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Are there distributions where no conjugate prior exists?
Is there a necessary and/or sufficient condition which guarantees the existence of a conjugate prior?

Edit:
Why has this question been closed? None of the two questions which are supposed to answer this question actually answers this.

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  • $\begingroup$ I think that conjugate priors exist only for distribution in the exponential family. (I am sure that any exponential family distribution has a conjugate priore, but I'm not as sure about the other way..) $\endgroup$
    – Pohoua
    Nov 27, 2020 at 10:13
  • $\begingroup$ See stats.stackexchange.com/questions/192554/… $\endgroup$
    – Christoph Hanck
    Nov 27, 2020 at 10:25
  • $\begingroup$ @ChristophHanck I believe that question answers the following: "membership of an exponential family is not a necessary condition for the existence of a conjugate prior". Is there a sufficient condition? Or a distribution that doesn't have a conjugate prior? $\endgroup$
    – helperFunction
    Nov 27, 2020 at 10:30
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    $\begingroup$ I am not sure what the consensus among really knowledgeable Bayesians is here, but such a result may be complicated by the fact that a lot hinges on what how the "family" is defined to which priors and posteriors must both belong for the prior to be conjugate for a given likelihood. If you define it to be the family of all probability distributions, then a conjugate prior exists, but it is of course not a very interesting result. $\endgroup$
    – Christoph Hanck
    Nov 27, 2020 at 11:32

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