknown sites nbatch=1e4 library(mcmc) ans = metrop(d.rel.conn.multinomial.unnorm, initial=phis0,nbatch=nbatch,scale=0.1, log=TRUE,ps=ps,ks=ks,n.sample=n.sample) # A more serious test would adjust blen and scale to improve results, and would repeat # multiple times to get results from multiple MCMC chains. # Plot marginal distribution of relative connectivity from first site h=hist(ans\$batch[,1],xlab="Rel. Conn., Site 1", main="Relative Connectivity for Source Site 1") # For comparison, add on curve that would correspond to single site calculation phi = seq(0,1,length.out=40) d1 = d.rel.conn.beta.prior(phi,ps,ks,n.sample) lines(phi,d1*nbatch*diff(h\$breaks),col="red",lwd=5) # Image plot of bivariate probability density t=table(cut(ans\$batch[,1],phi),cut(ans\$batch[,2],phi)) image(t,col=heat.colors(12)[12:1],xlab="Rel. Conn., Site 1",ylab="Rel. Conn., Site 2") # Add line indicate region above which one can never find results as that would # lead to a total connectivity great than 1 abline(1,-1,col="black",lty="dashed",lwd=3) ```