# Tag Info

Accepted

### Bayesian updating with new data

The basic idea of Bayesian updating is that given some data $X$ and prior over parameter of interest $\theta$, where the relation between data and parameter is described using likelihood function, you ...
• 115k

### Can anyone explain conjugate priors in simplest possible terms?

A prior for a parameter will almost always have some specific functional form (written in terms of the density, generally). Let's say we restrict ourselves to one particular family of distributions, ...
• 261k
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### What is the origin of the name "conjugate prior"?

The Oxford English Dictionary defines "conjugate" as an adjective meaning "joined together, esp. in a pair, coupled; connected, related." It's not a huge stretch to imagine that a ...
• 19.4k
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### Justification for conjugate prior?

Maybe satisfying the category "heuristic" justification, conjugate priors are useful because, among others, of the "fictitious sample interpretation". For example, in the Beta-Bernoulli case, the ...
• 26.7k
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### Aside from the exponential family, where else can conjugate priors come from?

As explained for example in Section 3.3.3 of book "The Bayesian choice" by Christian Robert, there is indeed a narrow connection between exponential families and conjugate priors, but there are ...
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### beta-binomial as conjugate to hypergeometric

The problem with the Wikipedia article and the reference therein (Fink D., 1997) is that there is some key information missing. Specifically, the given posterior is for $M-x$ (i.e. the number of ...
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### Gaussian is conjugate of Gaussian?

If we take your question to mean whether the product of the densities are Gaussian, then the answer is "yes" (P.A. Bromiley. Tina Memo No. 2003-003. "Products and Convolutions of ...
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### Understanding the Beta conjugate prior in Bayesian inference about a frequency

The point is that we know what the posterior is proportional to and it so happens that we do not need to do the integration to get the (constant) denominator, because we recognise that a distribution ...
• 22.9k
Accepted

### Why is the mixtures of conjugate priors important?

Calculating posteriors with general/arbitrary priors directly may be a difficult task. On the other hand, calculating posteriors with mixtures of conjugate priors is relatively simple, since a given ...
• 261k

### What are the parameters of a Wishart-Wishart posterior?

Ok, thanks to @Xi'an answer I could make the whole derivation. I will write it for a general case: \begin{align} \mathcal{W}(\mathbf{W} | \upsilon, \mathbf{S^{-1}} ) \times \mathcal{W}(\mathbf{S} | \...
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### Understanding the Beta conjugate prior in Bayesian inference about a frequency

The setup You have this model: \begin{align*} p & \, \sim \, \text{beta}(\alpha, \beta) \\ x \, | \, p & \, \sim \, \text{binomial}(n, p) \end{align*} The densities for which are \begin{...
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• 92.6k
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• 92.6k

### Dirichlet conjugate update derivation

There is nothing wrong with this derivation \begin{align} p({\alpha}|{\theta},{\nu},\eta) &\propto p({\alpha},{\theta}|{\nu},\eta)\\ &= f({\theta}|{\alpha})p({\alpha}|{\nu},\eta)\\ &\...
• 92.6k
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### Overestimation of the noise precision in Bayesian linear regression when $n\gtrsim p$

This problem turns out to be well-known in the frequentist literature. In particular, if we use an impropr prior $\Lambda_0=b_0=0$, the posterior scale hyperparameter for the distribution on $\tau$ is ...
Accepted

### conjugate prior: is ever the best choice?

As you said, conjugate priors make things easier and an additional nice property is that when you refresh the model using the posterior as the new prior things are nice and consistent. For instance ...
• 1,335
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### Why do we use inverse Gamma as prior on variance, when empirical variance is Gamma (chi square)

No reconciliation is needed. In one case you are referring to the sampling distribution of the maximum likelihood estimator, which is a function of the data. In the other, you are referring to the ...
• 33.4k