# SFS Inference

## Computing likelihoods

Following [Sawyer1992] the distribution of mutation frequencies is treated as a Poisson random field, so that composite likelihoods (in which we assume mutations are independent) are computed by taking Poisson likelihoods over bins in the SFS. We typically work with log-likelihoods, so that the log-likelihood of the data ($$D$$) given the model ($$M$$) is

$\log{\mathcal{L}} = \sum_{i} D_i \log{M_i} - M_i - \log{D_i !}$

where $$i$$ indexes the bins of the SFS.

Likelihoods can be computed from moments.Inference:

import moments
import numpy as np

theta = 1000
model = theta * moments.Demographics1D.snm([10])

data = model.sample()

print(model)
print(data)

[-- 1000.0 499.9999999999999 333.33333333333326 250.0 200.0
166.66666666666666 142.85714285714286 125.0 111.1111111111111 --]
[-- 974 483 316 266 210 159 135 122 123 --]

print(moments.Inference.ll(model, data))

-35.832644905426605


When simulating under some demographic model, we usually use the default theta of 1, because the SFS scales linearly in the mutation rate. When comparing to data in this case, we need to rescale the model SFS. It turns out that the maximum-likelihood rescaling is that which makes the total number of segregating sites in the model equal to the total number in the data:

data = moments.Spectrum([0, 3900, 1500, 1200, 750, 720, 600, 400, 0])
model = moments.Demographics1D.two_epoch((2.0, 0.1), [8])

print("Number of segregating sites in data:", data.S())
print("Number of segregating sites in model:", model.S())
print("Ratio of segregating sites:", data.S() / model.S())

opt_theta = moments.Inference.optimal_sfs_scaling(model, data)
print("Optimal theta:", opt_theta)

Number of segregating sites in data: 9070.0
Number of segregating sites in model: 2.7771726368386327
Ratio of segregating sites: 3265.911481226729
Optimal theta: 3265.911481226729


Then we can compute the log-likelihood of the rescaled model with the data, which will give us the same answer as moments.Inference.ll_multinom using the unscaled data:

print(moments.Inference.ll(opt_theta * model, data))
print(moments.Inference.ll_multinom(model, data))

-59.880644681554486
-59.880644681554486


## Optimization

moments optimization is effectively a wrapper for scipy optimization routines, with some features specific to working with SFS data. In short, given a demographic model defined by a set of parameters, we try to find those parameters that minimize the negative log-likelihood of the data given the model. There are a number of optimization functions available in moments.Inference:

• optimize and optimize_log: Uses the BFGS algorithm.

• optimize_lbfgsb and optimize_log_lbfgsb: Uses the L-BFGS-B algorithm.

• optimize_log_fmin: Uses the downhill simplex algorithm on the log of the parameters.

• optimize_powell and optimize_log_powell: Uses the modified Powell’s method, which optimizes slices of parameter space sequentially.

With each method, we require at least three inputs: 1) the initial guess, 2) the data SFS, and 3) the model function that returns a SFS of the same size as the data.

Additionally, it is common to set the following:

• lower_bound and upper_bound: Constraints on the lower and upper bounds during optimization. These are given as lists of the same length of the parameters.

• fixed_params: A list of the same length of the parameters, with fixed values given matching the order of the input parameters. None is used to specify parameters that are still to be optimized.

• verbose: If an integer greater than 0, prints updates of the optimization procedure at intervals given by that spacing.

For a full description of the various inference functions, please see the SFS inference API.

### Single population example

As a toy example, we’ll generate some fake data from a demographic model and then reinfer the input parameters of that demographic model. The model is an instantaneous bottleneck followed by exponential growth, implemented in moments.Demographics1D.bottlegrowth, which takes parameters [nuB, nuF, T] and the sample size. Here nuB is the bottleneck size (relative to the ancestral size), nuF is the relative final size, and T is the time in the past the bottleneck occurred (in units of $$2N_e$$ generations).

nuB = 0.2
nuF = 3.0
T = 0.4

n = 60  # the haploid sample size

fs = moments.Demographics1D.bottlegrowth([nuB, nuF, T], [n])

theta = 2000  # the scaled mutation rate (4*Ne*u*L)
fs = theta * fs
data = fs.sample()


The input demographic model (assuming an $$N_e$$ of 10,000), plotted using demesdraw:

We then set up the optimization inputs, including the initial parameter guesses, lower bounds, and upper bounds, and then run optimization. Here, I’ve decided to use the log-L-BFGS-B method, though there are a number of built in options (see previous section).

