{
"cells": [
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"source": [
"# rf604_constraints\n",
"Likelihood and minimization: fitting with constraints\n",
"\n",
"\n",
"\n",
"\n",
"**Author:** Clemens Lange, Wouter Verkerke (C++ version) \n",
"This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Tuesday, May 19, 2026 at 08:33 PM."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "f1f97dab",
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"execution": {
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},
"outputs": [],
"source": [
"import ROOT"
]
},
{
"cell_type": "markdown",
"id": "8298d218",
"metadata": {},
"source": [
"Create model and dataset\n",
"----------------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "589c1526",
"metadata": {},
"source": [
"Construct a Gaussian pdf"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "4877c595",
"metadata": {
"collapsed": false,
"execution": {
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"shell.execute_reply": "2026-05-19T20:33:07.301877Z"
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"outputs": [],
"source": [
"x = ROOT.RooRealVar(\"x\", \"x\", -10, 10)\n",
"\n",
"m = ROOT.RooRealVar(\"m\", \"m\", 0, -10, 10)\n",
"s = ROOT.RooRealVar(\"s\", \"s\", 2, 0.1, 10)\n",
"gauss = ROOT.RooGaussian(\"gauss\", \"gauss(x,m,s)\", x, m, s)"
]
},
{
"cell_type": "markdown",
"id": "c899b6a2",
"metadata": {},
"source": [
"Construct a flat pdf (polynomial of 0th order)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "dc68dfca",
"metadata": {
"collapsed": false,
"execution": {
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"shell.execute_reply": "2026-05-19T20:33:07.415372Z"
}
},
"outputs": [],
"source": [
"poly = ROOT.RooPolynomial(\"poly\", \"poly(x)\", x)"
]
},
{
"cell_type": "markdown",
"id": "88d7f98b",
"metadata": {},
"source": [
"model = f*gauss + (1-f)*poly"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "1bf98fdc",
"metadata": {
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"shell.execute_reply": "2026-05-19T20:33:07.599527Z"
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"outputs": [],
"source": [
"f = ROOT.RooRealVar(\"f\", \"f\", 0.5, 0.0, 1.0)\n",
"model = ROOT.RooAddPdf(\"model\", \"model\", [gauss, poly], [f])"
]
},
{
"cell_type": "markdown",
"id": "01c3d537",
"metadata": {},
"source": [
"Generate small dataset for use in fitting below"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "d8511145",
"metadata": {
"collapsed": false,
"execution": {
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"shell.execute_reply": "2026-05-19T20:33:07.755922Z"
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},
"outputs": [],
"source": [
"d = model.generate({x}, 50)"
]
},
{
"cell_type": "markdown",
"id": "7eecb557",
"metadata": {},
"source": [
"Create constraint pdf\n",
"-----------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "7fe1e579",
"metadata": {},
"source": [
"Construct Gaussian constraint pdf on parameter f at 0.8 with\n",
"resolution of 0.1"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "30c9c5ca",
"metadata": {
"collapsed": false,
"execution": {
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"shell.execute_reply": "2026-05-19T20:33:07.872343Z"
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},
"outputs": [],
"source": [
"fconstraint = ROOT.RooGaussian(\"fconstraint\", \"fconstraint\", f, 0.8, 0.1)"
]
},
{
"cell_type": "markdown",
"id": "d56c8483",
"metadata": {},
"source": [
"Method 1 - add internal constraint to model\n",
"-------------------------------------------------------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "40b30c8a",
"metadata": {},
"source": [
"Multiply constraint term with regular pdf using ROOT.RooProdPdf Specify in\n",
"fitTo() that internal constraints on parameter f should be used"
]
},
{
"cell_type": "markdown",
"id": "dec799af",
"metadata": {},
"source": [
"Multiply constraint with pdf"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "fc38ab86",
"metadata": {
"collapsed": false,
"execution": {
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"shell.execute_reply": "2026-05-19T20:33:07.987106Z"
}
},
"outputs": [],
"source": [
"modelc = ROOT.RooProdPdf(\"modelc\", \"model with constraint\", [model, fconstraint])"
]
},
{
"cell_type": "markdown",
"id": "edc17abb",
"metadata": {},
"source": [
"Fit model (without use of constraint term)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "2259999e",
"metadata": {
"collapsed": false,
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#1] INFO:Fitting -- RooAbsPdf::fitTo(model) fixing normalization set for coefficient determination to observables in data\n",
"[#1] INFO:Fitting -- using generic CPU library compiled with no vectorizations\n",
"[#1] INFO:Fitting -- Creation of NLL object took 769.964 μs\n",
"[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_model_modelData) Summation contains a RooNLLVar, using its error level\n",
"[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only\n"
]
}
],
"source": [
"r1 = model.fitTo(d, Save=True, PrintLevel=-1)"
]
},
{
"cell_type": "markdown",
"id": "53ff8306",
"metadata": {},
"source": [
"Fit modelc with constraint term on parameter f"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "0f854c9d",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:33:08.182097Z",
