# A Measure of the goodness of fit in unbinned likelihood fits; end of Bayesianism? Page: 3 of 8

This
**article**
is part of the collection entitled:
Office of Scientific & Technical Information Technical Reports and
was provided to Digital Library
by the UNT Libraries Government Documents Department.

#### Extracted Text

The following text was automatically extracted from the image on this page using optical character recognition software:

PHYSTAT2003, SLAC, Stanford, California, September 8-11,2003

v

6

N

0

v

w100

90

0

70

60

50

40

30

20

10

02002/06/06 12.54

Generated events and PDE comparison

T

II

t...

0 0.5 1 1,5 2 2.5 3 3.5 4 4.5 5

Time (arbitrary units)Figure 1: Figure shows the histogram (with errors) of

generated events. Superimposed is the theoretical curve

P(cls) and the PDE estimator (solid) histogram with no

errors.v

.5

6

c

s200

160

160

140

120

100

80

60

40

20

02002/06/06 12.53

Generated events and PDE comparison

- I

-0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (arbitrary units)Figure 2: Figure shows the histogram (with errors) of

1000 events in the fiducial interval 1.0 < c < 5.0

generated as an exponential with decay constant s=1.0

with a superimposed Gaussian of 500 events centered at

c=2.0 and width=0.2. The PDE estimator is the (solid)

histogram with no errors.

to estimate the number of -'s the observed KLICR. is

from the null case. Table I also gives the results of

a binned fit on the same "data". It can be seen that

the unbinned fit gives a 3- discrimination when the

number of Gaussian events is 85, where as the binned

fit gives a y2/ndf of 42/39 for the same case. We in-

tend to make these tests more sophisticated in future

work.

Figure 3 shows the variation of -log P(cn s) and -log

PPDE(cn) for an ensemble of 500 experiments each

with the number of events n = 1000 in the exponen-

tial and no events in the Gaussian (null hypothesis).Table I Goodness of fit results from unbinned likelihood

and binned likelihood fits for various data samples. The

negative values for the number of standard deviations in

some of the examples is due to statistical fluctuation.

Number of Unbinned fit Unbinned fit Binned fit X2

Gaussian events NCR Na 39 d.o.f.

500 189. 103 304

250 58.6 31 125

100 11.6 4.9 48

85 8.2 3.0 42

75 6.3 1.9 38

50 2.55 -0.14 30

0 0.44 -1.33 24

It can be seen that -log P(culs) and -log PPDE(cn)

are correlated with each other and the difference be-

tween the two (-log KZZJZ) is a much narrower dis-

tribution than either and provides the goodness of fit

discrimination.2002/08/29 14.23

40 35

35 35 -

25 (a) 25 (b)

20 -20 -

15 15 -

25 -5

a 1 L1....

800 850 900 950 1000 800 850 900 950 1000

1000 -lo theory likelihood -lo PDE likelihood

9 15 5 .., 70 --

o 960

940 -()50 d

- 9 2 0 )4 0 -- (d )

900-

Q- 880 - tx 30 -

860 20

845

80 10 -

800 0 ' ' ' ' ' ' ' ' O ' '

800 850 900 950 1000 -5 0 5 10 15

-log theory likelihood -log likelihood ratioFigure 3: (a) shows the distribution of the negative

log-likelihood -loge(P(cn0s)) for an ensemble of

experiments where data and experiment are expected to

fit. (b) Shows the negative log PDE likelihood

-loge(P(cn)) for the same data (c) Shows the correlation

between the two and (d) Shows the negative

log-likelihood ratio NVLR that is obtained by

subtracting (b) from (a) on an event by event basis.

5.1. Improving the PDE

The PDE technique we have used so far suffers from

two drawbacks; firstly, the smoothing parameter has

to be iteratively adjusted significantly over the full

range of the variable c, since the distribution P(cls)

changes significantly over that range; and secondly,

there are boundary effects at c=0 as shown in fig-

ure 1. Both these flaws are remedied if we define theMOCT003

3

## Upcoming Pages

Here’s what’s next.

## Search Inside

This article can be searched. **Note: **Results may vary based on the legibility of text within the document.

## Tools / Downloads

Get a copy of this page or view the extracted text.

## Citing and Sharing

Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.

### Reference the current page of this Article.

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/m1/3/: accessed May 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.