:orphan:

=====================================
Parameter Inference with MCMC
=====================================

This example (``ex_minf_sketch.py``) walks through a complete Bayesian
parameter inference workflow for a 1-D model using adaptive MCMC.  It
covers data generation, inference configuration, MCMC sampling, and
rich post-processing visualisation.

Source: `ex_minf_sketch.py <https://github.com/sandialabs/pytuq/blob/main/examples/ex_minf_sketch.py>`_


Synthetic Data
--------------

A ``tanh`` truth model is evaluated on a grid and noisy observations are
generated at 15 equally-spaced points:

.. code-block:: python

   true_model = tanh_model          # tanh(3*(x - 0.3))
   npt = 15
   xdata = 0.7 * np.linspace(-1., 1., npt).reshape(-1, 1)
   ydata = true_model(xdata) + 0.1 * np.random.randn(npt, 1)

.. image:: /_static/examples/ex_minf_sketch_data.png
   :width: 50%
   :align: center
   :alt: Synthetic noisy data from the tanh truth model


Calibration Model
-----------------

An exponential model with two parameters is used as the calibration
surrogate:

.. code-block:: python

   def exp_model_single(par, xx):
       return par[1] * np.exp(par[0] * xx) - 2

The model is deliberately simpler than the truth, so the inference must
compensate through its posterior uncertainty.


Inference Configuration
-----------------------

The inference is configured through a single dictionary that controls
every aspect of the calibration:

.. code-block:: python

   model_infer_config = {
       "inpdf_type" : 'pci',          # PC-based internal PDF
       "pc_type"    : 'LU',           # Legendre-Uniform basis
       "outord"     : 1,              # first-order output PC
       "rndind"     : [0, 1],         # both params are random
       "lik_type"   : 'gausmarg',     # Gaussian marginal likelihood
       "pr_type"    : 'uniform',      # uniform prior (unbounded)
       "dv_type"    : 'var_fixed',    # known data variance
       "dv_params"  : [0.01],         # sigma^2 = 0.1^2
       "calib_type" : 'amcmc',        # adaptive MCMC
       "calib_params": {
           'nmcmc': 10000, 't0': 100,
           'tadapt': 100, 'gamma': 0.1
       },
   }


Running MCMC
------------

A single call launches the full inference pipeline — BFGS initialisation
followed by adaptive MCMC:

.. code-block:: python

   calib_results = minf.model_infer(
       ydata, calib_model, xdata, calib_model_pdim,
       **model_infer_config)

The chain, log-posterior, and acceptance rates are returned in
``calib_results``.


Post-Processing
---------------

Posterior model samples are generated by evaluating the calibration model
at the MCMC chain points and collecting predictive statistics:

.. code-block:: python

   checkmode = [calib_results['post'].model, xgrid,
                range(len(xgrid)), md_transform]
   ycheck_pred, samples = minf.model_infer_postp(
       calib_results, checkmode=checkmode,
       nburn=len(calib_results['chain']) // 10,
       nevery=10, nxi=1)


Predictive Fit
--------------

The shaded predictive envelope shows the calibrated model uncertainty
against data and the true model:

.. code-block:: python

   fsamples_2d = np.reshape(samples['fmcmc'], (-1, ngrid), order='F')
   minf.plot_1dfit_shade(xgrid, fsamples_2d.T, xydata=(xdata, ydata))

.. image:: /_static/examples/ex_minf_sketch_fit_shade.png
   :width: 50%
   :align: center
   :alt: Shaded predictive envelope of the calibrated exponential model

A parity plot compares the true model output per grid point with the
calibrated posterior mean:

.. image:: /_static/examples/ex_minf_sketch_parity.png
   :width: 50%
   :align: center
   :alt: Parity plot of true vs predicted values


Parameter Posteriors
--------------------

Marginal and joint posterior distributions of the two exponential-model
parameters are visualised:

.. code-block:: python

   pl.plot_pdfs(samples_=psamples_2d, plot_type='inds')

.. image:: /_static/examples/ex_minf_sketch_posteriors.png
   :width: 50%
   :align: center
   :alt: Joint and marginal posterior distributions of model parameters