In Bayesian statistics, we usually assume that data $X_{1},...,X_{n}$ are independent to each other. But the predictive distribution shows that they are not independent because we have $p(X_{n+1}|X_{1},...,X_{n})$ depends on the historical data. So, is this a contradiction?

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    $\begingroup$ The assumption of independence when made is conditional on the parameter. The predictive integrates out the parameter, so there is no contradiction. $\endgroup$ – Xi'an May 23 '18 at 10:20
  • $\begingroup$ @Xi'an Thanks for your comments professor. So, from the perspective of a user who updates with Bayesian techniques, $X_1,...,X_n$ are dependent (since he thinks the unknown parameters are random and should be inferred from the historical data). But given the true parameter, we can assume they are in fact independent. Am i right? $\endgroup$ – IronMan May 24 '18 at 2:51
  • $\begingroup$ Check out this answer: stats.stackexchange.com/a/34474/87365 $\endgroup$ – HStamper May 25 '18 at 6:27

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