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"shell.execute_reply": "2026-05-19T20:33:08.292289Z"
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#1] INFO:Minimization -- Including the following constraint terms in minimization: (fconstraint)\n",
"[#1] INFO:Minimization -- The global observables are not defined , normalize constraints with respect to the parameters (f)\n",
"[#1] INFO:Fitting -- RooAbsPdf::fitTo(modelc) fixing normalization set for coefficient determination to observables in data\n",
"[#1] INFO:Fitting -- Creation of NLL object took 1.06354 ms\n",
"[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_modelc_modelData) Summation contains a RooNLLVar, using its error level\n",
"[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only\n"
]
}
],
"source": [
"r2 = modelc.fitTo(d, Constrain={f}, Save=True, PrintLevel=-1)"
]
},
{
"cell_type": "markdown",
"id": "0682c487",
"metadata": {},
"source": [
"Method 2 - specify external constraint when fitting\n",
"------------------------------------------------------------------------------------------"
]
},
{
"cell_type": "markdown",
"id": "4aa925cb",
"metadata": {},
"source": [
"Construct another Gaussian constraint pdf on parameter f at 0.8 with\n",
"resolution of 0.1"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "257408bc",
"metadata": {
"collapsed": false,
"execution": {
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"shell.execute_reply": "2026-05-19T20:33:08.397019Z"
}
},
"outputs": [],
"source": [
"fconstext = ROOT.RooGaussian(\"fconstext\", \"fconstext\", f, 0.2, 0.1)"
]
},
{
"cell_type": "markdown",
"id": "b7320ead",
"metadata": {},
"source": [
"Fit with external constraint"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "5e223a72",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:33:08.399095Z",
"iopub.status.busy": "2026-05-19T20:33:08.398969Z",
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"shell.execute_reply": "2026-05-19T20:33:08.510083Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[#1] INFO:Minimization -- Including the following constraint terms in minimization: (fconstext)\n",
"[#1] INFO:Minimization -- The global observables are not defined , normalize constraints with respect to the parameters (f,m,s)\n",
"[#1] INFO:Fitting -- RooAbsPdf::fitTo(model) fixing normalization set for coefficient determination to observables in data\n",
"[#1] INFO:Fitting -- Creation of NLL object took 227.822 μs\n",
"[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_model_modelData) Summation contains a RooNLLVar, using its error level\n",
"[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only\n"
]
}
],
"source": [
"r3 = model.fitTo(d, ExternalConstraints={fconstext}, Save=True, PrintLevel=-1)"
]
},
{
"cell_type": "markdown",
"id": "585d4674",
"metadata": {},
"source": [
"Print the fit results"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "5e230466",
"metadata": {
"collapsed": false,
"execution": {
"iopub.execute_input": "2026-05-19T20:33:08.511976Z",
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},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"fit result without constraint (data generated at f=0.5)\n",
"fit result with internal constraint (data generated at f=0.5, is f=0.8+/-0.2)\n",
"fit result with (another) external constraint (data generated at f=0.5, is f=0.2+/-0.1)\n",
"\n",
" RooFitResult: minimized FCN value: 134.849, estimated distance to minimum: 2.11658e-05\n",
" covariance matrix quality: Full, accurate covariance matrix\n",
" Status : MINIMIZE=0 HESSE=0 \n",
"\n",
" Floating Parameter InitialValue FinalValue +/- Error GblCorr.\n",
" -------------------- ------------ -------------------------- --------\n",
" f 5.0000e-01 6.4987e-01 +/- 1.21e-01 \n",
" m 0.0000e+00 7.1824e-01 +/- 4.97e-01 \n",
" s 2.0000e+00 2.1880e+00 +/- 4.61e-01 \n",
"\n",
"\n",
" RooFitResult: minimized FCN value: 133.901, estimated distance to minimum: 5.96672e-06\n",
" covariance matrix quality: Full, accurate covariance matrix\n",
" Status : MINIMIZE=0 HESSE=0 \n",
"\n",
" Floating Parameter InitialValue FinalValue +/- Error GblCorr.\n",
" -------------------- ------------ -------------------------- --------\n",
" f 6.4987e-01 7.3784e-01 +/- 7.53e-02 \n",
" m 7.1824e-01 6.6350e-01 +/- 5.04e-01 \n",
" s 2.1880e+00 2.3885e+00 +/- 4.95e-01 \n",
"\n",
"\n",
" RooFitResult: minimized FCN value: 137.195, estimated distance to minimum: 0.000153955\n",
" covariance matrix quality: Full, accurate covariance matrix\n",
" Status : MINIMIZE=0 HESSE=0 \n",
"\n",
" Floating Parameter InitialValue FinalValue +/- Error GblCorr.\n",
" -------------------- ------------ -------------------------- --------\n",
" f 7.3784e-01 3.6215e-01 +/- 8.17e-02 \n",
" m 6.6350e-01 7.0071e-01 +/- 5.88e-01 \n",
" s 2.3885e+00 1.6987e+00 +/- 4.78e-01 \n",
"\n"
]
}
],
"source": [
"print(\"fit result without constraint (data generated at f=0.5)\")\n",
"r1.Print(\"v\")\n",
"print(\"fit result with internal constraint (data generated at f=0.5, is f=0.8+/-0.2)\")\n",
"r2.Print(\"v\")\n",
"print(\"fit result with (another) external constraint (data generated at f=0.5, is f=0.2+/-0.1)\")\n",
"r3.Print(\"v\")"
]
}
],
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