# A Measure of the goodness of fit in unbinned likelihood fits; end of Bayesianism? Page: 4 of 8

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

PDE in hypercube space. After we find the maxi-

mum likelihood point s*, for which the PDE is not

needed, we transform the variable c -> c', such that

the distribution P(c'ls*) is flat and 0 < c' < 1. The

hypercube transformation can be made even if c is

multi-dimensional by initially going to a set of vari-

ables that are uncorrelated and then making the hy-

percube transformation. The transformation can be

such that any interval in c space maps on to the inter-

val (0,1) in hypercube space. We solve the boundary

problem by imposing periodicity in the hypercube. In

the one dimensional case, we imagine three "hyper-

cubes", each identical to the other on the real axis

in the intervals (-1,0), (0,1) and (1,2). The hyper-

cube of interest is the one in the interval (0, 1). When

the probability from an event kernel leaks outside the

boundary (0, 1), we continue the kernel to the next hy-

percube. Since the hypercubes are identical, this im-

plies the kernel re-appearing in the middle hypercube

but from the opposite boundary. Put mathematically,

the kernel is defined such that(c'- c') = 9(c' - c' - 1); c' > 1

(c'- c') =g(c' - c' +1); c' < 00

N6o

50

4020

102004/01/22 11.24

* h=0

Sh0.10

L s-in -s -5 -4 -2 e 2 4 6 8 10

Negative Log Likelihood RatioFigure 4: The distribution of the negative log likelihood

ratio KCC7Z for the null hypothesis for an ensemble of

500 experiments each with 1000 events, as a function of

the smoothing factor h=0.1, 0.2 and 0.3(13)

that it is also easier to arrive at an analytic theory of

(14) NZR with the choice of this simple kernel.Although a Gaussian Kernel will work on the hyper-

cube, the natural kernel to use considering the shape

of the hypercube would be the function 9(c')9(c') = -; c' < -

9(c') = 0; 1c' > h0

N

w'(15)

(16)This kernel would be subject to the periodic boundary

conditions given above, which further ensure that ev-

ery event in hypercube space is treated exactly as ev-

ery other event irrespective of their co-ordinates. The

parameter h is a smoothing parameter which needs to

be chosen with some care. However, since the theory

distribution is flat in hypercube space, the smoothing

parameter may not need to be iteratively determined

over hypercube space to the extent that data distri-

bution is similar to the theory distribution. Even if

iteration is used, the variation in h in hypercube space

is likely to be much smaller.

Figure 4 shows the distribution of the NER for

the null hypothesis for an ensemble of 500 experiments

each with 1000 events as a function of the smoothing

factor h. It can be seen that the distribution narrows

considerably as the smoothing factor increases. We

choose an operating value of 0.2 for h and study the

dependence of the KNLR as a function of the number

of events ranging from 100 to 1000 events, as shown in

figure 5. The dependence on the number of events is

seen to be weak, indicating good behavior. The PDE

thus arrived computed with h=0.2 can be transformed

from the hypercube space to c space and will repro-

duce data smoothly and with no edge effects. We note2003/11/28 19.51

ize=0.20Hvpereebe smasthina lacter bins

-8

Figure 5: The distribution of the negative log likelihood

ratio NLR for the null hypothesis for an ensemble of

500 experiments each with the smoothing factor h=0.2,

as a function of the number of events

6. End of Bayesianism?

By Bayesianism, we mean the practice of "guess-

ing" a prior distribution and introducing it into the

calculations. In what follows we will show that what

is conventionally thought of as a Bayesian prior dis-

tribution is in reality a number that can be calculated

from the data. We are able to do this since we use

two pdf's, one for theory and one for data. In what4

50 -

25

20

35

30

-10+ 1000 events

4 500 events

o 250 events

X 100 events

0~ . . --6 -4 -2 0 2 4 6 8 10

Negative Log Likelihood RatioMOCT003

0

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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/m1/4/: accessed May 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.