{ "cells": [ { "cell_type": "code", "execution_count": 1, "id": "0", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:42.917324Z", "iopub.status.busy": "2026-06-30T22:39:42.916958Z", "iopub.status.idle": "2026-06-30T22:39:42.924551Z", "shell.execute_reply": "2026-06-30T22:39:42.922660Z" }, "tags": [ "hide-in-docs" ] }, "outputs": [], "source": [ "# Check whether easydiffraction is installed; install it if needed.\n", "# Required for remote environments such as Google Colab.\n", "import importlib.util\n", "\n", "if importlib.util.find_spec('easydiffraction') is None:\n", " %pip install easydiffraction==0.19.1" ] }, { "cell_type": "markdown", "id": "1", "metadata": {}, "source": [ "# Bayesian Analysis: Tb2TiO7 (`emcee`), HEiDi\n", "\n", "This tutorial demonstrates a practical two-stage workflow for single-crystal\n", "diffraction analysis with EasyDiffraction.\n", "\n", "In the first stage, we run a fast local refinement to obtain a sensible\n", "point estimate and parameter uncertainties. In the second stage, we use\n", "these refined values to define fit bounds and then sample the posterior\n", "distribution with emcee.\n", "\n", "The example uses constant-wavelength neutron single-crystal diffraction data\n", "for Tb2TiO7 measured on HEiDi at FRM II.\n", "\n", "The goal is not only to obtain a good fit, but also to answer Bayesian\n", "questions such as:\n", "\n", "- Which parameter values are most probable?\n", "- How broad are the credible intervals?\n", "- Which parameters are strongly correlated?\n", "- How much uncertainty propagates into the calculated reflection\n", " intensities?" ] }, { "cell_type": "markdown", "id": "2", "metadata": {}, "source": [ "## 🛠️ Import Library" ] }, { "cell_type": "code", "execution_count": 2, "id": "3", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:42.926477Z", "iopub.status.busy": "2026-06-30T22:39:42.926259Z", "iopub.status.idle": "2026-06-30T22:39:46.878423Z", "shell.execute_reply": "2026-06-30T22:39:46.877548Z" } }, "outputs": [], "source": [ "import easydiffraction as edi" ] }, { "cell_type": "markdown", "id": "4", "metadata": {}, "source": [ "## 📦 Define Project\n", "\n", "The project object keeps structures, experiments, fit settings, and\n", "plotting utilities together in a single place. We will build the full\n", "workflow inside this object." ] }, { "cell_type": "code", "execution_count": 3, "id": "5", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:46.880211Z", "iopub.status.busy": "2026-06-30T22:39:46.879899Z", "iopub.status.idle": "2026-06-30T22:39:47.117667Z", "shell.execute_reply": "2026-06-30T22:39:47.116882Z" } }, "outputs": [], "source": [ "project = edi.Project(name='tbti_heidi_emcee')" ] }, { "cell_type": "code", "execution_count": 4, "id": "6", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.119809Z", "iopub.status.busy": "2026-06-30T22:39:47.119637Z", "iopub.status.idle": "2026-06-30T22:39:47.237461Z", "shell.execute_reply": "2026-06-30T22:39:47.236725Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\u001b[1;36mSaving project 📦 \u001b[0m\u001b[32m'tbti_heidi_emcee'\u001b[0m\u001b[1;36m to \u001b[0m\u001b[32m'../../../projects/bayesian-emcee-tbti-heidi'\u001b[0m\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "├── 📄 project.edi\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "├── 📁 structures/\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "├── 📁 experiments/\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "├── 📁 analysis/\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "│ └── 📄 analysis.edi\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "└── 📁 reports/\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ " └── 📄 tbti_heidi_emcee.html\n" ] } ], "source": [ "project.save_as(dir_path='projects/bayesian-emcee-tbti-heidi')" ] }, { "cell_type": "markdown", "id": "7", "metadata": {}, "source": [ "## 🧩 Define Structure\n", "\n", "For this example we start from a CIF file describing the Tb2TiO7\n", "pyrochlore structure. Loading the structure from CIF is convenient\n", "because it preserves a realistic starting\n", "model without rebuilding the full structure by hand." ] }, { "cell_type": "code", "execution_count": 5, "id": "8", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.238919Z", "iopub.status.busy": "2026-06-30T22:39:47.238774Z", "iopub.status.idle": "2026-06-30T22:39:47.244577Z", "shell.execute_reply": "2026-06-30T22:39:47.244171Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\u001b[1;36mGetting data\u001b[0m\u001b[1;36m...\u001b[0m\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "Data \u001b[32m'struct-tbti'\u001b[0m: Tb2Ti2O7 \u001b[1m(\u001b[0mcrystal structure\u001b[1m)\u001b[0m\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "✅ Data \u001b[32m'struct-tbti'\u001b[0m already present at \u001b[32m'../../../data/struct-tbti.cif'\u001b[0m. Keeping existing.