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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Inconsistency between normal-wishart marginal and marginal covariance computed using tower rule
Murphy's result, section 8.3 on normal conjugacy states (substituting $\mathbf{y}$ for $\mu$) that if:
$$\mathbf{y}|\Lambda \sim \mathcal N(\mathbf{0}, (\kappa\Lambda)^{-1}) $$
$$ \Lambda \sim \...
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Integral of normal likelihood and multivariate normal prior
I'm updating cluster assignments in the context of a non-parametric Bayesian mixture model. When computing the probability of starting a new cluster, in the absence of cluster parameters (and using a ...
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Posterior predictive distribution for Bernoulli (and categorical)
I'm trying to confirm something I've tried to figure out about the posterior predictive distribution for Bernoulli vs. Binomial (and categorical vs. multinomial) random variables after a Bayesian ...
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Bayesian estimation of iid sample from Uniform$[0,\theta]$ and a Pareto$(\alpha,\beta)$ prior for $\theta$
I am working on Bayesian estimation: suppose that $X_1,\dots, X_n$ is an iid sample from Uniform$[0,\theta]$. Assume a Pareto prior for $\theta\sim Pareto(\alpha,\beta)$, i.e.
$$
f(\theta)=\frac{\...
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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) ...
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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 ...
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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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Is updating gamma conjugate distributions always increasing?
In Bayesian inference, the gamma distribution is the conjugate prior to many likelihood distributions: Poisson, exponential, normal (with known mean), etc.
The update rule seems to always add to $\...
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What is the conjugate prior for the Von Mises distribution's precision
Does the Von Mises distribution have a conjugate prior for its precision/variance?
Update:
The concentration parameter $\kappa$ (Kappa) seems to control the variance of the Von Mises distribution. If $...
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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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What is the origin of the name "conjugate prior"?
I know what a conjugate prior is. But I'm confused by the name itself. Why is it called "conjugate"? A complex conjugate $z^\ast$ has a reciprocal relationship with $z$, i.e., ${z^\ast}^\ast ...
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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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Updating conjugate priors with noisy observations
I'm considering a problem that has been partially addressed elsewhere:
Bayesian updating with conjugate priors using the closed form expressions
but now I have an added twist. My samples are drawn ...
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Proving Matrix-Normal-Inverse-Wishart distribution is a conjugate prior for a Linear Model
How does one prove that the Matrix-Normal-Inverse-Wishart distribution is a conjugate prior for a Linear Model? This prior is a generalization of the Normal-Inverse-Wishart Distribution.
By Matrix-...
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A re-formalization of a conjugate prior?
It is quite easy to prove that if $p(\theta)$ is a conjugate prior to some likelihood then the following:
$$q(\theta') \propto p(\theta)I(\theta \in A)$$
where $A$ is a subset of the parameter space ...
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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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Specific step in the proof of conjugate prior for normal distribution with unknown mean and variance
I'm struggling to follow a specific step in the proof that
$$
\tau \sim \text{Gamma}(\alpha, \beta), \quad \mu | \tau \sim \mathcal{N}(\nu, \frac{1}{k\tau})
$$
is a conjugate prior distribution for ...
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Conjugate prior for a Gaussian model with shifted variance
Consider a set of observations $ \{ y_i \}$ and assume a Gaussian model for these data: $y_i \sim \mathcal{N}(\mu, \sigma^2)$. Suppose the mean parameter $\mu$ is known, but the variance parameter $\...
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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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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) ...
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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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Should the updated posterior for a Poisson distribution be discretized if based on the Gamma distribution as the prior?
I know that the Gamma distribution is the conjugate prior of the Poisson distribution, such that given $\alpha$ and $\beta$ that describe the prior distribution, the posterior distribution is $Gamma(\...
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What is the conjugate prior for the hypoexponential distribution?
Can't find it anywhere. I know Gamma is the conjugate prior for the exponential distribution (one parameter) but for the sum of exponential distributions (the hypoexponential distribution), I can't ...
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Does the beta negative binomial (BNB) distribution have a conjugate prior?
BNB distribution is constructed using negative binomial and beta distributions, which are both exponential family, so my guess would be yes, there shoudl exist a conjugate prior in theory. But what is ...
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Why is the mixtures of conjugate priors important?
I have questions about the mixture of conjugate priors. I learned and saw the mixture of conjugate priors a couple of times when I am learning bayesian. I am wondering why this theorem is such ...
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Use the gamma prior to obtain the gamma posterior values
I have the following information for the ages of individuals:
Sample size = 5.
Data: $$ x_i = (10, 12, 15, 16, 14) $$
The population mean previously accurately estimated is 12.
Prior information ...
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Find a conjugate prior for the Weibull distribution under reparametrization
Consider the Weibull sampling model for $X_1,\ldots,X_n$ iid, where
$$p(x|\lambda,k)=k\lambda^kx^{k-1}e^{-\lambda^kx^k}$$
for $x>0$. Assume $k$ is known and $\lambda$ is unknown. First, if I adopt ...
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the reason for using InverseGamma and LogNormal as prior for covariance matrix or variance
In the Bayesian analysis, sometimes we can see that InverseGamma and LogNormal distributions are used as prior for variance or covariance matrix respectively. What are the logic or explanations of ...
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Proportional to Gamma means the posterior is gamma
I'm reading through these lecture notes on posteriors and conjugate priors.
https://web.stanford.edu/class/stats200/Lecture20.pdf
In particular, it asserts that: "This is proportional to the PDF ...
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Dirichlet Process posterior with partially observed data
Suppose I dipose of a set of independant observed couples $(x_1,y_1),...,(x_N, y_N)$ from a joint distribution $P(x,y)$. Furthermore, I suppose that the random distribution $P$ as a Dirichlet prior
$P\...
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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 ...
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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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Posterior density problem
I am preparing for an exam and I have stumbled upon this exercise, which I am not certain of.
a)
You are given i.i.d. data $x_1, \dots, x_n$ from a continuous distribution with density $\frac{\alpha ...
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Identifying the value of parameters of the prior distribution. Arbitrarily?
Referring to this Question, let's not use Jeffrey's prior for $\theta$ but use $Gamma(\alpha,\beta)$ as the conjugate prior for $\theta$. Under quadratic loss function, the bayes estimator for $\theta$...
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Conjugate Prior for Multivariate Normal Variances and Correlations
Is there a way to separately specify conjugate priors for the variance and correlations of a multivariate normal? The inverse Wishart is conjugate if you want to specify the covariance, but covariance ...
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mean-field variational inference in Nonconjugate Models
I'm trying to conduct mean-field Variational Inference (VI) with Nonconjugate model.
I found Variational Inference in ...
