tokens: the sources. Prior information of such properties can be summarized in a joint density distribution P (). If the corresponding sub-hypothesis does not include a multi-messenger detection, then there may not be a requirement for a common source and the source parameters . In that case, if the detectors are independent from each other, the detection outcomes’ probabilities can be expanded as a product.where subscript i runs over different detectors and are the detection outcomes from the i th detector. Similarly, when there is a common source we can expand the detection outcomes’ probabilities for a fixed source as a product for different detectors.There we used the notation for representing a possible set of sources. There can be an additional level of complication related to combinatorics if the sub-hypothesis can be satisfied with different groupings of the detections. To illustrate this, consider the sub-hypothesis of having five detected particles, three of them from a source and all the rest being noise originated. In this case the sub-hypothesis can be satisfied with combinations. In such cases we expand the probabilities aswhere the sum is over all the combinations of detection outcomes satisfying the sub-hypothesis , the sets and are the detection outcomes of individual detections from different sources and is equal to the reciprocal of the total possible combinations for arising from detector i . For example, for a sub-hypothesis with a single source and w particles emitted from that source, and total of W detections; . are found by physics and empirical data. For memoryless detectors, the detections from different sources are independent soThe likelihoods are found via the detector characteristics and emission models. Now we look at our second term in Equation ( 3 ), . We transform it by using the Bayes’ rule.The denominator of Equation ( 8 ) cancel with the same term in Equation ( 2 ). P () is the joint density of source parameters being integrated over. The sub-hypothesis probabilities are found via the expected counts from the sources or the noise origin by assuming a source density and using the empirically known noise trigger rate. 4. Use Cases—Example: A Joint Gravitational Wave—High-energy Neutrino—Short Gamma-ray Burst Event Search The method for multi-messenger searches introduced above can be used in all scenarios. Specifically, in high-energy astrophysics, one can search for sources which emit more than one messenger. Those messengers can be in any wavelength in the electromagnetic spectrum, and can be neutrinos, cosmic rays, gravitational waves, or any other messenger. Joint emissions of gravitational waves, high-energy neutrinos and short gamma-rays from a binary neutron star or a neutron star black hole merger (Kimura et al. 2017 ; Berger 2014 ), or a binary black hole merger in a dense medium, such as an AGN disk, or surrounded by an accretion disk can be such examples (Ford et al. 2019a , 2019b ). In this section, we
ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-EntityOfFutureInterest B-EntityOfFutureInterest I-EntityOfFutureInterest B-EntityOfFutureInterest I-EntityOfFutureInterest O O O O O O O B-EntityOfFutureInterest I-EntityOfFutureInterest B-EntityOfFutureInterest I-EntityOfFutureInterest O O B-Wavelength O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O