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=Overview =

=Overview =

The EI metric is primarily used to measure the strength of causal effects in Markov dynamics. Unlike general causal inference theories, EI is used in cases where the dynamics (the Markov transition probability matrix) are known and no unknown variables (i.e., [[Confounders]]) are present. Its core objective is to measure the strength of causal connections, rather than the existence of causal effects. This means EI is more suitable for scenarios where a causal relationship between variables X and Y is already established.

The EI metric is primarily used to measure the strength of causal effects in Markov dynamics. Unlike general causal inference theories, EI is used in cases where the dynamics (the Markov transition probability matrix) are known and no unknown variables (i.e., [[Confounders]]) are present. Its core objective is to measure the strength of causal effects, rather than the existence of causal effects. This means EI is more suitable for scenarios where a causal relationship between variables X and Y is already established.

Formally, EI is a function of the causal mechanism (in a discrete-state [[Markov Chain]], this is the [[Probability Transition Matrix]] of the Markov chain) and is independent of other factors. The formal definition of EI is:

Formally, EI is a function of the causal mechanism (in a discrete-state [[Markov Chain]], this is the [[Probability Transition Matrix]] of the Markov chain) and is independent of other factors. The formal definition of EI is:

It is important to note that unlike EI calculations for Markov chains, the EI here measures the causal connections between two parts of the system, rather than the strength of causal connections across two different time points in the same system.

It is important to note that unlike EI calculations for Markov chains, the EI here measures the causal connections between two parts of the system, rather than the strength of causal connections across two different time points in the same system.

==EI and Other Causal Metrics==

==EI and Other Causal Metrics==

EI is a metric used to measure the strength of causal connections in a causal mechanism. Before the introduction of EI, several causal metrics had already been proposed. So, what is the relationship between EI and these causal measures? As Comolatti and Hoel pointed out in their 2022 paper, many causal metrics, including EI, can be expressed as combinations of two basic elements <ref name=":0">Comolatti, R., & Hoel, E. (2022). Causal emergence is widespread across measures of causation. ''arXiv preprint arXiv:2202.01854''.</ref>. These two basic elements are called "Causal Primitives", which represent '''Sufficiency''' and '''Necessity''' and in causal relationships.

EI is a metric used to measure the strength of causal effects in a causal mechanism. Before the introduction of EI, several causal metrics had already been proposed. So, what is the relationship between EI and these causal measures? As Comolatti and Hoel pointed out in their 2022 paper, many causal metrics, including EI, can be expressed as combinations of two basic elements <ref name=":0">Comolatti, R., & Hoel, E. (2022). Causal emergence is widespread across measures of causation. ''arXiv preprint arXiv:2202.01854''.</ref>. These two basic elements are called "Causal Primitives", which represent '''Sufficiency''' and '''Necessity''' and in causal relationships.

===Definition of Causal Primitives===

===Definition of Causal Primitives===

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