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# Pre-requisites

You should be familiar with the Metropolis–Hastings Algorithm, introduced here, and elaborated here.

# Caveat on code

Note: the code here is designed to be readable by a beginner, rather than “efficient”. The idea is that you can use this code to learn about the basics of MCMC, but not as a model for how to program well in R!

# Example 1: sampling from an exponential distribution using MCMC

Any MCMC scheme aims to produce (dependent) samples from a target" distribution. In this case we are going to use the exponential distribution with mean 1 as our target distribution. Here we define this function (on log scale):

log_exp_target = function(x){
return(dexp(x,rate=1, log=TRUE))
}

The following code implements a simple MH algorithm. (Note that the parameter log_target is a function which computes the log of the target distribution; you may be unfamiliar with the idea of passing a function as a parameter, but it works just like any other type of parameter…):

easyMCMC = function(log_target, niter, startval, proposalsd){
x = rep(0,niter)
x[1] = startval
for(i in 2:niter){
currentx = x[i-1]
proposedx = rnorm(1,mean=currentx,sd=proposalsd)
A = exp(log_target(proposedx) - log_target(currentx))
if(runif(1)<A){
x[i] = proposedx       # accept move with probabily min(1,A)
} else {
x[i] = currentx        # otherwise "reject" move, and stay where we are
}
}
return(x)
}

Now we run the MCMC three times from different starting points and compare results:

z1=easyMCMC(log_exp_target, 1000,3,1)
z2=easyMCMC(log_exp_target, 1000,1,1)
z3=easyMCMC(log_exp_target, 1000,5,1)

plot(z1,type="l")
lines(z2,col=2)
lines(z3,col=3)

Version Author Date
1543692 Matthew Stephens 2022-04-26
plot(log_exp_target(z1))
lines(log_exp_target(z2),col=2)
lines(log_exp_target(z3),col=3)

Version Author Date
1543692 Matthew Stephens 2022-04-26
par(mfcol=c(3,1)) #rather odd command tells R to put 3 graphs on a single page
maxz=max(c(z1,z2,z3))
hist(z1,breaks=seq(0,maxz,length=20))
hist(z2,breaks=seq(0,maxz,length=20))
hist(z3,breaks=seq(0,maxz,length=20))

## Exercise

Use the function easyMCMC to explore the following:

1. how do different starting values affect the MCMC scheme? (try some extreme starting points)
2. what is the effect of having a bigger/smaller proposal standard deviation? (again, try some extreme values)
3. try changing the (log-)target function to the following
log_target_bimodal = function(x){
log(0.8* dnorm(x,-4,1) + 0.2 * dnorm(x, 4, 1))
}

What does this target distribution look like? What happens if the proposal sd is too small here? (try e.g. 1 and 0.1)

# Example 2: Estimating an allele frequency

A standard assumption when modelling genotypes of bi-allelic loci (e.g. loci with alleles $$A$$ and $$a$$) is that the population is “randomly mating”. From this assumption it follows that the population will be in “Hardy Weinberg Equilibrium” (HWE), which means that if $$p$$ is the frequency of the allele $$A$$ then the genotypes $$AA$$, $$Aa$$ and $$aa$$ will have frequencies $$p^2, 2p(1-p)$$ and $$(1-p)^2$$ respectively.

A simple prior for $$p$$ is to assume it is uniform on $$[0,1]$$. Suppose that we sample $$n$$ individuals, and observe $$n_{AA}$$ with genotype $$AA$$, $$n_{Aa}$$ with genotype $$Aa$$ and $$n_{aa}$$ with genotype $$aa$$.

The following R code gives a short MCMC routine to sample from the posterior distribution of $$p$$. Try to go through the code to see how it works.

log_prior = function(p){
if((p<0) || (p>1)){  # || here means "or"
return(-Inf)}
else{
return(0)}
}

log_likelihood = function(p, nAA, nAa, naa){
return((2*nAA)*log(p)  + nAa * log (2*p*(1-p)) + (2*naa)*log(1-p))
}

psampler = function(nAA, nAa, naa, niter, pstartval, pproposalsd){
p = rep(0,niter)
p[1] = pstartval
for(i in 2:niter){
currentp = p[i-1]
newp = currentp + rnorm(1,0,pproposalsd)
A = exp(log_prior(newp) + log_likelihood(newp,nAA,nAa,naa) - log_prior(currentp) - log_likelihood(currentp,nAA,nAa,naa))
if(runif(1)<A){
p[i] = newp       # accept move with probabily min(1,A)
} else {
p[i] = currentp        # otherwise "reject" move, and stay where we are
}
}
return(p)
}

Running this sample for $$n_{AA}$$ = 50, $$n_{Aa}$$ = 21, $$n_{aa}$$=29.

