# A Measure of the goodness of fit in unbinned likelihood fits; end of Bayesianism? Page: 1 of 8

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FERMILAB-Conf-04/012-E

1

A Measure of the Goodness of Fit in Unbinned Likelihood Fits; End of

Bayesianism?

Rajendran Raja

Fermilab, Batavia, IL 60510, USA

Maximum likelihood fits to data can be done using binned data (histograms) and unbinned data. With binned

data, one gets not only the fitted parameters but also a measure of the goodness of fit. With unbinned data,

currently, the fitted parameters are obtained but no measure of goodness of fit is available. This remains, to

date, an unsolved problem in statistics. Using Bayes' theorem and likelihood ratios, we provide a method by

which both the fitted quantities and a measure of the goodness of fit are obtained for unbinned likelihood fits,

as well as errors in the fitted quantities. The quantity, conventionally interpreted as a Bayesian prior, is seen in

this scheme to be a number not a distribution, that is determined from data.1. Introduction

2. Likelihood ratios

As of the Durham conference [1], the problem of

obtaining a goodness of fit in unbinned likelihood fits

was an unsolved one. In what follows, we will de-

note by the vector s, the theoretical parameters (s for

"signal") and the vector c, the experimentally mea-

sured quantities or "configurations". For simplicity,

we will illustrate the method where both s and c are

one dimensional, though either or both can be multi-

dimensional in practice. We thus define the theo-

retical model by the conditional probability density

P(c s). Then an unbinned maximum likelihood fit to

data is obtained by maximizing the likelihood [2],

i=n

L = 17 P(cils) (1)

i=1

where the likelihood is evaluated at the n observed

data points ci,i = 1,n. Such a fit will determine

the maximum likelihood value s* of the theoretical

parameters, but will not tell us how good the fit is.

The value of the likelihood L at the maximum like-

lihood point does not furnish a goodness of fit, since

the likelihood is not invariant under change of vari-

able. This can be seen by observing that one can

transform the variable set c to a variable set c' such

that P(c's*) is uniformly distributed between 0 and

1. Such a transformation is known as a hypercube

transformation, in multi-dimensions. Other datasets

will yield different values of likelihood in the variable

space c when the likelihood is computed with the orig-

inal function P(cls*). However, in the original hyper-

cube space, the value of the likelihood is unity regard-

less of the dataset c', i = 1, n, thus the likelihood

cannot furnish a goodness of fit by itself, since neither

the likelihood, nor ratios of likelihoods computed us-

ing the same distribution P(cls*) is invariant under

variable transformations. The fundamental reason for

this non-invariance is that only a single distribution,

namely, P(cls*) is being used to compute the goodness

of fit.In binned likelihood cases, where one is comparing

a theoretical distribution P(cls) with a binned his-

togram, there are two distributions involved, the theo-

retical distribution and the data distribution. The pdf

of the data is approximated by the bin contents of the

histogram normalized to unity. If the data consists of

n events, the pdf of the data Pdta (c) is defined in the

frequentist sense as the normalized density distribu-

tion in c space of n events as n -> oc. In the binned

case, we can bin in finer and finer bins as n -+ oc and

obtain a smooth function, which we define as the pdf

of the data Pdete(c). In practice, one is always lim-

ited by statistics and the binned function will be an

approximation to the true pdf. We can now define a

likelihood ratio G7, such thatGam, = I7j P(iIs) - P(CnIs)

llJ=1 pdate(C) Pdat(cn)(2)

where we have used the notation Cn to denote the

event set ci, i = 1, n. Let us now note that Lz is

invariant under the variable transformation c -+ c',

sinceP(c'Is) = I d P(cIs)

pdate(c') = dc (Pdata(c)(3)

(4)

(5)and the Jacobian of the transformation - cancels

in the numerator and denominator in the ratio. This

is an extremely important property of the likelihood

ratio ZR that qualifies it to be a goodness of fit vari-

able. Since the denominator Pdete(Cn) is independent

of the theoretical parameters s, both the likelihood ra-

tio and the likelihood maximize at the same point s*.

One can also show [3] that the maximum value of the

likelihood ratio occurs when the theoretical likelihood

P(ci~s) and the data likelihood Pdate(ci) are equal for

all ci.MOCT003

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Raja, Rajendran. A Measure of the goodness of fit in unbinned likelihood fits; end of Bayesianism?, article, March 12, 2004; Batavia, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc783207/m1/1/: accessed May 25, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.