Questions tagged [posterior]

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

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Notation for conditional density

Are $p(\mu \mid \sigma)$ and $p(\mu ; \sigma)$ equivalent? I've seen the notation $p(b_i \mid T_i, \delta_i, y_i ; \theta)$ used to represent the posterior distribution for $b_i$. I am assuming that ...
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Shrinkage effects in a hierarchical model

I am working on the chimpanzees dataset from Richard McElreath's text, "Statistical Rethinking", edition 2. I have built 2 simple models, one a fixed effects model and the other a hierarchical model. ...
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What distributions are conjugate to themselves, besides the normal?

I know the normal distribution is conjugate to itself; are there others? Is there some sort of intuition behind why a given distribution would be conjugate to itself?
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Estimate random effects for a new individual with a linear mixed effects model

Consider repeated observations $\mathcal{Y} = (y_{i,j})_{i,j}$ obtained for $p$ individuals ($1 \leq i \leq p$), at different time points $t_{i,j}$ $(1 \leq j \leq n_i$). The "random slope and ...
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exact form for the marginal posterior

I have a question that I come across for practicing. Basically the question is this: Consider a random sample from the normal distribution with unknown mean and variance $Y_i \sim^{i.i.d.} N(\mu, \...
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106 views

Why in Hamiltonian MCMC do we multiply the posterior distribution by the likelihood?

So maybe I am misunderstanding what the author is staying, but I am reading Chapter 14 of Kruschke's Doing Bayesian Analysis. I am reading about the software Stan and how it uses the Hamiltonian MCMC ...
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72 views

mismatch in sampling between t distribution and normal-inverse-gamma distribution

I am looking at equivalence of sampling between t distribution and normal-inverse-gamma (NIG) distribution in python. The results don't match, and I want to see if there's a mistake in how I am ...
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Bayesian inference problem with three parameters

Assume that I observe a yearly time series for the number of certain events occurring per year. Also, assume that the data points come from a Poisson process with parameter $\lambda$. The dataset in ...
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Marginal probability in Gaussian Process

Let $\mathbf{a} \sim \mathcal{GP}(\mathbf{m},\mathbf{C})$ where $\mathbf{a} \in \mathbb{R}^T$ is modeled as Gaussian process with mean $\mathbf{m} \in \mathbb{R}^T$ and prior covariance $\mathbf{C} \...
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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 ...
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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) = \...
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91 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 ...
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$X\sim \mathcal{N}(\theta,\sigma^2)$, $\pi(\theta,\sigma^2)\propto 1/\sigma^2$, $Y\sim \mathcal{N}(\rho X,\sigma^2)$, $\rho$ fixed. $f(y|x)$?

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,\...
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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 ...
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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 ...
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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 ...
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51 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 ...
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203 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 ...
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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 ...
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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 ...
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284 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 $...
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Is there a derivation for the Posterior Predictive Distribution?

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 ...
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Posterior distribution of Bernoulli distribution

The pdf of X | $\theta$ is given by $\theta^x (1- \theta)^{1-x}$ and its prior distribution is given by $p(\theta) \frac {1} {B(\alpha, \beta)} \theta^{\alpha - 1} (1 - \theta)^{\beta - 1}$ where $...
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Representativity of a sample to a reference population with a mixed distribution

I a have a series of small samples taken not randomly from a reference population with a complex distribution over a categoric variable with 21 levels and a continuous one in the x > 0 domain. I ...
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Reducing dimension $P(\theta|y) = P(\theta|s)$ in the posterior distribution

Given a sample of $n$ independent observations $\boldsymbol{y}$. Let $S(\boldsymbol{y})$ be a sufficient statistic for the underlying parameter $\boldsymbol{\theta}$ so that the density can be ...
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Threshold crossing: Integrating posterior distribution over a forecast period

Suppose I have some time series $\{Y_t\}_{t=1}^T$ and I have come up with some probabilistic model: $$p(Y_{t+h} \mid Y_{1:t}, \theta)$$ for any horizon $h$, and some parameters $\theta$. Now ...
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Estimation of joint errors

Suppose we have multiple normal random variables, each which is a normal distribution with it's own mean but joint variance: $$X_i \sim N(\mu_i, \sigma^2)$$ Now suppose we collect data from these ...
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Nested sampling: estimate of bulk posterior support over prior

Going through the details of the Nested sampling Skilling paper, and I've encountered an estimate in Section 5 which I cannot reproduce. Rephrasing what's mentioned in the paper: we assume to have a ...
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Bayesian inference in independent but not identically distributed examples

Let data $\lbrace X_i \rbrace _{i=1}^m$ be examples sampled independently from normals, $X_i \sim N(\mu,\sigma_i)$. Variances are known but mean $\mu$ is unknown. Let prior on $\mu$ be also a Gaussian,...
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1answer
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Approximating p-values from Bayesian posterior distribution

