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I kind of understand the meaning of conjugate to something but can't really get the clear picture of the concept. Some video lectures on youtube talk about it but they all seem to just say "Normal Distribution is conjugate to Normal" but for me to understand, it wasn't enough to get it. Hope to hear with least mathematical terms.

user122358
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    Possible duplicate of [Gaussian is conjugate of Gaussian?](http://stats.stackexchange.com/questions/124623/gaussian-is-conjugate-of-gaussian) see also http://stats.stackexchange.com/questions/204920/do-conjugate-priors-just-lead-to-a-posterior-that-is-a-modification-of-the-param/204942#204942 and http://stats.stackexchange.com/questions/192554/aside-from-the-exponential-family-where-else-can-conjugate-priors-come-from – Sycorax Feb 04 '17 at 00:17
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    In Bayesian Statistics the normal distribution is a conjugate prior for the normal – Michael R. Chernick Feb 04 '17 at 00:42
  • The [table of conjugate priors in the Wikipedia conjugate prior article](https://en.wikipedia.org/wiki/Conjugate_prior#Table_of_conjugate_distributions) is quite instructive. The table lists the prior distribution, the likelihood distribution, and the parameterization of the posterior distribution. It may be a bit confusing at first, but working through a few of these examples may really help in understanding conjugate priors and the whole Bayesian approach. – Matthew Gunn Feb 04 '17 at 00:57

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Saying it in plain English: if you have some Bayesian model to estimate parameter of interest, then you use some prior distribution for it, likelihood to describe the probability of observing your data conditional on the parameter, so to obtain the posterior distribution. If you have a conjugate prior this means that the prior comes from the same family of distributions and there is a closed-form solution for such problem, so the posterior distribution is directly available. This is exactly the case when you use normal prior for mean parameter of normal distribution. Otherwise you would need some kind of numerical optimization or Markov Chain Monte Carlo to estimate the parameter.

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Tim
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    I believe the concept of a conjugate prior is much stronger than merely having a "closed form": the posterior must be in the same finitely parameterized family as the prior. – whuber Feb 04 '17 at 00:21
  • @whuber I added such information but as the question asks about basic understanding I don't think that going into details is needed in here. – Tim Feb 04 '17 at 00:24
  • Although I still need to understand the relation between bernoulli and beta distribution for conjugate prior, somehow, your explanation helps me get closer to the point. – user122358 Feb 04 '17 at 00:49
  • I believe that the closure of the family under updating is the essence of the concept of "conjugate prior," not the analytical tractability of the update. Of course its *utility* derives from the fact that we choose to use conjugate priors that lead to simple calculations. – whuber Feb 04 '17 at 17:35