Normal Conjugate Prior, Known Mean and Unknown Variance?
Bayesian Inference for Normal with Unknown Mean and Variance
conjugate prior normal distribution unknown variance
conjugate prior normal distribution unknown variance - win
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?
Setting up Markov chain that has the posterior distribution as its equilibrium distribution.
I am doing undergrad curse in statistics. Latest module was an introduction to Bayesian statistics. One topic that got my interest was MCMC, but unfortunately, it was described only very briefly. One example used in a textbook was estimation of the parameters from an unknown distribution X ~ N(mu, 1/tau). The interest lies in finding the marginal distribution mu | data, hence finding most probable value of mu. The textbook went through two ways of doing it: First was using a closed form - conjugate analysis for a normal mean and variance. It was shown that the conjugate joined prior pdf can be written in a form f(mu, tau) = f(tau)f(mu | tau), then with the priors mu | tau ~ N(a, b/tau), tau ~ Gamma(c,d) the joined prior distribution will have a form mu, tau ~ Ngamma(a,b,c,d) and given the sample of data from a normal distribution the posterior distribution is mu, tau | data ~ Ngamma(A, B, C, D). It then went though showing how the parameters a,b,c,d, A,B,C,D could be calculated. We then used WinBUGS to sample from mu, tau | data ~ Ngamma(A, B, C, D) and finally find the marginal distribution for mu. Second approach was using MCMC - to show that the full simulation method is much simpler. The textbook argued that similar effect could be achieved using independent priors and MCMC. So instead of finding and using mu, tau ~ Ngamma(a,b,c,d) we can define the priors independently mu ~ N(a,b), tau ~ Gamma(c,d) with a simple program using WinBugs and MCMC: ``` model {
for (i in 1:n){ x\[i\] \~ dnorm(mu,tau) } mu \~ dnorm(4,1) tau \~ dgamma(2.2,0.15)
} ``` Data is 29 observations. Do few more clicks and here we go. My question is how can I do the MCMC simulation my self from scratch? Either by hand or by using program written by myself? I usually get a better understanding of a problem when I will build it from scratch. It doesn't seem to be to complicated, for now, but I don't know where to start. I am fairly ok with linear algebra and I do have some understanding on how Markov chain works. As far as I understand it I would need to find the initial values and create transition matrix. So: How can I construct a Markow chain which in equilibrium state simulate the posterior distribution given data ~ N(mu,1/tau) with priors mu ~ N(4,1) and tau ~ Gamma(2.2,0.15)?
Looking for video lectures / MOOC(s) for a rigorous statistics course
I'm doing a statistics course that's pretty rigorous. I'm meant to understand and be able to prove most of the theorems on the exam. Unfortunately I ended up missing a lot of the lectures (illness) so there are some great big gaps in my knowledge. The lecture notes are complete, but they're rather terse and difficult to follow (definition-theorem-proof style, motivation is sometimes unclear). I went through them and made a list of all the topics covered: Intro and key concepts Likelihood, log-likelihood Score statistic Fisher information Exponential family Sufficiency, minimal sufficiency Neyman Factorisation Theorem Completeness Bahadur Theorem Point Estimation Method of moments Maximum likelihood estimation Desirable properties of estimators: unbiasedness, small volatility, consistency Minimum-variance unbiased estimator Properties of MLEs: invariance, consistency, asymptotic normality MLE and sufficiency Cramér-Rao Lower Bound Rao-Blackwell Theorem Lehmann-Scheffé Theorem Hypothesis Testing Type I and Type II errors Significance level and power Most Powerful Test Simple and composite hypotheses Randomised Tests Nyman-Pearson Lemma Uniformly Most Powerful Tests Likelihood Ratio tests Nested hypotheses Wilke's theorem Confidence sets, pivotal quantities Slutsky's lemma Duality (of?) confidence sets -- hypothesis testing The Delta Method Univariate case Multivariate case Multivariate normal distribution Bayesian Statistics The Bayesian paradigm Prior, posterior, likelihood ratio, updating Sufficiency Choice of prior distribution Non-informative priors, ignorant prior, vague prior, Jeffrey's prior Conjugate priors, conjugate families Point estimation: summaries of location, connection with loss function Interval estimation: Bayesian Credible Region, Highest Posterior