z=psampler(50,21,29,10000,0.5,0.01)

Now some R code to compare the sample from the posterior with the theoretical posterior (which in this case is available analytically; since we observed 121 $$A$$s, and 79 $$a$$s, out of 200, the posterior for $$p$$ is Beta(121+1,79+1).

x=seq(0,1,length=1000)
hist(z,prob=T)
lines(x,dbeta(x,122, 80))  # overlays beta density on histogram

Version Author Date
1543692 Matthew Stephens 2022-04-26

You might also like to discard the first 5000 z’s as “burnin”. Here’s one way in R to select only the last 5000 z’s

hist(z[5001:10000])

Version Author Date
1543692 Matthew Stephens 2022-04-26

## Exercise

Investigate how the starting point and proposal standard deviation affect the convergence of the algorithm.

# Example 3: Estimating an allele frequency and inbreeding coefficient

A slightly more complex alternative than HWE is to assume that there is a tendency for people to mate with others who are slightly more closely-related than “random” (as might happen in a geographically-structured population, for example). This will result in an excess of homozygotes compared with HWE. A simple way to capture this is to introduce an extra parameter, the “inbreeding coefficient” $$f$$, and assume that the genotypes $$AA$$, $$Aa$$ and $$aa$$ have frequencies $$fp + (1-f)p*p, (1-f) 2p(1-p)$$, and $$f(1-p) + (1-f)(1-p)(1-p)$$.

In most cases it would be natural to treat $$f$$ as a feature of the population, and therefore assume $$f$$ is constant across loci. For simplicity we will consider just a single locus.

Note that both $$f$$ and $$p$$ are constrained to lie between 0 and 1 (inclusive). A simple prior for each of these two parameters is to assume that they are independent, uniform on $$[0,1]$$. Suppose that we sample $$n$$ individuals, and observe $$n_{AA}$$ with genotype $$AA$$, $$n_{Aa}$$ with genotype $$Aa$$ and $$n_{aa}$$ with genotype $$aa$$.

## Exercise:

• Write a short MCMC routine to sample from the joint distribution of $$f$$ and $$p$$.

Hint: here is a start; you’ll need to fill in the …

fpsampler = function(nAA, nAa, naa, niter, fstartval, pstartval, fproposalsd, pproposalsd){
f = rep(0,niter)
p = rep(0,niter)
f[1] = fstartval
p[1] = pstartval
for(i in 2:niter){
currentf = f[i-1]
currentp = p[i-1]
newf = currentf + ...
newp = currentp + ...
...
}
return(list(f=f,p=p)) # return a "list" with two elements named f and p
}
• Use this sample to obtain point estimates for $$f$$ and $$p$$ (e.g. using posterior means) and interval estimates for both $$f$$ and $$p$$ (e.g. 90% posterior credible intervals), when the data are $$n_{AA} = 50, n_{Aa} = 21, n_{aa}=29$$.

You could also tackle this problem with a Gibbs Sampler (see vignettes here and here).

To do so you will want to use the following “latent variable” representation of the model: $z_i \sim Bernoulli(f)$ $p(g_i=AA | z_i=1) = p; p(g_i=AA | z_i=0) = p^2$ $p(g_i=Aa | z_i = 1)= 0; p(g_i=Aa | z_i=0) = 2p(1-p)$ $p(g_i=aa | z_i = 1) = (1-p); p(g_i =aa | z_i=0) = (1-p)^2$

Summing over $$z_i$$ gives the same model as above: $p(g_i=AA) = fp + (1-f)p^2$

## Exercise:

Using the above, implement a Gibbs Sampler to sample from the joint distribution of $$z,f,$$ and $$p$$ given genotype data $$g$$.

Hint: this requires iterating the following steps

1. sample $$z$$ from $$p(z | g, f, p)$$
2. sample $$f,p$$ from $$p(f, p | g, z)$$

sessionInfo()
R version 4.1.0 Patched (2021-07-20 r80657)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Monterey 12.2

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base

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[9] fansi_0.5.0      highr_0.9        stringr_1.4.0    tools_4.1.0
[13] xfun_0.28        utf8_1.2.2       git2r_0.29.0     jquerylib_0.1.4
[17] htmltools_0.5.2  ellipsis_0.3.2   rprojroot_2.0.2  yaml_2.2.1
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[25] later_1.3.0      sass_0.4.1       vctrs_0.3.8      fs_1.5.0
[29] promises_1.2.0.1 glue_1.5.0       evaluate_0.14    rmarkdown_2.11
[33] stringi_1.7.5    bslib_0.3.1      compiler_4.1.0   pillar_1.6.4
[37] jsonlite_1.7.2   httpuv_1.6.3     pkgconfig_2.0.3 

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