@misc {pub4752,
	title = {A partial information decomposition for discrete and continuous variables},
	author = {Kyle Poland AND Abdullah Makkeh AND Aaron Gutknecht AND Patricia Wollstadt AND Anja Sturm AND Michael Wibral},
	year = {2021},
	abstract = {Understanding the information mechanisms inside dynamical complex systems often poses intricate questions. In neural systems, information is often represented by an ensemble of agents. Knowledge about how this information is distributed amongst those agents can lead to insights about how to distribute relevant information about a problem with respect to the available agents. These agents can, for instance, be taken to be neurons which are recorded during stimulation, one may imagine spike trains connected to neurophysical behavior in a classical center-out-task.
Determining the exact nature of the information that arises when varying the composition of agents is thoroughly answered by partial information decomposition (PID), modelling the agents as so-called source and target random variables.
The framework of PID intends to decompose the multivariate mutual information according to information contributions, such as shared information, unique information, and synergistic information. These categories represent the distinct ways in which a collection of source random variables might contribute information about a specific target random variable, and can vastly enhance the knowledge and understanding of neural systems.
However, in its conceptual generality, particular propositions for PID quantities have so far mostly been defined for systems of purely discrete random variables. 
While recently a quantification for PID in continuous settings for two or three source variables was introduced, no ansatz for quantifying a continuous PID has managed to both cover more than three variables and at the same time assume general, measure-theoretic random variables, such as mixed discrete-continuous, or continuous random variables yet. In this work we will propose such an information quantity, defining the terms of a PID, which is well-defined for an arbitrary finite number or type of source and target random variable. This proposed quantity is tightly related to a recently developed local shared information quantity for discrete random variables based on the idea of shared exclusions. Further, we prove that this newly proposed measure fulfills various desirable properties, crucial for the applicability of this new measure in particular settings of interest for neuroscientists and physicists. We demonstrate that this new measure satisfies (i) a set of local PID axioms, (ii) invariance under invertible transformations, guaranteeing independence of the precise experimental setup and measurement unit of neural recordings, (iii) smoothness with respect to the underlying probability density enabling gradient descent methods for learning in neural networks, and (iv) admitting a target chain rule quantifying the simultaneous treatment multiple neurons for investigating i.e. their statistical cross-dependence on the neurons past.
},
	publisher = {ICMS},
	booktitle = {2021 International Conference on Mathematical Neuroscience (ICMNS2021)}
}