Density Region, Frequentist vs Bayesian intervals Large sample properties of the posterior Bayesian model comparison, Bayes factor Predictive inference, Law of Total Probability The Case of Categorical Data (Bayesian) Binary data Categorical data Dirichlet distribution, limitations thereof Marginal distribution, model comparison Application: contingency tables The Case of Normal Data (Bayesian) Likelihood Variance known, mean unknown Mean known, variance unknown, inverse gamma distribution Mean unknown, variance unknown, normal inverse gamma distribution Generalised t-distribution Conjugate analysis in the general case Marginal inference about the mean, variance Predictive inference Weak prior information Introduction to Decision Theory Utility funcitons, loss functions Expected utility Utility scale Choosing the utility function Prioposterior expected utilities Bayes decision rule Value of information Perfect information, value thereof Risk, admissibility, Bayes rules Wald Theorem I'd like to be able to watch some video lectures for these topics, because I find I learn much better when there's a friendly voice explaining stuff to me on a blackboard. Does anyone have any recommendations for Youtube videos or MOOCs that go into detail on these things? Failing that, what are some good textbooks for learning these topics from scratch?
Looking for video lectures / MOOC(s) for a rigorous statistics course
I'm doing a statistics course that's pretty rigorous. I'm meant to understand and be able to prove most of the theorems on the exam. Unfortunately I ended up missing a lot of the lectures (illness) so there are some great big gaps in my knowledge. The lecture notes are complete, but they're rather terse and difficult to follow (definition-theorem-proof style). I went through them and made a list of all the topics covered: Intro and key concepts Likelihood, log-likelihood Score statistic Fisher information Exponential family Sufficiency, minimal sufficiency Neyman Factorisation Theorem Completeness Bahadur Theorem Point Estimation Method of moments Maximum likelihood estimation Desirable properties of estimators: unbiasedness, small volatility, consistency Minimum-variance unbiased estimator Properties of MLEs: invariance, consistency, asymptotic normality MLE and sufficiency Cramér-Rao Lower Bound Rao-Blackwell Theorem Lehmann-Scheffé Theorem Hypothesis Testing Type I and Type II errors Significance level and power Most Powerful Test Simple and composite hypotheses Randomised Tests Nyman-Pearson Lemma Uniformly Most Powerful Tests Likelihood Ratio tests Nested hypotheses Wilke's theorem Confidence sets, pivotal quantities Slutsky's lemma Duality (of?) confidence sets -- hypothesis testing The Delta Method Univariate case Multivariate case Multivariate normal distribution Bayesian Statistics The Bayesian paradigm Prior, posterior, likelihood ratio, updating Sufficiency Choice of prior distribution Non-informative priors, ignorant prior, vague prior, Jeffrey's prior Conjugate priors, conjugate families Point estimation: summaries of location, connection with loss function Interval estimation: Bayesian Credible Region, Highest Posterior Density Region, Frequentist vs Bayesian intervals Large sample properties of the posterior Bayesian model comparison, Bayes factor Predictive inference, Law of Total Probability The Case of Categorical Data (Bayesian) Binary data Categorical data Dirichlet distribution, limitations thereof Marginal distribution, model comparison Application: contingency tables The Case of Normal Data (Bayesian) Likelihood Variance known, mean unknown Mean known, variance unknown, inverse gamma distribution Mean unknown, variance unknown, normal inverse gamma distribution Generalised t-distribution Conjugate analysis in the general case Marginal inference about the mean, variance Predictive inference Weak prior information Introduction to Decision Theory Utility funcitons, loss functions Expected utility Utility scale Choosing the utility function Prioposterior expected utilities Bayes decision rule Value of information Perfect information, value thereof Risk, admissibility, Bayes rules Wald Theorem I'd like to be able to watch some video lectures for these topics, because I find I learn much better when there's a friendly voice explaining stuff to me on a blackboard. Does anyone have any recommendations for Youtube videos or MOOCs that go into detail on these things? Failing that, what are some good textbooks for learning these topics from scratch?