\n" ] } ], "source": [ "structure_path = edi.download_data('struct-tbti', destination='data')" ] }, { "cell_type": "code", "execution_count": 6, "id": "9", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.246391Z", "iopub.status.busy": "2026-06-30T22:39:47.246234Z", "iopub.status.idle": "2026-06-30T22:39:47.258868Z", "shell.execute_reply": "2026-06-30T22:39:47.258068Z" } }, "outputs": [], "source": [ "project.structures.add_from_cif_path(structure_path)" ] }, { "cell_type": "code", "execution_count": 7, "id": "10", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.260509Z", "iopub.status.busy": "2026-06-30T22:39:47.260348Z", "iopub.status.idle": "2026-06-30T22:39:47.263415Z", "shell.execute_reply": "2026-06-30T22:39:47.262607Z" } }, "outputs": [], "source": [ "structure = project.structures['tbti']" ] }, { "cell_type": "markdown", "id": "11", "metadata": {}, "source": [ "Render the structure to confirm the pyrochlore model loaded from CIF as\n", "expected before configuring the experiment." ] }, { "cell_type": "code", "execution_count": 8, "id": "12", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.264774Z", "iopub.status.busy": "2026-06-30T22:39:47.264614Z", "iopub.status.idle": "2026-06-30T22:39:47.470144Z", "shell.execute_reply": "2026-06-30T22:39:47.468822Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\u001b[1;36mStructure 🧩 \u001b[0m\u001b[32m'tbti'\u001b[0m\u001b[1;36m \u001b[0m\u001b[1;36m(\u001b[0m\u001b[1;36mAtom view type: \u001b[0m\u001b[32m'covalent'\u001b[0m\u001b[1;36m)\u001b[0m\n" ] }, { "data": { "text/html": [ "
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Keeping existing.\n" ] } ], "source": [ "data_path = edi.download_data('meas-tbti-heidi', destination='data')" ] }, { "cell_type": "code", "execution_count": 10, "id": "15", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.479498Z", "iopub.status.busy": "2026-06-30T22:39:47.479348Z", "iopub.status.idle": "2026-06-30T22:39:47.605577Z", "shell.execute_reply": "2026-06-30T22:39:47.604858Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\u001b[1;36mData loaded successfully\u001b[0m\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "Experiment 🔬 \u001b[32m'heidi'\u001b[0m. Number of data points: \u001b[1;36m220\u001b[0m.\n" ] } ], "source": [ "project.experiments.add_from_data_path(\n", " name='heidi',\n", " data_path=data_path,\n", " sample_form='single crystal',\n", " beam_mode='constant wavelength',\n", " radiation_probe='neutron',\n", ")" ] }, { "cell_type": "code", "execution_count": 11, "id": "16", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.613173Z", "iopub.status.busy": "2026-06-30T22:39:47.612953Z", "iopub.status.idle": "2026-06-30T22:39:47.616621Z", "shell.execute_reply": "2026-06-30T22:39:47.615972Z" } }, "outputs": [], "source": [ "experiment = project.experiments['heidi']" ] }, { "cell_type": "markdown", "id": "17", "metadata": {}, "source": [ "Link the crystal structure to the experiment and set its scale factor." ] }, { "cell_type": "code", "execution_count": 12, "id": "18", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.619643Z", "iopub.status.busy": "2026-06-30T22:39:47.619459Z", "iopub.status.idle": "2026-06-30T22:39:47.628377Z", "shell.execute_reply": "2026-06-30T22:39:47.624864Z" } }, "outputs": [], "source": [ "experiment.linked_structure.structure_id = 'tbti'\n", "experiment.linked_structure.scale = 1.0" ] }, { "cell_type": "markdown", "id": "19", "metadata": {}, "source": [ "Set the instrument wavelength and starting extinction parameters.\n", "These values provide the initial experiment description for the local\n", "refinement." ] }, { "cell_type": "code", "execution_count": 13, "id": "20", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.630307Z", "iopub.status.busy": "2026-06-30T22:39:47.630115Z", "iopub.status.idle": "2026-06-30T22:39:47.635502Z", "shell.execute_reply": "2026-06-30T22:39:47.633083Z" } }, "outputs": [], "source": [ "experiment.instrument.setup_wavelength = 0.793" ] }, { "cell_type": "code", "execution_count": 14, "id": "21", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.637308Z", "iopub.status.busy": "2026-06-30T22:39:47.637139Z", "iopub.status.idle": "2026-06-30T22:39:47.642367Z", "shell.execute_reply": "2026-06-30T22:39:47.640962Z" } }, "outputs": [], "source": [ "experiment.extinction.mosaicity = 35000\n", "experiment.extinction.radius = 10" ] }, { "cell_type": "markdown", "id": "22", "metadata": {}, "source": [ "## 🚀 Initial Refinement\n", "\n", "Before Bayesian sampling, it is useful to run a deterministic fit. This\n", "gives us:\n", "\n", "- a good point estimate near the best-fit region,\n", "- uncertainties from the local optimizer,\n", "- a quick check that the model and experiment are configured\n", " sensibly.\n", "\n", "In this tutorial we refine a small set of structural and extinction\n", "parameters while keeping occupancies fixed." ] }, { "cell_type": "code", "execution_count": 15, "id": "23", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.644108Z", "iopub.status.busy": "2026-06-30T22:39:47.643928Z", "iopub.status.idle": "2026-06-30T22:39:47.652684Z", "shell.execute_reply": "2026-06-30T22:39:47.651947Z" } }, "outputs": [], "source": [ "structure.atom_sites['O1'].fract_x.free = True\n", "\n", "structure.atom_sites['Ti'].occupancy.free = False\n", "structure.atom_sites['O1'].occupancy.free = False\n", "structure.atom_sites['O2'].occupancy.free = False\n", "\n", "structure.atom_sites['Tb'].adp_iso.free = True\n", "structure.atom_sites['Ti'].adp_iso.free = True\n", "structure.atom_sites['O1'].adp_iso.free = True\n", "structure.atom_sites['O2'].adp_iso.free = True" ] }, { "cell_type": "code", "execution_count": 16, "id": "24", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.654395Z", "iopub.status.busy": "2026-06-30T22:39:47.654226Z", "iopub.status.idle": "2026-06-30T22:39:47.660295Z", "shell.execute_reply": "2026-06-30T22:39:47.657927Z" } }, "outputs": [], "source": [ "experiment.linked_structure.scale.free = True\n", "experiment.extinction.radius.free = True" ] }, { "cell_type": "markdown", "id": "25", "metadata": {}, "source": [ "We keep using the default LMFIT Levenberg-Marquardt minimizer as a fast local\n", "optimizer. Its main purpose here is to provide a stable starting point\n", "and uncertainty estimates for the Bayesian run." ] }, { "cell_type": "code", "execution_count": 17, "id": "26", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:47.662191Z", "iopub.status.busy": "2026-06-30T22:39:47.661875Z", "iopub.status.idle": "2026-06-30T22:39:47.671511Z", "shell.execute_reply": "2026-06-30T22:39:47.670463Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\u001b[1;36mMinimizer types\u001b[0m\n" ] }, { "data": { "text/html": [ "
TypeDescription
1bumpsBUMPS library using the default Levenberg-Marquardt method
2bumps (amoeba)BUMPS library with Nelder-Mead simplex method
3bumps (de)BUMPS library with differential evolution method
4bumps (dream)BUMPS library with DREAM Bayesian sampling
5bumps (lm)BUMPS library with Levenberg-Marquardt method
6dfolsDFO-LS library for derivative-free least-squares optimization
7emceeemcee affine-invariant ensemble Bayesian sampling
8lmfitLMFIT library using the default Levenberg-Marquardt method
9lmfit (least_squares)LMFIT library with SciPy's trust region reflective algorithm
10*lmfit (leastsq)LMFIT library with Levenberg-Marquardt least squares method
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5370.7712.7432.9% ↓
6621.0712.71
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "🏆 Best goodness-of-fit \u001b[1m(\u001b[0mreduced χ²\u001b[1m)\u001b[0m is \u001b[1;36m12.71\u001b[0m at iteration \u001b[1;36m61\u001b[0m\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "✅ Fitting complete.\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "\u001b[1;36mSaving project 📦 \u001b[0m\u001b[32m'tbti_heidi_emcee'\u001b[0m\u001b[1;36m to \u001b[0m\u001b[32m'../../../projects/bayesian-emcee-tbti-heidi'\u001b[0m\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "├── 📄 project.edi\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "├── 📁 structures/\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "│ └── 📄 tbti.edi\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "├── 📁 experiments/\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "│ └── 📄 heidi.edi\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "├── 📁 analysis/\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "│ └── 📄 analysis.edi\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "└── 📁 reports/\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ " └── 📄 tbti_heidi_emcee.html\n" ] } ], "source": [ "project.analysis.fit()" ] }, { "cell_type": "markdown", "id": "28", "metadata": {}, "source": [ "The fit-results display summarizes the locally refined values and their\n", "estimated uncertainties." ] }, { "cell_type": "code", "execution_count": 19, "id": "29", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:49.187888Z", "iopub.status.busy": "2026-06-30T22:39:49.187686Z", "iopub.status.idle": "2026-06-30T22:39:49.544753Z", "shell.execute_reply": "2026-06-30T22:39:49.542087Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "⚙️ Settings used:\n" ] }, { "data": { "text/html": [ "
NameValueDescription
1max_iterations1000Maximum solver iterations.
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "📋 Least-squares fit results:\n" ] }, { "data": { "text/html": [ "
MetricValue
1🧪 Minimizerlmfit (leastsq)
2✅ Overall statussuccess
3⏱️ Fitting time (seconds)1.07
4🔁 Iterations59
5📏 Goodness-of-fit (reduced χ²)12.71
6📏 R-factor (Rf, %)7.67
7📏 R-factor squared (Rf², %)8.12
8📏 Weighted R-factor (wR, %)8.52
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "📈 Refined parameters:\n" ] }, { "data": { "text/html": [ "
datablockcategoryentryparameterunitsstartvalues.u.change
1tbtiatom_siteTbadp_isoŲ0.53000.53190.01900.36 % ↑
2tbtiatom_siteTiadp_isoŲ0.48000.47760.03110.50 % ↓
3tbtiatom_siteO1fract_x0.32800.32800.00010.00 % ↓
4tbtiatom_siteO1adp_isoŲ0.45000.45040.01650.09 % ↑
5tbtiatom_siteO2adp_isoŲ0.23000.23750.02593.25 % ↑
6heidiextinctionradiusμm10.000026.46071.0003164.61 % ↑
7heidilinked_structurescale1.00002.92360.0488192.36 % ↑
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "
• start = parameter value before refinement
• value = refined value from least-squares minimization
• s.u. = standard uncertainty (one sigma), from the covariance matrix
• change = relative change from start, in %; ↑ = increase, ↓ = decrease
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "project.display.fit.results()" ] }, { "cell_type": "markdown", "id": "30", "metadata": {}, "source": [ "The correlation plot shows how strongly the refined parameters move\n", "together in the local refinement. The measured-vs-calculated plot shows\n", "how well the refined crystal model reproduces the measured reflection\n", "intensities." ] }, { "cell_type": "code", "execution_count": 20, "id": "31", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:49.547363Z", "iopub.status.busy": "2026-06-30T22:39:49.546630Z", "iopub.status.idle": "2026-06-30T22:39:49.581333Z", "shell.execute_reply": "2026-06-30T22:39:49.580276Z" } }, "outputs": [ { "data": { "text/html": [ "\n", "\n", "
\n", "
Loading plot…
\n", "
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "project.display.fit.correlations()" ] }, { "cell_type": "code", "execution_count": 21, "id": "32", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:49.582848Z", "iopub.status.busy": "2026-06-30T22:39:49.582674Z", "iopub.status.idle": "2026-06-30T22:39:49.598556Z", "shell.execute_reply": "2026-06-30T22:39:49.597667Z" } }, "outputs": [ { "data": { "text/html": [ "
Loading plot…
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "project.display.pattern(expt_name='heidi')" ] }, { "cell_type": "markdown", "id": "33", "metadata": {}, "source": [ "## 🎲 Prepare Sampling\n", "\n", "Bayesian samplers require finite bounds for the free parameters. Instead of\n", "setting them manually, we derive them from the uncertainties estimated\n", "in the local refinement.\n", "\n", "The helper method `set_fit_bounds_from_uncertainty` centers the bounds\n", "on the current parameter value and expands them by a chosen multiple of\n", "the reported uncertainty.\n", "\n", "The default `multiplier` is 4. In this single-crystal tutorial we use\n", "a tighter value of `1.5` to keep the sampling window closer to the\n", "locally refined solution.\n", "\n", "Show unset fit bounds before setting them from the local refinement\n", "uncertainties." ] }, { "cell_type": "code", "execution_count": 22, "id": "34", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:49.600042Z", "iopub.status.busy": "2026-06-30T22:39:49.599874Z", "iopub.status.idle": "2026-06-30T22:39:49.703147Z", "shell.execute_reply": "2026-06-30T22:39:49.699621Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\u001b[1;36mFree parameters for both structures \u001b[0m\u001b[1;36m(\u001b[0m\u001b[1;36m🧩 data blocks\u001b[0m\u001b[1;36m)\u001b[0m\u001b[1;36m and experiments \u001b[0m\u001b[1;36m(\u001b[0m\u001b[1;36m🔬 data blocks\u001b[0m\u001b[1;36m)\u001b[0m\n" ] }, { "data": { "text/html": [ "
datablockcategoryentryparametervalueuncertaintyminmaxunits
1tbtiatom_siteTbadp_iso0.531880.01900-infinfŲ
2tbtiatom_siteTiadp_iso0.477590.03113-infinfŲ
3tbtiatom_siteO1fract_x0.328030.00009-infinf
4tbtiatom_siteO1adp_iso0.450430.01652-infinfŲ
5tbtiatom_siteO2adp_iso0.237470.02588-infinfŲ
6heidiextinctionradius26.460711.00034-infinfμm
7heidilinked_structurescale2.923560.04884-infinf
