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In a previous question, I explored the derivation of joint likelihood using only conditional independence assumptions - Derivation of Joint Likelihood with Only Conditional Independence Assumptions (Without Marginal Independence) .

Now, I am interested in understanding real-world applications or datasets where the property of conditional independence with marginal dependence is observed.

Questions:

  1. Are there well-known real-world use cases or datasets where this type of relationship appears?
  2. What kinds of statistical models or domains (e.g., economics, medicine, social sciences) commonly encounter such relationships?

I’d appreciate examples, references, or any relevant discussions on this topic to better understand its implications in practice.

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Conditional independence with marginal dependence arises in many real-world scenarios where variables share a common cause or are influenced by latent factors. For example, human symptoms of a disease are often marginally dependent but conditionally independent given the disease. Therefore if the labels $\mathbf{y}_i$ as symptoms such as fever, cough, migraine, and sore throat are marginally dependent within a population from common experience, and for demonstrative simplicity purpose here assuming each symptom $\mathbf{y}_i$ has its own physiological causal condition $\mathbf{X}_i$ which are also marginally dependent from common experience and caused by a common disease, and you have conditional probabilities $p_i(\mathbf{y}_i|\mathbf{X}_i)$ for each pair, then for a specific sick person all physiological conditions $\mathbf{X}_i$ become conditionally independent due to the assumed confounding effect of the said person's disease. You may further read the many references mentioned the linked confounding concept which is a common terminology in statistics.

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  • $\begingroup$ Thanks for the example @cinch that makes sense. I recently saw stats.stackexchange.com/a/214315/408276 and I wanted to check whether "The assumption holds as a consequence of the identically distributed part under the i.i.d. assumption, but it is weaker because we don't make any assumptions about the 𝐗𝑖's." means the same thing ie. conditional independence but marginal dependence? $\endgroup$
    – spie227
    Commented Dec 5 at 17:08
  • $\begingroup$ Indeed your concerned assumptions here are more general than the referenced post. Note in my above example you could have different conditional probabilities $p_i(\mathbf{y}_i|\mathbf{X}_i)$ for each pair as the most general case, while the referenced supervised learning or textbook linear regression model assume a common conditional probability $p(\mathbf{y}_i|\mathbf{X}_i)$ as the discriminative model for each pair to be learnable. I guess it's due to the most general abstract case you envisioned, so few people answer your such related questions. Hope this clarifies this answer. $\endgroup$
    – cinch
    Commented Dec 5 at 20:02

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