Approximate
Bayesian Computation (ABC) is a simulation based method for Bayesian
inference. It is commonly used in evolutionary biology to estimate
parameters of demographic models. Coala makes it easy to conduct the
simulations for an ABC analysis and works well together with the
`abc`

package for doing the estimation. To demonstrate the
principle, we will estimate the parameter of an over-simplified toy
model.

Let’s assume that we have 50 genetic loci from 10 individuals from a
panmictic population. We will use the site frequency spectrum of the
data as a set of summary statistics to estimate the scaled mutation
rate, `theta`

. Let’s assume we get the following frequency
spectrum from the data:

`<- c(112, 57, 24, 34, 16, 29, 8, 10, 15) sfs `

We can now use coala to set the model up:

```
library(coala)
<- coal_model(10, 50) +
model feat_mutation(par_prior("theta", runif(1, 1, 5))) +
sumstat_sfs()
```

Note that we used `par_prior`

to set a uniform prior
between 1 and 5 for `theta`

.

We can now easily simulate the model:

`<- simulate(model, nsim = 2000, seed = 17) sim_data `

For this toy model, we ran just 2000 simulations (to keep the time
for building this document within a reasonable range). A real analysis
will need many more, but you can reduce the simulation time by
parallelizing them with the `cores`

argument.

We need to prepare the simulation data for the `abc`

package. We can use the `create_abc_param`

and
`create_abc_sumstat`

functions for this purpose:

```
# Getting the parameters
<- create_abc_param(sim_data, model)
sim_param head(sim_param, n = 3)
```

```
## theta
## 1 1.620203
## 2 1.216988
## 3 2.410258
```

```
# Getting the summary statistics
<- create_abc_sumstat(sim_data, model)
sim_sumstat head(sim_sumstat, n = 3)
```

```
## sfs1 sfs2 sfs3 sfs4 sfs5 sfs6 sfs7 sfs8 sfs9
## [1,] 74 54 32 30 13 14 10 6 2
## [2,] 63 30 12 12 12 12 11 8 7
## [3,] 121 49 45 36 12 28 12 7 8
```

Now, we can estimate the posterior distribution of
`theta`

:

```
suppressPackageStartupMessages(library(abc))
<- abc(sfs, sim_param, sim_sumstat, 0.05, method = "rejection")
posterior hist(posterior, breaks = 20)
```

Due to the low number of simulations, this posterior should only be treated as a rough approximation. We have prepared a version using a million simulations instead. Generating it took significantly longer (about two hours on a modern laptop), but we get a smoother estimate of the posterior distribution: