Bayesian uncertainty analysis#

R.A. Collenteur, Hydroconsult, May, 2026

In this notebook it is shown how the MCMC-algorithm can be used to estimate the model parameters and quantify the (parameter) uncertainties for a Pastas model using a Bayesian approach. For this purpose the EmceeSolver is introduced, based on the emcee Python package.

Besides Pastas the following Python Packages have to be installed to run this notebook:

emcee
corner

Note: The EmceeSolver is still an experimental feature and some of the arguments might be changed in the future (2026/05/22). We welcome testing and feedback on this new feature!

import corner
import emcee
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

import pastas as ps

ps.set_log_level("ERROR")
ps.show_versions()

Pastas     : 2.0.0
Python     : 3.14.0
Numpy      : 2.4.6
Pandas     : 3.0.3
Scipy      : 1.17.1
Matplotlib : 3.10.9
Numba      : 0.65.1

1. Create a Pastas Model#

The first step is to create a Pastas Model, including the RechargeModel to simulate the effect of precipitation and evaporation on the heads. Here, we first estimate the model parameters using the standard least-squares approach. Although not necessary, this is a convenient approach to obtain initial parameter values to generate the priors of the parameters later on.

head = pd.read_csv(
    "data/B32C0639001.csv", parse_dates=["date"], index_col="date"
).squeeze()

evap = pd.read_csv("data/evap_260.csv", index_col=0, parse_dates=[0]).squeeze()
rain = pd.read_csv("data/rain_260.csv", index_col=0, parse_dates=[0]).squeeze()

ml = ps.Model(head)
ml.add_noisemodel(ps.ArNoiseModel())

# Select a recharge model
rch = ps.rch.Linear()

rm = ps.RechargeModel(rain, evap, recharge=rch, rfunc=ps.Exponential(), name="rch")
ml.add_stressmodel(rm)

ml.solve(tmin="1990", report="full")

ax = ml.plot(figsize=(10, 3))

Fit report solver                   Fit Statistics
==================================================
nfev     34                     EVP          81.20
nobs     351                    R2            0.81
noise    True                   RMSE          0.09
tmin     1990-01-01 00:00:00    AICc      -1957.61
tmax     2005-10-14 00:00:00    BIC       -1938.48
freq     D                      Obj            nan
freq_obs None                   ___               
warmup   3650 days 00:00:00     Interp.         No

Parameters (5 optimized)
==================================================
               optimal    initial  vary   stderr
rch_A         0.271051   0.198424  True   ±7.98%
rch_a        88.306407  10.000000  True   ±8.48%
rch_f        -0.544182  -1.000000  True  ±11.92%
constant_d    0.942309   1.359779  True   ±5.42%
noise_alpha  51.787216  15.000000  True  ±16.44%

Parameter correlations |rho| > 0.5
==================================================
rch_A rch_a       0.85
rch_A constant_d -0.84
rch_a constant_d -0.63
rch_f constant_d -0.79

../_images/4297d781e785269b6b9d7edc5afb0ac075a5dff6d3bd317709d4a846e83b7a2b.png

2. Use the EmceeSolver#

We will now use the EmceeSolve solver to estimate the model parameters and their uncertainties. This solver wraps the Emcee package, which implements different versions of MCMC. A good understanding of Emcee helps when using this solver, so it is recommended to read their documentation as well.

To set up the solver, a number of decisions need to be made:

Determine the priors of the parameters
Choose a (log) likelihood function
Choose the number of steps and thinning

2a. Create the solver instance#

The first step is to create an instance of the EmceeSolve solver class. At this stage all the settings need to be provided on how the Ensemble Sampler is created (https://emcee.readthedocs.io/en/stable/user/sampler/). Important settings are the nwalkers, the moves, the objfunction. More advanced options are to parallelize the MCMC algorithm (parallel=True), and to set a backend to store the results. The solver should be added before using ml.solve to estimate the parameters. Here’s an example:

ml_mc = ml.copy("ml_mc")
ml_mc.del_noisemodel()

# Choose the objective function
ln_prob = ps.likelihood.GaussianLikelihoodAr1()

# Create the EmceeSolver with some settings
s = ps.solver.Emcee(
    nwalkers=20,
    moves=emcee.moves.DEMove(),
    objfunction=ln_prob,
    progress_bar=True,
    parallel=False,
)
ml_mc.add_solver(s)

In the above code we created an EmceeSolve instance with 20 walkers, which take steps according to the DEMove move algorithm (see Emcee docs), and a Gaussian likelihood function that assumes AR1 correlated errors. Different objective functions are available, see the Pastas documentation on the different options.

ml_mc.parameters

	initial	pmin	pmax	vary	name	sigma	dist	optimal
rch_A	0.271051	1.000000e-05	19.842364	True	rch	NaN	NaN	0.271051
rch_a	88.306407	1.000000e-02	10000.000000	True	rch	NaN	NaN	88.306407
rch_f	-0.544182	-2.000000e+00	0.000000	True	rch	NaN	NaN	-0.544182
constant_d	0.942309	NaN	NaN	True	constant	NaN	NaN	0.942309
solver_var	0.050000	1.000000e-10	1.000000	True	solver	1.0	norm	NaN
solver_phi	0.500000	1.000000e-10	0.999990	True	solver	1.0	norm	NaN

Depending on the likelihood function, a number of additional parameters need to be inferred. These parameters are added to the Pastas Model instance. The solver also add new columns to the parameter DataFrame with information necessary for the MCMC method. Using the set_parameter method of the model, these parameters can be changed. It is important to ensure that the ml.parameters is filled with information before using ml.solve.

In this example where we use the GaussianLikelihoodAr1 function the var and phi are estimated; the unknown standard deviation of the errors and the autoregressive parameter.

ml_mc.set_parameter("solver_var", initial=0.0028, vary=False, dist="norm", sigma=0.01)
ml_mc.parameters

	initial	pmin	pmax	vary	name	sigma	dist	optimal
rch_A	0.271051	1.000000e-05	19.842364	True	rch	NaN	NaN	0.271051
rch_a	88.306407	1.000000e-02	10000.000000	True	rch	NaN	NaN	88.306407
rch_f	-0.544182	-2.000000e+00	0.000000	True	rch	NaN	NaN	-0.544182
constant_d	0.942309	NaN	NaN	True	constant	NaN	NaN	0.942309
solver_var	0.002800	1.000000e-10	1.000000	False	solver	0.01	norm	NaN
solver_phi	0.500000	1.000000e-10	0.999990	True	solver	1.00	norm	NaN

2b. Choose and set the priors#

The next step is to choose and set the priors of the parameters. This is done by using the ml.set_parameter method and the dist argument (from distribution). Any distribution from the scipy.stats can be chosen (https://docs.scipy.org/doc/scipy/tutorial/stats/continuous.html), for example uniform, norm, or lognorm. Here, for the sake of the example, we set all prior distributions to a normal distribution. We also set the sigma to the standard error of the parameters. The values for loc and scale in the scipy distributions are taken from the optimal or initial en sigma columns, respectively.

# Set the initial parameters to a normal distribution
ml_mc.set_parameter("constant_d", pmin=0.0, pmax=2.0)

for name in ml_mc.parameters.index:
    ml_mc.set_parameter(name, dist="norm")

for name in ml.parameters.index:
    if name in ml_mc.parameters.index:
        ml_mc.set_parameter(name, sigma=ml.parameters.at[name, "stderr"])

ml_mc.parameters

	initial	pmin	pmax	vary	name	sigma	dist	optimal
rch_A	0.271051	1.000000e-05	19.842364	True	rch	0.021633	norm	0.271051
rch_a	88.306407	1.000000e-02	10000.000000	True	rch	7.491552	norm	88.306407
rch_f	-0.544182	-2.000000e+00	0.000000	True	rch	0.064851	norm	-0.544182
constant_d	0.942309	0.000000e+00	2.000000	True	constant	0.051073	norm	0.942309
solver_var	0.002800	1.000000e-10	1.000000	False	solver	0.010000	norm	NaN
solver_phi	0.500000	1.000000e-10	0.999990	True	solver	1.000000	norm	NaN

Pastas will use the initial value of the parameter for the loc argument of the distribution (e.g., the mean of a normal distribution), and the stderr as the scale argument (e.g., the standard deviation of a normal distribution). Only for the parameters with a uniform distribution, the pmin and pmax values are used to determine a uniform prior. By default, all parameters are assigned a uniform prior.

Note: This means that either the `pmin` and `pmax` should be set for uniform distributions, or the `sigma` for any other distribution. That is why in this example model was first solved using LeastSquares, in order to obtain estimates for the `stderr` to set the `sigma`. In practice, these could also be set based on expert judgement or information about the parameters.

2c. Run the solver and solve the model#

After setting the parameters and creating a EmceeSolve solver instance we are now ready to run the MCMC analysis. We can do this by running ml.solve. We can pass the same parameters that we normally provide to this method (e.g., tmin or fit_constant). Here we use the initial parameters from our least-square solve. Note that we do not fit a noise model, because we already take autocorrelated errors into account through the likelihood function.

All the arguments that are not used by ml.solve, for example steps and tune, are passed on to the run_mcmc method from the sampler (see Emcee docs). The most important is the steps argument, that determines how many steps each of the walkers takes.

# Use the solver to run MCMC
ml_mc.solve(
    initial=False,
    tmin="1990",
    steps=100,
    tune=True,
)

Fit report solver                        Fit Statistics
=======================================================
nwalkers 20                     EVP               81.78
nsteps   100                    R2                 0.82
nobs     351                    RMSE               0.09
tmin     1990-01-01 00:00:00    AICc           -1687.89
tmax     2005-10-14 00:00:00    BIC            -1668.76
freq     D                      Obj                 nan
freq_obs None                   ___                    
warmup   3650 days 00:00:00     Interp.              No

Parameters (5 optimized)
=======================================================
              optimal    initial   vary     sigma  dist
rch_A        0.263206   0.271051   True  0.021633  norm
rch_a       89.199882  88.306407   True  7.491552  norm
rch_f       -0.563485  -0.544182   True  0.064851  norm
constant_d   0.956584   0.942309   True  0.051073  norm
solver_var   0.002800   0.002800  False  0.010000  norm
solver_phi   0.721739   0.500000   True  1.000000  norm

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3. Posterior parameter distributions#

The results from the MCMC analysis are stored in the sampler object, accessible through ml.solver.sampler variable. The object ml.solver.sampler.flatchain contains a Pandas DataFrame with \(n\) the parameter samples, where \(n\) is calculated as follows:

\(n = \frac{\left(\text{steps}-\text{burn}\right)\cdot\text{nwalkers}}{\text{thin}} \)

Corner.py#

Corner is a simple but great python package that makes creating corner graphs easy. A couple of lines of code suffice to create a plot of the parameter distributions and the covariances between the parameters.

# Corner plot of the results
fig = plt.figure(figsize=(8, 8))

labels = list(ml_mc.parameters.index[ml_mc.parameters.vary])

labels = [label.split("_")[1] for label in labels]

best = list(ml_mc.parameters[ml_mc.parameters.vary].optimal)

axes = corner.corner(
    ml_mc.solver.sampler.get_chain(flat=True, discard=50),
    quantiles=[0.025, 0.5, 0.975],
    labelpad=0.1,
    show_titles=True,
    title_kwargs=dict(fontsize=10),
    label_kwargs=dict(fontsize=10),
    max_n_ticks=3,
    fig=fig,
    labels=labels,
    truths=best,
)

plt.show()

../_images/7d51b499819380df00732a73606e18725febdbb39b35e585cab68c362fc47b65.png

4. What happens to the walkers at each step?#

The walkers take steps in different directions for each step. It is expected that after a number of steps, the direction of the step becomes random, as a sign that an optimum has been found. This can be checked by looking at the autocorrelation, which should be insignificant after a number of steps. Below we just show how to obtain the different chains, the interpretation of which is outside the scope of this notebook.

fig, axes = plt.subplots(len(labels), figsize=(10, 7), sharex=True)

samples = ml_mc.solver.sampler.get_chain(flat=True)
for i in range(len(labels)):
    ax = axes[i]
    ax.plot(samples[:, i], "k", alpha=0.5)
    ax.set_xlim(0, len(samples))
    ax.set_ylabel(labels[i])
    ax.yaxis.set_label_coords(-0.1, 0.5)

axes[-1].set_xlabel("step number")

Text(0.5, 0, 'step number')

../_images/c773a8d336c6278a717dd67e890e9f318ffbf09277fdf1026515f0360f624e56.png

5. Plot some simulated time series to display uncertainty?#

We can now draw parameter sets from the chain and simulate the uncertainty in the head simulation.

# Plot results and uncertainty
ax = ml_mc.plot(figsize=(10, 3))
plt.title(None)

chain = ml_mc.solver.sampler.get_chain(flat=True, discard=50)
inds = np.random.randint(len(chain), size=100)
for ind in inds:
    params = chain[ind]
    p = ml.parameters.optimal.copy().to_numpy(copy=True)
    p[ml.parameters.vary] = params[: ml.parameters.vary.sum()]
    _ = ml.simulate(p, tmin="1990").plot(c="gray", alpha=0.1, zorder=-1)

plt.legend(["Measurements", "Simulation", "Ensemble members"], numpoints=3);

../_images/978aac81de01add37fd6b1f950d9e1bdde0b69e0d7491099b474e7f59421fce1.png