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Conjugate prior relationships - John Cook
共轭先验（conjugate prior） - 司马_羽鹤 - 博客园
Conjugate Priors, Uninformative Priors
Chapter 9 The exponential family: Conjugate priors
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Why are conjugate prior distributions usually the beta or gamma distribution?
I am working through a textbook on statistical inference and working through the section about conjugate prior distributions. I noticed that for the majority of the distributions, all of the theorems given and proven use a beta (bernoulli, binomical, negative binomial, geometric) or gamma (piosson, normal, pareto, and gamma) prior. Even for a normal with known mean but unknown variance, the book uses a gamma rather than a normal. Why do we use these two functions? Is there something more fundamental that I am missing?
I'm reviewing a proof for the poisson-gamma conjugate prior proof (See YouTube link below.) Everything makes sense except in when consolidating terms (time 5:31). The author claims that:
g(r) * (y)! = g(r+y)
Now, I'm a bit confused, it's my understanding that gamma is defined as
g(n) = (n-1)! g(4) = (4-1)! = 3*2*1 = 6
So, the first code block doesn't make sense to me...
r = 3 y = 4 g(3) * (4)! = (3-1)! * (4)! = (2*1) * (4*3*2*1) = 48
Lecture slide: https://imgur.com/NdSmXQW So I understand that conjugate priors allow the posterior to be of the same algebraic form as the posterior. We have a Bernoulli likelihood P(D | u) and since u is the model parameter we include it in the conjugate prior so you get p(u | 0). However, what I don't understand is where the actual equation for the conjugate prior came from? Is it just the x replaced by various functions of whatever distribution theta is? Also, is theta the model parameters of mu? EDIT: Never mind I understand now
Bayesian Q: What is a good reference for conjugate prior/posterior distributions?
I am interested in slice sampling and Gibbs sampling to estimate parameters for a complex HMM, and I am seeking a good reference on conjugate prioposterior relationships to set up the samplers. I'm looking for something beyond the depth of the wiki page on the topic. What's your favorite go-to to look up conjugate priors?
Hello, Does anyone know how to prove that Beta(a,b) is the conjugate prior of Bernoulli? I really don’t see where this result comes from. Thank you in advance!
So I understand the usefulness of using a conjugate prior, makes the posterior super easy to compute. But if your real-world likelihood function is not going to be a pretty exponential family distribution anyway, is having a conjugate prior ever actually useful in practice?
[University Statistics] Help understanding conjugate priors and probability distributions in general.
Include instructor prompts. What does your instructor want you to accomplish?
. [deleted] (a) Using a uniform prior on q, find the posterior distribution
Tell us what is holding you up. So I understand that this requires a binomial distribution, as such: [;p(q|y) = p(q)p(y|q)`;] [;p(q) = 1;] [;p(q|y) = {n \choose y} qy (1-q)n-y;] And this sums to 1 when keeping q constant and checking for values of y, so that makes sense. But we want to have a posterior on q, so does this mean the integral has to sum to 1? if so, howww? I think we may need a conjugate prior, which the book says is the beta distribution, and we divide by that. Giving us [;p(q|y) = \frac{{n \choose y} qy (1-q)n-y}{B(something?)};] What exactly is a conjugate prior? Because the next question is to find the Expected value of Q, would this just be the integral from 0 to 1 of the posterior with respect to q? With or without the beta prior?
Bayesian Testing question regarding which conjugate prior to use
I'm working with data that is pretty close to binomial but not quite. Not all observations are 0 or 1, there are VERY few that have 2, 3, even 4. I have quite a bit of data points and feel that I could essentially use a beta-binomial model to model my conversion rates even if the data isn't exactly binomial. This is a table breakdown of what I'm talking about: control 0 1 2 3 4 .935 .060 .0045 .0003 0.0001 Is it reasonable to just default to a beta-binomial? What kind of issues could I face making this assumption?
I am rarely at a loss for what to use as a search keyword, but this one has me stumped... Consider some probability distribution g(x). I'd like to know all of (or at least some of) the other distributions f(x) such that the integral of f(x)g(x) from 0 to infinity (or whatever the relevant domain is) has a tidy closed form. If the integral of f(x)g(x) is in the same family as g(x), we say that f is a conjugate prior. But this is something more general - I just want the posterior to be something I can work with easily, not necessarily something similar in form to g and not necessarily a named distribution. Is there a name for these? "Generalized conjugate prior" and "integrable non-conjugate distribution" didn't get me much of anywhere.
Need help understanding what conjugate priors are and how they are chosen.
I asked a question here with more detail but thought I'd ask y'all as well. What exactly is a conjugate prior? When do you use it? How does one decide which one to use?
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: $$p(\thetax) = \frac{p(x\theta)p(\theta)}{\int{p(x\theta)p(\theta)d\theta}}$$ Conjugate prior is here to rescue us from this misery. We can obtain exact posterior distribution if our prior distribution (in case of the example above, it will be the distribution of weights) is conjugate prior for the likelihood function. Let’s see with a mischievous example: The conjugate prior is an initial probability assumption expressed in the same distribution type (parameterization) as the posterior probability or likelihood function. In the most common case of Bayesian inference, the probability and likelihood functions are essentially the same thing when assigning initial degrees of belief ( prior probability ... Chapter 9 The exponential family: Conjugate priors Within the Bayesian framework the parameter θ is treated as a random quantity. This requires us to specify a prior distribution p(θ), from which we can obtain the posterior Sadly, there are only a few problems we can solve with conjugate priors; in fact, this chapter includes most of the ones that are useful in practice. For the vast majority of problems, there is no conjugate prior and no shortcut to compute the posterior distribution. This post is an introduction to conjugate priors in the context of linear regression. Conjugate priors are a technique from Bayesian statistics/machine learning. The reader is expected to have some basic knowledge of Bayes’ theorem, basic probability (conditional probability and chain rule), machine learning and a pinch of matrix algebra. Conjugate prior relationships. The following diagram summarizes conjugate prior relationships for a number of common sampling distributions. Arrows point from a sampling distribution to its conjugate prior distribution. The symbol near the arrow indicates which parameter the prior is unknown. Beta is the conjugate prior of Binomial. Dirichlet is the conjugate prior of multinomial. posted @ 2016-01-19 21:53 司马_羽鹤 阅读( 8565 ) 评论( 0 ) 编辑 收藏 Conjugate priors Mixture of conjugate prior Uninformative priors Jeffreys prior. The Exponential Family A probability mass function (pmf) or probability distribution function (pdf) p(Xj ), for X= (X 1;:::;X m) 2Xm and Rd, is said to be in theexponential familyif it is the form: p(Xj ) = 1 Z( ) h(X)exp[ T ˚(X)] (1) Conjugate prior in essence. For some likelihood functions, if you choose a certain prior, the posterior ends up being in the same distribution as the prior.Such a prior then is called a Conjugate Prior. It is a lways best understood through examples. Below is the code to calculate the posterior of the binomial likelihood. θ is the probability of success and our goal is to pick the θ that ...
Demonstration that the gamma distribution is the conjugate prior distribution for poisson likelihood functions.These short videos work through mathematical d... Skip navigation Sign in. Search In Bayesian probability theory, if the posterior distributions p(θx) are in the same family as the prior probability distribution p(θ), the prior and poster... This video provides a short introduction to the concept of 'conjugate prior distributions'; covering its definition, examples and why we may choose to specif... Definition of conjugate priors, and a couple of examples. For more detailed examples, see the videos on the Beta-Bernoulli model, the Dirichlet-Categorical m... About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators ...
