Questions tagged [posterior]

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

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Finding the posterior mean

I have been trying to solve the following problem: Suppose $X_1,...,X_n$ are iid exponential random variables, with density $f(x;\theta) =\theta e^{-\theta x}$ ,and let us suppose that we have a prior ...
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1answer
38 views

Finding which distribution the posterior is

I'm trying to teach myself Bayesian statistics and am currently trying find the posterior distribution on the following problem: Suppose $X_1,...,X_n$ are iid exponential random variables, with ...
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Is the product of conditional posterior equal to the joint distribution?

Given conditional posterior probabilities of $x$ random variables, can we find the joint distribution by multiplying the conditional posteriors together ? An example would be distribution $$p(x_1,x_2,...
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general rules of the variance of the posterior distribution

In Bayesian posterior inference, we have the following equation relating posterior to prior and likelihood: $$\pi(\theta|D)\propto \pi(\theta)l(D|\theta).$$ Is there any general rules that quantify ...
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How do I read this posterior distribution? [closed]

This might be a very strange question, but I am having a bit of trouble. Here is a posterior distribution. If I am reading this passage out loud, when I get near the end to π(δ1), do I read it as pi ...
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137 views

Why do we do Bayesian Inference about function parameters?

I have been reading about Bayesian inference by Han Liu and Larry Wasserman. In the section 12.2.3 they defined a bayesian inference on a variable parameterised by a function. Given a random variable $...
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Joint and Posterior Distributions of Continuous and Discrete R.V.s

Consider random variables $P$ and $X$ where $P \sim Uniform(0,1)$ and $X|P \sim Binomial (1, P)$. For any $s \in [0,1]$, calculate both $\mathbb{P}(P \leq s, X = 0)$ and $\mathbb{P}(P \leq s, X = 1)$. ...
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Notation predictive posterior distribution and $x^*$, $y^*$

I often see the posterior predictive distribution in ML defined as follows: $$p(y^* \mid x^*, X, Y) = \int p(y^* \mid x^*, \omega)p(\omega, X, Y) d\omega$$ where $\omega$ are all parameters, $x^*$ is ...
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Posterior Distribution of a Normal Sample using Jeffreys Prior with a Known Parameter

Suppose I have a sample of $x_1, x_2, ... x_n$, where $X \sim N(\mu, \sigma^2)$, for some known $\sigma^2$, and that $\mu$ is defined only in $\mu \in [0, b]$, for some finite constant $b$. It then ...
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How to obtain a generalized bayes estimator when we have random sample from the uniform distribution with a Pareto prior and a improper hyperprior?

Let $\boldsymbol{X}=\left(X_{1}, \ldots, X_{n}\right)$ be a random sample from the uniform distribution on $(0, \theta),$ where $\theta>0$ is unknown. Let $$ \pi(\theta)=b a^{b} \theta^{-(b+1)}, a&...
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Bimodal posterior of ATE predicted via Bayesias Additive Regression Trees

I am using BART to estimate the ATE on a large (10k obs x 224 p) dataset with a binomial outcome. In short, I first model the risk of the outcome $Y$ given the $(Z,X)$ covariates, then, for a chosen ...
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Interpretation of concentration of posteriors in the limit of infinitely many independent versus dependent random variables

Disclaimer: the setup and specific example may not be a minimal example to illustrate the point, but I am not well-versed in these topics enough to construct a smaller example without accidentally ...
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Finding posterior probability mass function of binomial parameter

Question: Suppose a lot containing 1000 items is received from a supplier containing parameter (unknown) defective items. The past experiences with this supplier suggest that 5% of items in a lot are ...
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1answer
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Normalize the posterior density for a Cauchy Distribution C ($\theta$,1) and a Uniform [0,100] prior [closed]

Using a Bayesian approach we have $$P(\theta|\text{data})= P(\text{data}|\theta) \frac{P(\theta)}{P(\text{data})}$$ Therefore, the posterior distribution will be proportional to $$\frac{1}{N} (1+(y+\...
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Confused with the fundamental assumptions of Frequentist and Bayesian Linear Regression

In Frequentist Linear Regression, I have seen 2 approaches which lead to basically similar models. We have $W,y,X,\epsilon$ related as $y=W^TX+\epsilon$, where $y$ is the dependent random variable, ...
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compute posterior probability using specificity and sensitivity

Im trying to calculate the posterior and have this posterior_probability = (prevalence * sensitivity)/ ((prevalence * sensitivity) + ((1-prevalence) * specificity)) but am getting the wrong answer?
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Summarize a set of medians and 95% credibility intervals

I have a set of 10 medians and 95% credibility intervals summarizing posterior draws (from the carcass package in R). These 10 ...
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1answer
37 views

Reverse engineering Beta prior parameters from Binomial likelihood and posterior beta parameters

Suppose a friend has calculated a posterior distribution from a Beta prior and binomial likelihood, and you are interested in the prior parameters they used, but they won't give them to you. They only ...
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1answer
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Metropolis Hastings algorithm for joint posterior of probability of heads for 2 coins

I am trying to implement a simple metropolis hastings algorithm to simulate the joint posterior of the probability of flipping heads for 2 coins, $\theta_1,\theta_2$. I am following the problem ...
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Utility of the whole distribution (other than the mean) in Bayesian posterior predictive

In prediction task, when using Bayesian fashion of predictors, I think in most the cases, people just use posterior mean for each individual estimate. I wonder if there's any utility of the higher ...
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Why prior distribution is not conditioned on X?

