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Questions tagged [posterior]

Refers to the probability distribution of parameters conditioned on data in Bayesian statistics.

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What do these equations on Bayesian regression (MAP) from Chapter 3.3 in PRML by Bishop mean?

This was taken from Ch 3.3 on Bayesian Linear Regression from Pattern Recognition in Machine Learning by Bishop. Apparently the posterior can be described by eq 3.49. Eq 3.48 represents the prior ...
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Explanation of Equation 5.3 from Gaussian Processes for Machine Learning

I am currently reading through C. E. Rasmussen & C. K. I. Williams' Gaussian Processes for Machine Learning and was going through chapter 5. I could not exactly understand the derivation of ...
19 views

Find posterior distribution given prior distribution

Problem Be $X_1,...,X_n$ a random sample of $X$ ~ $Geometric(\theta)$, i.e., $f(x|\theta)=\theta(1-\theta)^x \forall x = 0,1,2,...$ Assuming a prior distribution for $\theta$ find the posterior ...
32 views

Posterior Predictive Distribution for Uniform Likelihood and Pareto Prior

I'm trying to find the posterior predictive distribution for data $X_i, \dots X_n$ from a a $Uniform [0, \theta]$ distribution. The prior distribution for $\theta$ is a $Pareto[\alpha, \beta]$ ...
9 views

Deriving full conditional of ordered probit model (Bayes)

I have a question regarding the following exercise: I am able to compute the complete (full) data likelihood function, the full conditionals of $y^{*}_i$ and $\beta$. However, I do not know how to ...
15 views

Dynamically updating posterior density in R

I want to redefine my function in a loop by calling the function from last iteration. However I know this is basically a recursive way which I don't want. To give an example, see the following ...
26 views

Range of integration for joint and conditional densities

Did I mess up the range of integration in my solution to the following problem ? Consider an experiment for which, conditioned on $\theta,$ the density of $X$ is \begin{align*} f_{\theta}(x) = \...
18 views

Generating marginal posteriors from MCMC output in two-factor model

Quick summary: if I have a MCMC sample of the posterior distribution of two factors and their interactions, can I marginalize out one factor simply by concatenating the posterior samples from each ...
11 views

Parameter estimation: Chi^2 vs. posterior density

Let's say I want to estimate the mean of a sample of i.i.d normal random variables. For this I use two methods: Compute Chi^2 as $\chi^2 = \sum (x_i - \mu)^2/\sigma^2$ for a range of possible means ...
30 views

Bayesian Linear Regression, trouble with posterior. Variance equal identity

I am trying to solve the following problem. If $y | \beta \sim N(X \beta, I_n)$ and $\beta \sim N(0, g^{-1}(X^t X)^{-1})$ for $g>0$. Find $\pi(\beta|y)$ and show that $E(\beta|y)$ is a function ...
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What is the conceptual difference between posterior and likelihood? [duplicate]

I have trouble discerning conceptually between these two notions. I am aware of their formal relations, proprieties and what not, but I just can't wrap my head around what they "mean", if that even ...
28 views

like in the title I have the following question. Let $X\sim \mathcal{N}(\theta,\sigma^2)$ with the improper prior $\pi(\theta,\sigma^2)\propto 1/\sigma^2$ and consider $Y\sim \mathcal{N}(\rho X,\... 1answer 111 views GIBBS sampling : do samples for the subset of variables approxiamate the related marginal distribution? I'm reading the page Gibbs sampling on wikipedia. I really don't understand why the following statement is true. "The marginal distribution of any subset of variables can be approximated by simply ... 0answers 46 views How to start coding for posterior inference I am trying to implement the model given in http://proceedings.mlr.press/v84/andersen18a/andersen18a.pdf where they have used mean-field variational inference for posterior inference, but I want to ... 0answers 24 views Reproducing a didactic example of Lindley (1993) Lindley (1993) discusses the following mixed discrete and continuous prior for the tea tasting lady experiment, where$\pi$is probability of a correct classification:$p(\pi=0.5) = 0.8$(discrete ... 1answer 23 views How does Thomson Sampling work in a real world application of Multi Armed Bandit Testing I understand the basics of Thomson Sampling, but how is it implemented in practice? If there are three variants each with a 1/3 of traffic allocated to them on day 1, how is traffic dynamically ... 0answers 21 views Calculating probability from posterior distribution with continuous parameter space Suppose that we are interested in the parameter$\theta$, and that our prior distribution for$\theta$is given by$Beta(a=.1, b=.9)$. For the binary random variable$Y$that takes value$1$with ... 1answer 38 views Finding the posterior distribution given an improper prior Let$X \sim N(\theta, \sigma^2)$where$\sigma^2$is known. Let the prior density$\pi(\theta) =1, \theta \in \mathbb{R}$to be the improper uniform density over the real line. Find the posterior ... 1answer 47 views Questions about the principles of Bayesian analysis + R [closed] Let's say I have a data of flywing lengths which is identically distributed (normal). (data: https://seattlecentral.edu/qelp/sets/057/s057.txt). I want to estimate the mean (theta). I have to choose ... 0answers 10 views Posterior probability and linear discriminant analysis Is it possible, using any form of the posterior probability calculations, to assign a cutoff point in linear discriminant analysis based on a posterior probability value? Hypothetical scenario: We ... 2answers 42 views Borrowing observations for prior probability in Bayesian Inference For the purposes of Bayesian Inference, is it assumed that the historical observations used for the prior probability values must be from the exact entity for which you are looking to calculate the ... 1answer 32 views Uniform Prior on Normal Mean with Known Variance Implies Truncated Normal Posterior? Let's say I have a uniform prior$\mu \sim \mathcal{U}(a,b)$, a normal likelihood$y|\mu \sim \mathcal{N}(\mu,\sigma^2)$with known variance$\sigma^2$, and one observation$y$. Is then the posterior$...
I came across this term in the deep learning book: $p(x_{m+1}|x_1 ... x_m) = \int p(x_{m+1}|\theta)p(\theta|x_1 ... x_m)d\theta$ After some research I find that this term is the definition of the ...