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Click here to download the full example code

# Emcee and the Model Interface

    import corner
    import matplotlib.pyplot as plt
    import numpy as np

    import lmfit

Set up a double-exponential function and create a Model:

    def double_exp(x, a1, t1, a2, t2):
        return a1*np.exp(-x/t1) + a2*np.exp(-(x-0.1) / t2)

    model = lmfit.Model(double_exp)

Generate some fake data from the model with added noise:

    truths = (3.0, 2.0, -5.0, 10.0)
    x = np.linspace(1, 10, 250)
    np.random.seed(0)
    y = double_exp(x, *truths)+0.1*np.random.randn(x.size)

Create model parameters and give them initial values:

    p = model.make_params(a1=4, t1=3, a2=4, t2=3)

Fit the model using a traditional minimizer, and show the output:

    result = model.fit(data=y, params=p, x=x, method='Nelder', nan_policy='omit')

    lmfit.report_fit(result)
    result.plot()

Calculate parameter covariance using `emcee`:

> - start the walkers out at the best-fit values

> - set `is_weighted` to `False` to estimate the noise weights

> - set some sensible priors on the uncertainty to keep the MCMC in check

    emcee_kws = dict(steps=5000, burn=500, thin=20, is_weighted=False,
                     progress=False)
    emcee_params = result.params.copy()
    emcee_params.add('__lnsigma', value=np.log(0.1), min=np.log(0.001), max=np.log(2.0))

run the MCMC algorithm and show the results:

    result_emcee = model.fit(data=y, x=x, params=emcee_params, method='emcee',
                             nan_policy='omit', fit_kws=emcee_kws)

    lmfit.report_fit(result_emcee)

    result_emcee.plot_fit()
    plt.plot(x, model.eval(params=result.params, x=x), '--', label='Nelder')
    plt.legend()

Check the acceptance fraction to see whether `emcee` performed well:

    plt.plot(result_emcee.acceptance_fraction, 'o')
    plt.xlabel('walker')
    plt.ylabel('acceptance fraction')

Try to compute the autocorrelation time:

    if hasattr(result_emcee, "acor"):
        print("Autocorrelation time for the parameters:")
        print("----------------------------------------")
        for i, p in enumerate(result.params):
            print(f'{p} = {result_emcee.acor[i]:.3f}')

Plot the parameter covariances returned by `emcee` using `corner`:

    emcee_corner = corner.corner(result_emcee.flatchain, labels=result_emcee.var_names,
                                 truths=list(result_emcee.params.valuesdict().values()))

    print("\nmedian of posterior probability distribution")
    print('--------------------------------------------')
    lmfit.report_fit(result_emcee.params)

Find the maximum likelihood solution:

    highest_prob = np.argmax(result_emcee.lnprob)
    hp_loc = np.unravel_index(highest_prob, result_emcee.lnprob.shape)
    mle_soln = result_emcee.chain[hp_loc]
    print("\nMaximum Likelihood Estimation (MLE):")
    print('----------------------------------')
    for ix, param in enumerate(emcee_params):
        print(f"{param}: {mle_soln[ix]:.3f}")

    quantiles = np.percentile(result_emcee.flatchain['t1'], [2.28, 15.9, 50, 84.2, 97.7])
    print(f"\n\n1 sigma spread = {0.5 * (quantiles[3] - quantiles[1]):.3f}")
    print(f"2 sigma spread = {0.5 * (quantiles[4] - quantiles[0]):.3f}")

Total running time of the script: ( 0 minutes 0.000 seconds)

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