# 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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### Conjugate Prior for Student T distribution with known degrees of freedom

Somebody asked a question about a conjugate prior distribution for Student-t distribution with unknown degrees of freedom. It was answered that there are no conjugate prior distribution in that case. ...
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### Form of conjugate prior on $(\mu,\sigma^2)$ for Normal $N(\mu,\sigma^2)$ distribution

I have seen many postings related to finding a conjugate prior for $(\mu,\sigma^2)$ for the normal distribution $N(\mu,\sigma^2)$. I am trying to derived an expression that has a rather simple ...
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### Conditionally conjugate prior in heteroskedastic model

I am researching a linear model where the noise is a function of the slope parameter as follows $$y_i = \beta_0 + \beta_1x_i + \beta_1\epsilon_i$$ $$\epsilon_i \sim N(0, \sigma^2 g)$$ where $g$ is ...
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### posterior predictive of a normal distribution with normal prior over mean and Gamma prior over precision

What is the posterior predictive of a normal distribution with normal prior over mean and Gamma prior over precision. Thus, what is the distribution of x given: x \sim \mathcal{N}(x; \...
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### 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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### How to calculate the posterior distribution with a normal likelihood function and a prior that involves sigma

In the problem, the data X follows a normal distribution, or $f(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp(-\frac{1}{2}(\frac{x-\mu}{\sigma})^2)$. Let's say I know the value of $\sigma^2$ and ...
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### Eliciting a Gamma informative prior in a Gamma–Poisson Bayesian problem

I employ the Gamma–Poisson conjugate family for my statistical model. I want to use an informative prior. From theory, I know that the values of the Gamma-distributed random variable lie within the ...
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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 ...
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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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### 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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### 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 ...
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