# Questions tagged [conjugate-prior]

A prior distribution in Bayesian statistics that is such that, when combined with the likelihood, the resulting posterior is from the same family of distributions.

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### Do conjugate priors just lead to a posterior that is a modification of the parameters of the prior?

We know, for example, that the conjugate relationship between the classic beta-binomial is as follows: \begin{align} y &∼ \mathcal{Bin}(n,\ θ) \\ θ &∼ \mathcal{Beta}(α,\ β) \\ θ|y &∼ \...
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### Bayesian A/B test analysis for non-binary actions

When analyzing experiments with binary action (for example, will a user convert or not), then Beta-Binomial distribution is apt for that. For example, to model Click Through Rate (CTR): ...
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### Does the beta distribution have a conjugate prior?

I know that the beta distribution is conjugate to the binomial. But what is the conjugate prior of the beta? Thank you.
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### Conditionally conjugate prior for non-nested (i.e. crossed) normal model?

I am trying to write/understand a conditionally-conjugate Gibbs sampler for what is essentially a linear, mixed effects model. I more or less get the conditionally-conjugate posterior for the ...
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### Bayesian inference on mean of statistic from population

Suppose that a collection of time intervals $t_i$ have occurred, for $i=1,...,n$. These should be considered as samples from a population governed by some distribution. During these time intervals, ...
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### why isn't the marginal distribution needed when using a conjugate prior?

What is a good explanation as to why you wouldn't have to integrate to find the posterior when you use a conjugate prior. Most examples (like for instance: http://www.youtube.com/watch?v=0XD6C_MQXXE) ...
5k views

### Does there exist a conjugate prior for the Laplace distribution?

Does there exist a conjugate prior for the Laplace distribution? If not, is there a known closed form expression that approximates the posterior for the parameters of the Laplace distribution? I've ...
33 views

### Equivalent of Conjugate Priors for Marginal Probability Distributions

In probability, there are nice “conjugate prior” distributions that enable closed-form Bayesian updating – e.g. if you have a Normal likelihood and Normal prior (on the mean parameter), you get a ...
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### Posterior distribution of a $\text{Gamma}(\alpha,\beta)$ random variable given a Gamma prior for $\beta$

Let $Y$ be a $\text{Gamma}(\alpha,\beta)$ random variable with known shape parameter $\alpha$ and unknown scale parameter $\beta$. Suppose we assign a $\Gamma(\alpha_0,\beta_0)$ prior to $\beta$. I am ...
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### Pareto distribution with Gamma prior on parameter $\theta$

I want to calculate the posterior distribution of Pareto distribution with known parameter $X_m$ and unknown parameter $\theta$, with conjugate prior on $\theta$ the Gamma distribution: My effort is ...
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### Necessity of Metropolis Hastings algorithm for given posterior distribution

Let's say that we have calculated the posterior distribution of a parameter of interest given the data of a binomial experiment $N=70,x=34$ which the probability of event occurrence $\theta$ follows ...
1 vote
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### Is There a Conjugate Prior for a Multivariate Hypergeometric Likelihood?

I am working on a problem using a multivariate hypergeometric likelihood. The multivariate hypergeometric distribution does not belong to the exponential family of distributions, so (to my knowledge) ...
5k views

### Deriving the posterior density for a lognormal likelihood and Jeffreys's prior

The likelihood function of a lognormal distribution is: $$f(x; \mu, \sigma) \propto \prod_{i_1}^n \frac{1}{\sigma x_i} \exp \left ( - \frac{(\ln{x_i} - \mu)^2}{2 \sigma^2} \right )$$ and Jeffreys's ...
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### beta-binomial as conjugate to hypergeometric

According to the table of conjugate distributions on Wikipedia, the hypergeometric distribution has as conjugate prior a beta-binomial distribution, where the parameter of interest is "$M$, the number ...
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### Why not use the same distribution for the prior in Bayesian statistics?

I am wondering why introductory books on statistics use a conjugate distribution family for the prior instead of using the same pdf of the one we are trying to infer the parameters? For example, the ...
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### Why do we use inverse Gamma as prior on variance, when empirical variance is Gamma (chi square)

Let $$X_i\sim \mathcal{N}(0,\sigma^2)$$ than we know that $$\sum_{i=1}^N\frac{X_i^2}{N}\sim\Gamma(\frac{N}{2},\frac{2\sigma^2}{N})$$ that the empirical variance follows a Gamma distribution. How do ...
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### Conjugate Prior for Student T distribution

Does the Student T distribution have a conjugate prior distribution? If so, what is it and what are the parameters?
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### What is the posterior distribution of θ? Is the Gamma a conjugate prior for an exponential likelihood?

A manufacturer is interested in the time to failure of his batteries. Suppose the time to failure of the batteries has an exponential distribution: $$p(x│\theta)=\theta\exp(-\theta x)$$ Note that the ...
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### In what ways do conjugate priors compose?

A lot of conjugate priors are known for a lot of likelihood distributions (mostly the exponential family). But most Bayesian models in practice don't just consist of one distribution. Usually, you ...
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### How should I deduce the conjugate prior and corresponding posterior for a geometric distribution

The given pmf is for a geometric distribution and is $f(x_i|\theta) = (1-\theta)^{x_i - 1}\theta; ~x_i = 1, 2 ,\cdots,$ and the 1-parameter exponential family I have obtained is; f(x|\theta) = \exp ...
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### Prior on a dirichlet distribution [duplicate]

I would like to know if there is a "conjugate" prior that we can place on the Dirichlet distribution parameters. Thanks in advance,
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### Posterior distribution when the domain of the likelihood depends on the parameter

I am trying to calculate a posterior density given distribution and a prior. And I am a bit confused about how I should act as the domain of the distribution depends on the parameter. I am talking ...
1 vote
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### Sequential Bayesian updating of mean and variance of normal distribution

I am trying to write some code to learn the parameters of a normal distribution. I am new to this, and I have patched together the equations from various sources, which may be part of the problem. In ...
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1 vote