# Signal discovery, limits, and uncertainties with sparse on/off measurements: an objective bayesian analysis

## Abstract

For decades researchers have studied the On/Off counting problem where a measured rate consists of two parts. One part is due to a signal process and the other is due to a background process, the magnitudes for both of which are unknown. While most frequentist methods are adequate for large number counts, they cannot be applied to sparse data. Here, I want to present a new objective Bayesian solution that only depends on three parameters: the number of events in the signal region, the number of events in the background region, and the ratio of the exposure for both regions. First, the probability of the counts only being due to background is derived analytically. Second, the marginalized posterior for the signal parameter is also derived analytically. With this two-step approach it is easy to calculate the signal's significance, strength, uncertainty, or upper limit in a unified way. This approach is valid without restrictions for any number count, including zero, and may be widely applied in particle physics, cosmic-ray physics, and high-energy astrophysics. In order to demonstrate the performance of this approach, I apply the method to gamma-ray burst data.

- Authors:

- Institute for Particle Physics, ETH Zurich, 8093 Zurich (Switzerland)

- Publication Date:

- OSTI Identifier:
- 22365511

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Astrophysical Journal; Journal Volume: 790; Journal Issue: 2; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; ASTROPHYSICS; COSMIC GAMMA BURSTS; GAMMA RADIATION; MATHEMATICAL SOLUTIONS; PARTICLES; PROBABILITY

### Citation Formats

```
Knoetig, Max L., E-mail: mknoetig@phys.ethz.ch.
```*Signal discovery, limits, and uncertainties with sparse on/off measurements: an objective bayesian analysis*. United States: N. p., 2014.
Web. doi:10.1088/0004-637X/790/2/106.

```
Knoetig, Max L., E-mail: mknoetig@phys.ethz.ch.
```*Signal discovery, limits, and uncertainties with sparse on/off measurements: an objective bayesian analysis*. United States. doi:10.1088/0004-637X/790/2/106.

```
Knoetig, Max L., E-mail: mknoetig@phys.ethz.ch. Fri .
"Signal discovery, limits, and uncertainties with sparse on/off measurements: an objective bayesian analysis". United States.
doi:10.1088/0004-637X/790/2/106.
```

```
@article{osti_22365511,
```

title = {Signal discovery, limits, and uncertainties with sparse on/off measurements: an objective bayesian analysis},

author = {Knoetig, Max L., E-mail: mknoetig@phys.ethz.ch},

abstractNote = {For decades researchers have studied the On/Off counting problem where a measured rate consists of two parts. One part is due to a signal process and the other is due to a background process, the magnitudes for both of which are unknown. While most frequentist methods are adequate for large number counts, they cannot be applied to sparse data. Here, I want to present a new objective Bayesian solution that only depends on three parameters: the number of events in the signal region, the number of events in the background region, and the ratio of the exposure for both regions. First, the probability of the counts only being due to background is derived analytically. Second, the marginalized posterior for the signal parameter is also derived analytically. With this two-step approach it is easy to calculate the signal's significance, strength, uncertainty, or upper limit in a unified way. This approach is valid without restrictions for any number count, including zero, and may be widely applied in particle physics, cosmic-ray physics, and high-energy astrophysics. In order to demonstrate the performance of this approach, I apply the method to gamma-ray burst data.},

doi = {10.1088/0004-637X/790/2/106},

journal = {Astrophysical Journal},

number = 2,

volume = 790,

place = {United States},

year = {Fri Aug 01 00:00:00 EDT 2014},

month = {Fri Aug 01 00:00:00 EDT 2014}

}