conjugate prior normal distribution unknown variance video
Informative Prior for SPF Construct an informative prior distribution for : I Take prior median SPF to be 16 I P( > 64) = 0:01 I information in prior is worth 25 observations Solve for hyperparameters that are consistent with these quantiles: m0 = log(16), p0 = 25, v0 = p0 1 P( < log(64)) = 0:99 where m0 p SS0=(v0p0) ˘ tv0) SS0 = 185:7 I'm following these notes to compute the conjugate prior of a normal distribution with unknown mean and known variance. I'm following these notes to compute the conjugate prior of a normal distribution with unknown mean and known variance. At some point they claim: How can I compute the conjugate prior of a normal distribution with Useful distribution theory Conjugate prior is equivalent to (μ− γ) √ n0/σ ∼ Normal(0,1). Also 1/σ2|y ∼ Gamma(α,β) is equivalent to 2β/σ2 ∼ χ2 2α. Now if Z ∼Normal(0,1),X χ2ν/ν,thenZ/ √ X tν. Therefore the marginal prior distribution for μ in the bivariate conjugate prior is such that (μ− γ) n0α/β ∼ t2α 6-6 We now turn to another important example: the normal distribution is its own conjugate prior. In particular, if the likelihood function is normal with known variance, then a normal prior gives a normal posterior. Now both the hypotheses and the data are continuous. We have a conjugate prior if the posterior as a function of has the same form as the prior. Exponential/Normal posterior: f( jx) = c 1 e ( prior)2 2˙2 prior x The factor of before the exponential means this is not the pdf of a normal distribution. Therefore it is not a conjugate prior. Exponential/Gamma posterior: Note, we have never learned It states that a normal random variable with mean 0 and variance 1 divided by the square root of an independent chi‐squared random variable over its degrees of freedom will have the Student's t distribution. The product of independent conjugate priors is a perfectly acceptable prior, but it is not jointly conjugate. Chapter 2 Conjugate distributions. Conjugate distribution or conjugate pair means a pair of a sampling distribution and a prior distribution for which the resulting posterior distribution belongs into the same parametric family of distributions than the prior distribution. We also say that the prior distribution is a conjugate prior for this sampling distribution. It states that a normal random variable with mean 0 and variance 1 divided by the square root of an independent chi‐squared random variable over its degrees of freedom will have the Student's t distribution. The product of independent conjugate priors is a perfectly acceptable prior, but it is not jointly conjugate. The Conjugate Prior for the Normal Distribution 5 3 Both variance (˙2) and mean ( ) are random Now, we want to put a prior on and ˙2 together. We could simply multiply the prior densities we obtained in the previous two sections, implicitly assuming and ˙2 are independent. Unfortunately, if we did that, we would not get a conjugate prior. For Normal distribution, with know mean and unknown variance. When $\tau = 1/\sigma^2$ ~ Gamma(). In such has posterior of $\tau$ has the following distribution:
conjugate prior normal distribution unknown variance top
Provides an introduction to the example which will be used to describe inference for the case of a normal likelihood, with known variance, and a normal prior... This video provides a proof of the idea that a normal prior with a normal likelihood results in a normal posterior density. If you are interested in seeing m... This is a demonstration of how to show that an Inverse Gamma distribution is the conjugate prior for the variance of a normal distribution with known mean.Th... Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. This is a demonstration of how to show that a Gamma distribution is the conjugate prior for the precision of a normal distribution with known mean. These sho... Conjugate Prior for Variance of Normal Distribution with known mean - Duration: 9:21. ... Inverse Normal distribution: in the Natural Exponential Family - Duration: 10:28. Get more lessons like this at http://www.MathTutorDVD.com.In this lesson, we will cover what the normal distribution is and why it is useful in statistics. ... parameter estimation using maximum likelihood approach for Poisson mass function Putting a Gamma distribution prior on the inverse variance. Also a pre-cursor to Relevance Vector Machines You will need to solve a set of simultaneous equations for this one!