I'm preparing a manuscript for publication in which I have fit a mixed linear model using Bayesian regression. I'm assessing whether group A is bigger than group B. In the paper, I have reported the ...
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what is the step by step procedure of updating a posterior probability with new data coming in

I have a question about how to update posterior probability sequentially when new data comes in sequentially by say $x_1$, then $x_2$, then $x_3$,.... I understand this form when the first data $x_1$ ...
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Posterior with a much larger uncertainty than the prior

I have done an MCMC analysis with many variables. One of my nuisance parameters has a Normal prior distribution with mean 0 and standard deviation 1. The posterior distribution for this parameter has ...
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Help with Old exam questions on Bayesian Inference Problem [closed]

I've been trying to teach myself bayesian inference and I found a question sheet online ---> https://math.mit.edu/~dav/05.dir/ps6.pdf. I was attempting to solve question 4 but I'm not sure the method ...
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Is there an algebra error in this derivation in a Coursera class of the posterior of a Normal with a Student-distributed mean?

This is a screenshot from Coursera's class "Bayesian Statistics: Techniques and Models", Week 1, "Non-conjugate models" lecture (any one can audit the class and access the materials for free): This ...
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Bayesian predictions from posterior parameter distributions

I have two physical models $f(\theta)$ and $g(\theta)$ (not probability distributions) parameterized on the same set of parameters $\theta$. I also have data $y$ with measurement noise $\epsilon$ ...
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Multivariate bayesian inference: learnig about the mean of a variable by observing another variable

I want to derive a Bayesian learning procedure where I don't only learn from my own signal, but also from other signals which are correlated to mine. I thought it could simply work with Bayesian ...
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Bayesian Inference: Feeding Posterior back in as Prior

I've just started reading about Bayesian Inference, and one thing I've wondered about is if it's possible to feed the posterior in as a new prior for a new model, using the same data. Or would that ...
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Mechanics for combining likelihood and prior in non-trivial case

Bayes rule is simple enough on its face: $$ pr(B|A) = \frac{pr(A|B)pr(B)}{pr(A)} $$ If these things are known scalar probabilities, the answer is simple to compute. But I'm failing to understand ...
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MAP versus Component-Wise Maximum Marginal

Suppose I have the joint distribution: \begin{align} p(\mathbf{x}) = p(x_1, ... , x_n) \end{align} The maximum a posteriori (MAP) solution is given by: \begin{align} \mathbf{x}_{MAP} = \arg \max p(\...
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102 views

Posterior distribution from piecewise likelihood

Consider a hierarchical Bayesian model for analysing data from an inhomogeneous Poisson process that we observe in discrete time. Let $Y_i, i = 1,...,n$, be the number of events occurring in the time ...
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Is it valid to use samples drawn from posterior using MCMC as prior distribution in sequational updating?

Let's say there are two sets of parameters $x$ and $y$, and two sets of data $a$ and $b$. $x$ only depends on $a$, but $y$ depends on both $a$ and $b$. I first sample from the posterior $$P(x \mid a) ...
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Notation in definition of a quantity involving uncertainty and posterior probability

As a probs & stats noob, I'm still getting confused by notation. I would appreciate if someone could elaborate a bit on what's going on here, on p12 of Settles' Active Learning survey (2012). He ...
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Comparing posteriors when the same data are examined with different priors

I have two Bayesian regression models: both use the same data $D$ and all same specification $M$ but the different priors $p_1(\theta)$ and $p_2(\theta)$. Can I interpret the posteriors from the two ...
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79 views

Poisson-gamma posterior mean expectation

Let's have a gamma prior $\lambda\sim \operatorname{Gamma}(a,b)$ (mean: $\frac{a}{b}$) With Poisson data $Y\mid \lambda\sim \operatorname{Pois}(N\lambda)$ (mean: $N\lambda$) The posterior is $\...
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What is the posterior mean of $\mu$ given a randomly stopped i.i.d. observations from a Normal

Let's imagine I have a machine giving me an independent random number from a normal distribution $N(\mu,1)$ whenever I push a button. I have a stopping rule to decide how many samples to collect - I ...
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Residual Analysis

I have just reproduced an example regarding a regression model for fibre strength data. The data consisted of tensile strengths of silicon carbide fibre at four different lengths. From the data, a ...
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meaning of posterior distribution and credible internal

In Bayesian method, we can get a posterior distribution of a parameter. Now I want to do some simulation to know if the posterior distribution is the same as the true distribution. For example, mean ...
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54 views

how posterior function is calculated in JAGS

I have a theoretical question. I understand the JAGS samples from the posterior function of a model. But I don't understand (nor I can find in the documentation) how it calculates the posterior in the ...

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