I am new to the bayesian statistics and I most frequently see the conjugate prior distribution. Can you explain it with clear example? I would be very thankful.
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other related questions include https://stats.stackexchange.com/questions/90969/how-to-find-conjugate-prior-for-a-given-distribution and https://stats.stackexchange.com/questions/59363/having-a-conjugate-prior-deep-property-or-mathematical-accident – Glen_b Sep 29 '18 at 08:43
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Simple illustration of beta prior conjugate to binomial likelihood [here](https://math.stackexchange.com/questions/2182174/how-to-set-values-in-a-conjugate-prior/2182491#2182491). – BruceET Sep 29 '18 at 08:57
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A conjugate prior is a probability distribution that, when multiplied by the likelihood and divided by the normalizing constant, yields a posterior probability distribution that is in the same family of distributions as the prior.
In other words, in the formula:
$$p(\theta|x) = \frac{p(x|\theta)p(\theta)}{\int{p(x|\theta)p(\theta)d\theta}}$$
The prior $p(\theta)$ is conjugate to the posterior $p(\theta | x)$ if both are in the same family of distributions.
For example, the normal distribution is conjugate to itself, because if the likelihood and prior are normal, then so is the posterior.
Frans Rodenburg
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