Multivariate probability distribution

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Discussion Overview

The discussion centers on the properties of multivariate probability distributions, specifically focusing on the role and significance of higher-order cumulants compared to higher-order moments. Participants explore theoretical aspects and implications of these concepts.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants assert that higher-order cumulants in multivariate probability distributions contain information of decreasing significance, unlike higher-order moments.
  • One participant requests clarification on the context of the discussion, indicating a need for more detail about the source or application of the information.
  • It is noted that for normal distributions, cumulants above the second order vanish, suggesting they do not contribute significant information.
  • Another participant emphasizes that the first two cumulants represent means and covariances, and argues against the notion that higher-order moments can be disregarded, citing a mathematical inequality.
  • A later reply reiterates that for normal distributions, cumulants of order higher than two are zero, challenging the idea of decreasing significance by stating there is no significance at all for those cumulants.

Areas of Agreement / Disagreement

The discussion contains competing views regarding the significance of higher-order cumulants, with some participants arguing for decreasing significance while others contend that they hold no significance in certain cases, particularly for normal distributions. There is no consensus on the interpretation of higher-order cumulants.

Contextual Notes

Participants reference mathematical properties and inequalities related to moments and cumulants, but the discussion does not resolve the implications of these properties or the conditions under which they apply.

beman
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"In multivariate probability distribution higher-order cumulants contain information of decreasing significance, unlike higher-order moments".
 
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Can you provide more detail? is this for a course? did you read it in some book? what is the context of your question?
 
For normal distributions cumulants above those of second order vanish, which pretty much means they don't contribute much information.
 
Last edited:
In multivariate probability distribution the first two cumulants are the means and covariances.Higher-order cumulants contain information of decreasing significance, unlike higher-order moments.We cannot set all moments higher than a certain order equal to zero since E(X^2n)>=E(X^n)^2 and thus,all moments contain information about
the lower moments.
 
beman said:
In multivariate probability distribution the first two cumulants are the means and covariances.Higher-order cumulants contain information of decreasing significance, unlike higher-order moments.We cannot set all moments higher than a certain order equal to zero since E(X^2n)>=E(X^n)^2 and thus,all moments contain information about
the lower moments.

Again - for normal distributions, cumulants of order higher than two are zero: it isn't decreasing significance, it is no significance.
 

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