sim.Rmd

---
title: "Applying Gibbs Sampling"
description: |
  Gibbs Sampling can be used in population genetics research to build populations.
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

# Gibbs Sampling in statistical genetics {#sim}

Often times in population genetics research, it is useful to classify individuals in a sample into populations. While other areas of study may use factors like linguistic, cultural, or physical characters, it is not always easy to assign populations based on genotypes.  

[Pritchard et al. 2000](https://web.stanford.edu/group/pritchardlab/publications/pdfs/PritchardEtAl00.pdf) proposed a method to assign individuals to populations and simultaneously estimate allele frequencies: homozygous dominant, heterozygous, and homozygous recessive. This method can utilize various genetic markers as indication of alleles including SNPS, RFLPS, micro-satellites, etc. Additionally, they follow a few assumptions:  

+ Assumes markers are unlinked loci → can be drawn as independent samples  
  
+ Assumes Hardy Weinberg Equilibrium  
  
**Hardy Weinberg Equilibrium**
$$p^2+2pq+q^2 = 1$$
Where $p^2$ is the homozygous dominant allele frequency, $2pq$ is the heterozygous allele frequency, and $q^2$ is the homozygous recessive allele frequency. By assuming Hardy Weinberg Equilibrium, we are stating that the genetic variation in a population will remain constant from one generation to the next.

Under these assumptions, each allele at each locus in each genotype is an independent draw from the appropriate frequency distribution.

# Building a Model
Assume each population is modeled by a characteristic set of allele frequencies.

  * Let X be the genotypes of sampled individuals (our data)  
  * Let Z be the unknown population of origin of individuals  
  * Let P be the unknown allele frequency in all populations  

Information about P and Z is given by the posterior distribution:

$$\begin{aligned}P(Z, P |X) &= \frac{P(Z, P, X)}{P(X)} \\
&\propto P(P) P(Z) P(X|Z, P)
\end{aligned}$$

This is a great opportunity to use Gibbs Sampling! It is not possible to sample from the posterior, $P(Z, P|X)$, directly. We don't know what $P(P)$ or $P(Z)$ are because they are our unknown variables. However, it is possible to use conditional sampling to build an approximate distribution: $(Z_1,P_1), (Z_2, P_2), ..., (Z_n, P_n)$.

Start with randomly drawn initial value $Z_0$ as a hypothetical population of origin and $P_0$ as a hypothetical allele frequency. Then, iterate the following steps:

1. Sample $P_1$ from $P(P | X, Z_0)$
    + This estimates allele frequencies for each population assuming population of origin for the individual is known.
2. Sample $Z_1$ from $P(Z | X, P_1)$
    + This estimates the population of origin for each individual assuming that the population allele frequencies are known.
3. Continue the pattern $n$ times:
    + Sample $P_n$ from $P(P | X, Z_{n-1})$
    + Sample $Z_n$ from $P(Z | X, P_n)$