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "project.display.parameters.free()" ] }, { "cell_type": "markdown", "id": "35", "metadata": {}, "source": [ "Set fit bounds for all free parameters using `multiplier=1.5`. In this\n", "tutorial that means the posterior pair plot will later refer to a\n", "`±1.5 × uncertainty` region in its title. To widen the sampling window,\n", "increase the multiplier explicitly." ] }, { "cell_type": "code", "execution_count": 23, "id": "36", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:49.710114Z", "iopub.status.busy": "2026-06-30T22:39:49.709881Z", "iopub.status.idle": "2026-06-30T22:39:49.722632Z", "shell.execute_reply": "2026-06-30T22:39:49.721878Z" } }, "outputs": [], "source": [ "for param in project.free_parameters:\n", " param.set_fit_bounds_from_uncertainty(multiplier=1.5)" ] }, { "cell_type": "markdown", "id": "37", "metadata": {}, "source": [ "Displaying the free parameters again is a convenient way to confirm\n", "that the fit bounds have been assigned as expected before launching the\n", "sampler." ] }, { "cell_type": "code", "execution_count": 24, "id": "38", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:49.724885Z", "iopub.status.busy": "2026-06-30T22:39:49.724693Z", "iopub.status.idle": "2026-06-30T22:39:49.840569Z", "shell.execute_reply": "2026-06-30T22:39:49.839758Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\u001b[1;36mFree parameters for both structures \u001b[0m\u001b[1;36m(\u001b[0m\u001b[1;36m🧩 data blocks\u001b[0m\u001b[1;36m)\u001b[0m\u001b[1;36m and experiments \u001b[0m\u001b[1;36m(\u001b[0m\u001b[1;36m🔬 data blocks\u001b[0m\u001b[1;36m)\u001b[0m\n" ] }, { "data": { "text/html": [ "
datablockcategoryentryparametervalueuncertaintyminmaxunits
1tbtiatom_siteTbadp_iso0.531880.019000.503380.56038Ų
2tbtiatom_siteTiadp_iso0.477590.031130.430900.52428Ų
3tbtiatom_siteO1fract_x0.328030.000090.327890.32817
4tbtiatom_siteO1adp_iso0.450430.016520.425650.47520Ų
5tbtiatom_siteO2adp_iso0.237470.025880.198660.27629Ų
6heidiextinctionradius26.460711.0003424.9602027.96123μm
7heidilinked_structurescale2.923560.048842.850302.99682
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "project.display.parameters.free()" ] }, { "cell_type": "markdown", "id": "39", "metadata": {}, "source": [ "## 🎲 Run Sampling\n", "\n", "We now switch from the local minimizer to the Bayesian emcee sampler.\n", "\n", "The settings below are intentionally small so the tutorial runs\n", "quickly. For production analysis you would usually increase the number\n", "of steps and often the burn-in as well. emcee also lets you tune how\n", "walkers are initialized, how many walkers are used, and which proposal\n", "move drives the ensemble.\n", "\n", "The `burn` setting is auto-resolved when left unset. Here we override\n", "`steps` with a smaller value to keep the tutorial fast, and the\n", "effective burn-in is recomputed automatically." ] }, { "cell_type": "code", "execution_count": 25, "id": "40", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:49.842141Z", "iopub.status.busy": "2026-06-30T22:39:49.841967Z", "iopub.status.idle": "2026-06-30T22:39:49.848846Z", "shell.execute_reply": "2026-06-30T22:39:49.848040Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\u001b[1;36mMinimizer types\u001b[0m\n" ] }, { "data": { "text/html": [ "
TypeDescription
1bumpsBUMPS library using the default Levenberg-Marquardt method
2bumps (amoeba)BUMPS library with Nelder-Mead simplex method
3bumps (de)BUMPS library with differential evolution method
4bumps (dream)BUMPS library with DREAM Bayesian sampling
5bumps (lm)BUMPS library with Levenberg-Marquardt method
6dfolsDFO-LS library for derivative-free least-squares optimization
7emceeemcee affine-invariant ensemble Bayesian sampling
8lmfitLMFIT library using the default Levenberg-Marquardt method
9lmfit (least_squares)LMFIT library with SciPy's trust region reflective algorithm
10*lmfit (leastsq)LMFIT library with Levenberg-Marquardt least squares method
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "project.analysis.minimizer.show_supported()" ] }, { "cell_type": "code", "execution_count": 26, "id": "41", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:49.850279Z", "iopub.status.busy": "2026-06-30T22:39:49.850124Z", "iopub.status.idle": "2026-06-30T22:39:49.857914Z", "shell.execute_reply": "2026-06-30T22:39:49.857122Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "⚠️ Switching minimizer type removes these settings: \n", " • max_iterations \n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "⚠️ Switching minimizer type adds these settings with defaults: \n", " • burn_in_steps=1000 \n", " • initialization_method='ball' \n", " • parallel_workers=0 \n", " • population_size=32 \n", " • proposal_moves='de' \n", " • random_seed=None \n", " • sampling_steps=5000 \n", " • thinning_interval=1 \n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "\u001b[1;36mCurrent minimizer changed to\u001b[0m\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "emcee\n" ] } ], "source": [ "project.analysis.minimizer.type = 'emcee'" ] }, { "cell_type": "code", "execution_count": 27, "id": "42", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:49.859464Z", "iopub.status.busy": "2026-06-30T22:39:49.859329Z", "iopub.status.idle": "2026-06-30T22:39:49.862663Z", "shell.execute_reply": "2026-06-30T22:39:49.861766Z" } }, "outputs": [], "source": [ "project.analysis.minimizer.sampling_steps = 500 # lower than the default 3000\n", "project.analysis.minimizer.burn_in_steps = 100 # lower than the default 600\n", "project.analysis.minimizer.population_size = 16 # lower than the default 32\n", "project.analysis.minimizer.random_seed = 42 # fixed seed for reproducible output" ] }, { "cell_type": "code", "execution_count": 28, "id": "43", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:39:49.864015Z", "iopub.status.busy": "2026-06-30T22:39:49.863877Z", "iopub.status.idle": "2026-06-30T22:41:44.860663Z", "shell.execute_reply": "2026-06-30T22:41:44.860003Z" } }, "outputs": [ { "data": { "text/html": [], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/javascript": [ "\n", "(function() {\n", " const button = document.getElementById('ed-fit-stop-640429646d9446d6b287049c12ca910d-button');\n", " const status = document.getElementById('ed-fit-stop-640429646d9446d6b287049c12ca910d-status');\n", " const kernelId = '';\n", " if (!button) {\n", " return;\n", " }\n", "\n", " function setStatus(text) {\n", " if (status) {\n", " status.textContent = text;\n", " }\n", " }\n", "\n", " function pageConfig() {\n", " const element = document.getElementById('jupyter-config-data');\n", " if (!element || !element.textContent) {\n", " return {};\n", " }\n", " try {\n", " return JSON.parse(element.textContent);\n", " } catch (error) {\n", " return {};\n", " }\n", " }\n", "\n", " function baseUrl(config) {\n", " const configured = config.baseUrl || config.base_url ||\n", " (window.Jupyter && Jupyter.notebook && Jupyter.notebook.base_url);\n", " if (configured) {\n", " return configured.endsWith('/') ? configured : configured + '/';\n", " }\n", " const markers = ['/lab/', '/notebooks/', '/tree/'];\n", " for (const marker of markers) {\n", " const index = window.location.pathname.indexOf(marker);\n", " if (index >= 0) {\n", " return window.location.pathname.slice(0, index + 1);\n", " }\n", " }\n", " return '/';\n", " }\n", "\n", " function token(config) {\n", " return config.token || new URLSearchParams(window.location.search).get('token') || '';\n", " }\n", "\n", " function cookie(name) {\n", " const prefix = name + '=';\n", " for (const part of document.cookie.split(';')) {\n", " const trimmed = part.trim();\n", " if (trimmed.startsWith(prefix)) {\n", " return decodeURIComponent(trimmed.slice(prefix.length));\n", " }\n", " }\n", " return '';\n", " }\n", "\n", " function notebookPath() {\n", " const decoded = decodeURIComponent(window.location.pathname);\n", " const markers = ['/lab/tree/', '/notebooks/', '/tree/'];\n", " for (const marker of markers) {\n", " const index = decoded.indexOf(marker);\n", " if (index >= 0) {\n", " return decoded.slice(index + marker.length);\n", " }\n", " }\n", " return '';\n", " }\n", "\n", " async function kernelFromSessions(config) {\n", " const url = new URL(baseUrl(config) + 'api/sessions', window.location.origin);\n", " const authToken = token(config);\n", " if (authToken) {\n", " url.searchParams.set('token', authToken);\n", " }\n", " const response = await fetch(url, {credentials: 'same-origin'});\n", " if (!response.ok) {\n", " return '';\n", " }\n", " const sessions = await response.json();\n", " const path = notebookPath();\n", " const session = sessions.find((item) => item.path === path) || sessions[0];\n", " return session && session.kernel ? session.kernel.id : '';\n", " }\n", "\n", " async function interruptKernel(config, resolvedKernelId) {\n", " const url = new URL(\n", " baseUrl(config) + 'api/kernels/' + resolvedKernelId + '/interrupt',\n", " window.location.origin\n", " );\n", " const authToken = token(config);\n", " if (authToken) {\n", " url.searchParams.set('token', authToken);\n", " }\n", " const xsrfToken = cookie('_xsrf');\n", " const headers = {};\n", " if (xsrfToken) {\n", " headers['X-XSRFToken'] = xsrfToken;\n", " }\n", " const response = await