26.5.6. Lifetime 2D Fit with Bat

This directory contains an example of how to use BAT for a 2D fit. To run this code, you will need to install BAT. Follow the instructions in Installation instructions to install BAT.

The template for creating these examples was generated by BAT via a the command-line tool provided with BAT:

bat-project Lifetime2DFit Lifetime2DFit

This creates the directory Lifetime2DFit; the header and source files for a model class, Lifetime2DFit; the source code for a program to instantiate and run the model, runLifetime2DFit.cxx; and a Makefile for compiling everything.

To compile the code in this example, run

make

The Data

The data to be analyzed in this example is stored in a file in a directory above this one at ../../example-data/lifetime.root. In this ROOT file is a TTree (DtoKpipi0) with branches named Dz_t and Dz_t_err, containing the measured decay time and decay-time uncertainty for D0 mesons reconstructed via their decay to K pi pi0. This data is entirely signal with no background.

The Model

We perform a fit to the data to learn the D0 lifetime, \(\tau\). We model the two-dimensional data distribution with the likelihood

\[L(\mathrm{data}|\tau,\vec\nu) = \prod_i L(t_i, \sigma_i|\tau,\vec\nu)\]

where \(i\) runs over the data points, each containing a decay time, \(t\), and an uncertainty on it, \(\sigma\); and \(\vec\nu\) is a vector of nuisance parameters that help describe our data distribution. The 2D likelihood function for an individual data point is

\[L(t, \sigma|\tau,\vec\nu) \propto P(t|\sigma,\tau,\vec\nu) \times P(\sigma|\vec\nu).\]

It is the product of a probability density for \(t\) given \(\sigma\) and a probability density for \(\sigma\). The probability for \(t\) is the convolution of the standard decay time distribution, \(\exp(-t'/\tau)\) with a normal distribution for the true decay time \(t'\) given the measured \(t\) and \(\sigma\):

\[P(t | \sigma, \tau, b, s) \propto \int_0^{\infty} e^{-t'/\tau} \mathcal{N}(t-t'|b, s\sigma) \mathrm{d}t',\]

where \(b\) is a bias (offset) in \(t\) and \(s\) is a scaling of the projected uncertainty on \(t\). The integral evaluates to

\[P(t | \sigma, \tau, b, s) \propto e^{-t/\tau} \mathrm{erfc}([(b-t)/s\sigma + s\sigma/\tau]/\sqrt{2}),\]

where erfc is the complimentary error function; and the full function is normalized for fixed \(\sigma\) over the range of \(t\) in the data.

The probability for \(\sigma\) is a Johnson’s \(S_U\) distribution:

\[P(\sigma | \mu, \lambda, \gamma, \delta) \propto e^{-\frac12[\gamma + \delta \sinh^{-1}k(\sigma)]^2} / \sqrt{1 - k^2(\sigma)} \qquad \mathrm{with}\ k(\sigma) \equiv (\sigma - \mu) / \lambda\]

which is normalized over the range of \(\sigma\) in the data.

These functions and their integrals (and some of their logarithms) are contained in Functions.h.

The Code

We code the above model into our class Lifetime2DFit. We will store the data in a vector of data points, each data point an array of two double’s. We will pass this data set to our constructor along with a level of parallelization (to speed up our likelihood).

Inside the constructor (in Lifetime2DFit.cxx), we define the parameters of our model:

// define lifetime parameter
AddParameter("tau", 390e-3, 430e-3, "#tau", "[ps]");

// define decay time shape parameters
AddParameter("b", -10e-3, 10e-3, "bias", "[ps]");
AddParameter("s", 1.2, 1.45, "#sigma scaling");
AddParameter("mu", 25e-3, 33e-3, "#mu", "[ps]");
AddParameter("lambda", 7e-3, 12e-3, "#lambda", "[ps]");
AddParameter("gamma", -3.5, -2.5, "#gamma");
AddParameter("delta", 1.26, 1.56, "#delta");

For each parameter, we give it a name, lower and upper limits, a “pretty” name (in ROOT TeX) for plotting, and a units string for plotting, if applicable. After defining our parameters, we must set priors for them. For simplicity we will take flat priors for all the parameters:

SetPriorConstantAll();

In Lifetime2DFit::LogLikelihood(...) we will loop over all the data points and add their individual log-likelihood to the total. To help this, we define the lambda function add_ll that calculates the log-likelihood for a data point (with the unnormalized Johnson’s \(S_U\) function) and adds it into the sum of log-likelihoods.

We define the lambda function sum_ll which sums up the log-likelihood for a block of data. And then we divide the data set into _NThreads disjoint blocks and perform their sums in parallel using C++’s built in threading system.

Finally we need to normalize our Johnson’s \(S_U\) function, by dividing each data point by the normalization factor. That is: by subtracting from the log the log of the integral for each data point.

In runLifetime2DFit.cxx we load our data from the TTree in the file and instantiate our model with that data:

Lifetime2DFit m("Lifetime2DFit", data, 8)

where we have also specified to calculate our likelihood using eight-fold parallelization. We set our desired precision

m.SetPrecision(BCEngineMCMC::kMedium);

(you can use kQuick if you are impatient). Then we run our analysis, sampling from our posterior and improving the best-fit result with Minuit:

m.MarginalizeAll();
m.FindMode();

And finally print the results to files and to the screen:

m.PrintAllMarginalized(...);
m.PrintSummary();

The Output

The output plots show the posterior probability distributions for the individual parameters in 1D and for pairs of parameters in 2D. The (textual) output summary gives information on the best-fit point and the individual marginalizations. For more details consult the documentation in the Simple1DFit example.

The member function DrawFit(...) in Lifetime2DFit draws the data set as a 2D histogram and the fit in projections of \(t\) and \(\sigma\) at the best-fit parameter point. It also gives the marginalized mean and standard deviation for \(\tau\) and a \(\chi^2/\mathrm{ndf}\) for the data as binned in the displayed 2D histogram.

Prior Selection

In principle, our prior information on the lifetime is the current state of knowledge concerning it: the PDG value. We could set the prior to that knowledge by uncommenting-out the relevant highlighted code in runLifetime2DFit.cxx, most importantly

m.GetParameter("tau").SetPrior(new BCGaussianPrior(0.4101, 0.0015));

If you also uncomment out the block of code that prints the knowledge update, you can see how our data has changed our knowledge from the prior to the posterior. When we run the fit with a flat prior, the result is

\[\tau = (410.8 \pm 3.5)\,\mathrm{fs}\]

This is twice as wide as the prior we have just set, so we should expect only a small improvement from the prior to the posterior. This is indeed what we see

\[\tau = (410.1 \pm 1.5)\,\mathrm{fs} \to (410.3 \pm 1.2)\,\mathrm{fs},\]

which you can visually inspect in Lifetime2DFit_update.pdf.

Caution: If you want your result to be averaged with the results in the PDG (that is, included in the PDG itself), you should not give it a prior based on those results and instead opt for the flat prior. (Or report both in your analysis.)