p0 = [0.2, 3.0, 0.4]
lower_bound = [0, 0, 0]
upper_bound = [None, None, None]
p_guess = moments.Misc.perturb_params(p0, fold=1,
lower_bound=lower_bound, upper_bound=upper_bound)

model_func = moments.Demographics1D.bottlegrowth

opt_params = moments.Inference.optimize_log_lbfgsb(
p0, data, model_func,
lower_bound=lower_bound,
upper_bound=upper_bound)

model = model_func(opt_params, data.sample_sizes)
opt_theta = moments.Inference.optimal_sfs_scaling(model, data)
model = model * opt_theta


The reinferred parameters:

Params	nuB	nuF	T	theta
Input	0.2	3.0	0.4	2000
Refit	0.2468	3.057	0.4094	1.919e+03


We can also visualize the fit of the model to the data:

moments.Plotting.plot_1d_comp_Poisson(model, data)


#### Confidence intervals

We’re often interested in estimating the precision of the inferred parameters from our best fit model. To do this, we can compute a confidence interval for each free parameter from the model fit. Methods implemented in moments to compute, particularly the method based on the Godambe Information Matrix [Coffman2016], were first implemented in dadi by Alec Coffman, who’s paper should be cited if these methods are used.

See the API documentation for uncertainty functions for information on their usage.

### Two population example

Here, we will create some fake data for a two-population split-migration model, and then re-infer the input parameters to the model used to create that data. This example uses the optimize_log_fmin optimization function. We’ll also use the FIM_uncert function to compute uncertainties (reported as standard errors).

input_theta = 10000
params = [2.0, 3.0, 0.2, 2.0]
model_func = moments.Demographics2D.split_mig
model = model_func(params, [20, 20])
model = input_theta * model
data = model.sample()

p_guess = [2, 2, .1, 4]
lower_bound = [1e-3, 1e-3, 1e-3, 1e-3]
upper_bound = [10, 10, 1, 10]

p_guess = moments.Misc.perturb_params(
p_guess, lower_bound=lower_bound, upper_bound=upper_bound)

opt_params = moments.Inference.optimize_log_fmin(
p_guess, data, model_func,
lower_bound=lower_bound, upper_bound=upper_bound,
verbose=20) # report every 20 iterations

refit_theta = moments.Inference.optimal_sfs_scaling(
model_func(opt_params, data.sample_sizes), data)

uncerts = moments.Godambe.FIM_uncert(
model_func, opt_params, data)

print_params = params + [input_theta]
print_opt = np.concatenate((opt_params, [refit_theta]))

print("Params\tnu1\tnu2\tT_div\tm_sym\ttheta")
print(f"Input\t" + "\t".join([str(p) for p in print_params]))
print(f"Refit\t" + "\t".join([f"{p:.4}" for p in print_opt]))
print(f"Std-err\t" + "\t".join([f"{u:.3}" for u in uncerts]))

moments.Plotting.plot_2d_comp_multinom(
model_func(opt_params, data.sample_sizes), data)

80      , -2037.68    , array([ 2.23462    ,  1.44521    ,  0.158979   ,  2.76268    ])

100     , -1669.86    , array([ 1.88718    ,  1.63916    ,  0.215159   ,  3.56113    ])

120     , -1410.36    , array([ 1.71384    ,  1.97358    ,  0.253576   ,  3.10853    ])

140     , -1262.37    , array([ 1.83998    ,  2.58247    ,  0.210581   ,  2.03112    ])

160     , -1203.16    , array([ 2.00812    ,  3.0514     ,  0.20485    ,  1.9948     ])

180     , -1199.16    , array([ 1.97937    ,  2.86564    ,  0.197018   ,  2.09359    ])

200     , -1198.13    , array([ 1.98305    ,  2.92208    ,  0.200246   ,  2.11119    ])

220     , -1198.13    , array([ 1.98274    ,  2.92226    ,  0.200443   ,  2.11031    ])

240     , -1198.13    , array([ 1.98024    ,  2.91717    ,  0.20053    ,  2.11718    ])

260     , -1198.13    , array([ 1.98055    ,  2.91783    ,  0.200547   ,  2.11654    ])

Params	nu1	nu2	T_div	m_sym	theta
Input	2.0	3.0	0.2	2.0	10000
Refit	1.981	2.918	0.2005	2.116	1.001e+04
Std-err	0.0389	0.0649	0.00449	0.0725	70.2


Above, we can see that we recovered the parameters used to simulate the data very closely, and we used moments’s plotting features to visually compare the data to the model fit.

## References

Sawyer1992

Sawyer, Stanley A., and Daniel L. Hartl. “Population genetics of polymorphism and divergence.” Genetics 132.4 (1992): 1161-1176.

Coffman2016

Coffman, Alec J., et al. “Computationally efficient composite likelihood statistics for demographic inference.” Molecular biology and evolution 33.2 (2016): 591-593.