fetch(url, {\n", " method: 'POST',\n", " credentials: 'same-origin',\n", " headers: headers\n", " });\n", " return response.ok;\n", " }\n", "\n", " button.addEventListener('click', async function() {\n", " button.disabled = true;\n", " setStatus('Stopping...');\n", " const config = pageConfig();\n", " try {\n", " const resolvedKernelId = kernelId || await kernelFromSessions(config);\n", " if (!resolvedKernelId) {\n", " throw new Error('Could not resolve the current kernel id.');\n", " }\n", " const interrupted = await interruptKernel(config, resolvedKernelId);\n", " if (!interrupted) {\n", " throw new Error('Jupyter Server rejected the interrupt request.');\n", " }\n", " setStatus('Interrupt sent...');\n", " } catch (error) {\n", " button.disabled = false;\n", " setStatus('Use Kernel > Interrupt to stop this fit.');\n", " }\n", " });\n", "})();\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "\u001b[1;36mStandard fitting\u001b[0m\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "📋 Using experiment 🔬 \u001b[32m'heidi'\u001b[0m for \u001b[32m'single'\u001b[0m fitting\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "🚀 Starting fit process with \u001b[32m'emcee'\u001b[0m\u001b[33m...\u001b[0m\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "📈 Bayesian sampling progress:\n" ] }, { "data": { "text/html": [ "
stepprogresstime (s)log posteriorphase
10.34-1354.04pre-processing
234/6005.7%4.09-1354.48burn-in
367/60011.2%9.05-1354.44burn-in
4100/60016.7%12.75-1354.78burn-in
5101/60016.8%12.88-1354.78sampling
6126/60021.0%15.78-1355.34sampling
7151/60025.2%18.55-1354.79sampling
8176/60029.3%21.34-1354.79sampling
9201/60033.5%23.90-1355.65sampling
10226/60037.7%26.42-1354.92sampling
11251/60041.8%28.95-1356.35sampling
12276/60046.0%31.49-1354.44sampling
13301/60050.2%34.03-1354.57sampling
14326/60054.3%36.50-1354.68sampling
15350/60058.3%38.93-1355.26sampling
16375/60062.5%41.40-1355.31sampling
17400/60066.7%43.89-1355.14sampling
18425/60070.8%46.34-1355.43sampling
19450/60075.0%48.85-1355.05sampling
20475/60079.2%51.39-1355.61sampling
21500/60083.3%53.91-1354.88sampling
22525/60087.5%56.26-1354.86sampling
23550/60091.7%58.54-1355.59sampling
24575/60095.8%61.06-1354.60sampling
25600/600100.0%63.61-1354.48sampling
26114.69post-processing
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "✅ Bayesian sampling complete.\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "⚠️ Convergence diagnostics indicate the posterior may be poorly mixed. \n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "\u001b[1;36mSaving project 📦 \u001b[0m\u001b[32m'tbti_heidi_emcee'\u001b[0m\u001b[1;36m to \u001b[0m\u001b[32m'../../../projects/bayesian-emcee-tbti-heidi'\u001b[0m\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "├── 📄 project.edi\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "├── 📁 structures/\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "│ └── 📄 tbti.edi\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "├── 📁 experiments/\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "│ └── 📄 heidi.edi\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "├── 📁 analysis/\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "│ ├── 📄 analysis.edi\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "│ └── 📄 mcmc.h5\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ "└── 📁 reports/\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ " └── 📄 tbti_heidi_emcee.html\n" ] } ], "source": [ "project.analysis.fit()" ] }, { "cell_type": "markdown", "id": "44", "metadata": {}, "source": [ "## 📊 Inspect Results\n", "\n", "The fit-results display now includes sampler settings, convergence\n", "diagnostics, committed parameter values, and posterior summary\n", "statistics." ] }, { "cell_type": "code", "execution_count": 29, "id": "45", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:41:44.862617Z", "iopub.status.busy": "2026-06-30T22:41:44.862429Z", "iopub.status.idle": "2026-06-30T22:41:44.892422Z", "shell.execute_reply": "2026-06-30T22:41:44.891539Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "⚙️ Settings used:\n" ] }, { "data": { "text/html": [ "
NameValueDescription
1sampling_steps500Total sampler iterations per chain.
2burn_in_steps100Sampler iterations discarded as warm-up.
3thinning_interval1Sampler thinning interval.
4population_size16Number of chains or walkers.
5parallel_workers0Worker count; 0 uses all available CPUs.
6initialization_methodballemcee walker initialization method.
7random_seed42Random seed; None uses a system-derived seed.
