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This question already has an answer here:

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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marked as duplicate by Glen_b distributions Sep 29 '18 at 8:41

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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.

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