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  • $\begingroup$ I think there is an error in point a. : the posterior does depend on x1, in addition to z, via an indicator function that ensures x1 is positive (the pdf of an exponential random variable being defined for positive values only) ... that is, the posterior must be written as g(z) * 1{x1 > 0} ... or perhaps 1{x1 >= 0}, depending on how the exponential distribution is defined $\endgroup$ Commented Apr 16, 2022 at 16:38
  • $\begingroup$ Why should the posterior be multiplied by this indicator as a function of the parameter? The indicators all equal one, given the observations. $\endgroup$ Commented Apr 16, 2022 at 16:42
  • $\begingroup$ Say you have calculated that posterior, which is supposed to depend on z only. Then I come to you with a vector z, and ask you what's the value of the posterior for that z, at e.g. xi = 1. Then you tell me a positive value. And then I tell you that in fact I calculated z with only negative values for x1, ..., xn. : ) So the postive value should have been a 0 ! And hence knowing z only seems not suffcient for evaluating that posterior. ... and hence in fact we need all the indicators, not just the one for x1 $\endgroup$ Commented Apr 16, 2022 at 16:58
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    $\begingroup$ Please see stats.meta.stackexchange.com/questions/6304/my-upvoting-policy, when you find a question sufficiently clear to write an answer, consider to upvote the question! $\endgroup$ Commented Apr 21, 2022 at 15:04
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    $\begingroup$ If you found this answer helpful, then please consider upvoting and/or accepting it. $\endgroup$ Commented Apr 22, 2022 at 16:11