8proposal_movesdeSingle emcee proposal move; move mixtures are not persisted in v1.
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "📋 Bayesian fit results:\n" ] }, { "data": { "text/html": [ "
MetricValue
1🧪 Sampleremcee
2❌ Overall statusfailed
3💬 Engine messageemcee sampling completed
4⏱️ Fitting time (seconds)114.69
5📏 Goodness-of-fit (reduced χ²)12.71
6📏 R-factor (Rf, %)7.67
7📏 R-factor squared (Rf², %)8.11
8📏 Weighted R-factor (wR, %)8.57
9📉 Best log-posterior-1354.14
10📊 Convergence statusfailed
11📊 Max r-hat1.072
12📊 Min ess bulk376.618
13📊 Draws per chain501
14📊 Chains16
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "📈 Committed parameters:\n" ] }, { "data": { "text/html": [ "
datablockcategoryentryparameterunitsstartvalues.u.change
1tbtiatom_siteTbadp_isoŲ0.53190.53090.00520.18 % ↓
2tbtiatom_siteTiadp_isoŲ0.47760.47740.00840.03 % ↓
3tbtiatom_siteO1fract_x0.32800.32800.00000.00 % ↓
4tbtiatom_siteO1adp_isoŲ0.45040.44970.00440.15 % ↓
5tbtiatom_siteO2adp_isoŲ0.23750.23470.00681.15 % ↓
6heidiextinctionradiusμm26.460726.41500.27360.17 % ↓
7heidilinked_structurescale2.92362.92130.01380.08 % ↓
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• start = parameter value before sampling
• value = estimate written back to the project (best posterior sample)
• s.u. = standard uncertainty (one sigma), posterior standard deviation
• change = relative change from start, in %; ↑ = increase, ↓ = decrease
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "📊 Posterior distribution:\n" ] }, { "data": { "text/html": [ "
datablockcategoryentryparameterunitsmedian95% CIr-hatess bulk
1tbtiatom_siteTbadp_isoŲ0.5325[0.5229, 0.5432]1.060471.6
2tbtiatom_siteTiadp_isoŲ0.4783[0.4620, 0.4945]1.060376.6
3tbtiatom_siteO1fract_x0.3280[0.3280, 0.3281]1.060410.4
4tbtiatom_siteO1adp_isoŲ0.4509[0.4425, 0.4596]1.051396.6
5tbtiatom_siteO2adp_isoŲ0.2368[0.2241, 0.2508]1.044436.8
6heidiextinctionradiusμm26.4967[25.9381, 27.0226]1.072444.2
7heidilinked_structurescale2.9246[2.8984, 2.9529]1.056424.7
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• median = 50th percentile of the marginal posterior
• 95% CI = 95% credible interval (2.5%-97.5%, asymmetric)
• r-hat = Gelman-Rubin diagnostic (good convergence: r-hat <= 1.01)
• ess bulk = bulk effective sample size (typically >= 400)
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⚠️ r-hat > 1.01: Consider longer sampling, better initialization, or reparameterization.
⚠️ ess bulk < 400: Consider longer sampling or reparameterization.
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "project.display.fit.results()" ] }, { "cell_type": "markdown", "id": "46", "metadata": {}, "source": [ "The correlation and posterior-pair plots are complementary:\n", "\n", "- `plot_param_correlations` summarizes pairwise structure in a compact\n", " matrix.\n", "- `plot_posterior_pairs` shows marginal densities on the diagonal and\n", " posterior contours off-diagonal. In this tutorial its title also\n", " reminds you that the display region follows the `±1.5 × uncertainty`\n", " bounds defined above, while numeric subplot ranges are omitted to\n", " keep the grid readable." ] }, { "cell_type": "code", "execution_count": 30, "id": "47", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:41:44.894154Z", "iopub.status.busy": "2026-06-30T22:41:44.893969Z", "iopub.status.idle": "2026-06-30T22:41:44.916111Z", "shell.execute_reply": "2026-06-30T22:41:44.915393Z" } }, "outputs": [ { "data": { "text/html": [ "\n", "\n", "
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" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "project.display.posterior.distribution()" ] }, { "cell_type": "markdown", "id": "51", "metadata": {}, "source": [ "Finally, the posterior predictive plot propagates the sampled\n", "parameter uncertainty into the calculated single-crystal reflection\n", "intensities." ] }, { "cell_type": "code", "execution_count": 33, "id": "52", "metadata": { "execution": { "iopub.execute_input": "2026-06-30T22:41:45.502857Z", "iopub.status.busy": "2026-06-30T22:41:45.502697Z", "iopub.status.idle": "2026-06-30T22:41:45.517924Z", "shell.execute_reply": "2026-06-30T22:41:45.517181Z" } }, "outputs": [ { "data": { "text/html": [ "
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