I would like to know why in the below formula the prior distribution of theta is not conditioned on X (observations): In my understanding, the correct formula should be: P(theta | X, y) = P(y| X, ...
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118 views

How does Prior Variance Affect Discrepancy between MLE and Posterior Expectation

Suppose that $\theta\in R$ is a parameter of interest, $p(\theta)$ is our prior belief regarding $\theta$, and $\hat \theta$ is the MLE for theta derived from the data $x$. It is my understanding that ...
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Approximating Bayesian Posterior with Weighted Likelihood Bootstrap (WLB)

$\DeclareMathOperator*{\argmax}{arg\,max}$Given a set of $N$ i.i.d. observations $X=\left\{x_1, \ldots, x_N\right\}$, we train a model $p(x|\boldsymbol{\theta})$ by maximizing marginal log-likelihood $...
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26 views

Average treatment effect from matrix of individual posterior distributions

I'm trying to estimate the average treatment effect of an intervention using the potential outcomes framework in a classification problem. The analysis uses machine learning to learn $\hat{y} = f(Y, X,...
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68 views

How can I derive the distribution of parameters from data?

For example, I know that my data comes from some normal distribution. I have data - measures (5.5, 4.9, 4.4, 5.3). Is it possible to tell the probability that the mean is less than 5? More generally, ...
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1answer
52 views

posterior distribution of a Poisson mixture model

This is a Poisson-gamma model with mixture prior, thus mixture posterior. I am having some trouble finding the posterior weightings. I have the prior weightings $p_1=1/3$; $p_2=2/3$. The 2 component ...
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How can I insert multiple profiles resulting from a LPA as mediators in a mediation analysis?

I'm currently writing my master thesis and I'm a little stuck at one step of the data analysis process. Maybe you can help me: I'm going to do a Latent Profile Analysis with 12 continuous variables in ...
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Normal learning model

How can I go about calculating a posterior with this information? Suppose that $z_{t}$ is a stochastic signal about a variable $\eta$, $z_{t}=-\alpha\eta +\epsilon_{t}$, where $\eta$ is Normally ...
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43 views

How do I calculate posterior mean for a uniform prior

How do I calculate posterior mean for the following: prior: $m_i \sim U (0,100)$, observations: $X_i\sim N (m_i, 10)$ for $i=1\dots 100$? I understand the theoretics of the answer but having a hard ...
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Help with the prior distribution

The question is as follows: Consider an SDOF mass-spring system. The value of the mass is known and is equal to 1 kg. The value of the spring stiffness is unknow and based on the experience and ...
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Intuition on why Gibbs Sampling samples from the posterior distribution

I am new to Gibbs Sampling and I do understand how the algorithm works but I would also like to understand how sampling from the conditional distributions is equivalent to sampling from the joint. ...
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What is the Dirichlet Proces Mixture Models posterior

I am trying to understand Dirichlet Process Mixture models. One of the videos I have been watching is by Tamara Broderick. I think it is a very good introductory video to Dirichlet Process mixture ...
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Uniform posterior on bounded space vs unbounded space

According to this answer: There's no problem with a flat posterior on a bounded space, as here. You just have to start out with a prior that's more spread out than a flat one. What you can't have is ...
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What is the probability that a parameter is greater than other parameters given we know the posterior distribution

Suppose that we have have some form of compositional data $x_i, i\in[1, n]$ which we propose comes from a Dirichlet distribution such that $$ x_i \sim \mbox{Dir}(\lambda \alpha),$$ where $x_i=(x_1^{(i)...
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Uniform posterior on bounded space [duplicate]

In a particular Bayesian problem, I have encountered a choice of parameters that leads to a uniform posterior distribution. Given prior \begin{equation} p(\boldsymbol{\pi}) =Dirichlet(\boldsymbol{\...
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Is the density of the unnormalised posterior distribution the same as the density for normalised posterior? [duplicate]

The posterior distribution is proportional to the likelihood times the prior distribution $p(\theta|D) \propto p(D|\theta)p(\theta)$. Computing $p(D|\theta)p(\theta)$ would give the un-normalised ...
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Can we directly use the likelihood function for Bayesian inference? [duplicate]

I'm new to statistics, so this might be obvious. When doing full Bayesian inference, we first compute the parameter posterior $P(\theta | \text{data})$ using Bayes' rule. Then, to make a prediction, ...
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Why is the normalisation constant in Bayesian not a marginal distribution

The formula for Baye's rule is as follows $$p(\theta |D) = \frac{p(D|\theta)p(\theta)}{\int p(D|\theta)p(\theta)d\theta}$$ where $\int p(D|\theta)p(\theta)d\theta$ is the normalising constant $z$. ...
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Posterior distribution of $\sigma^2$

In chapter 9 of Jim Albert's Bayesian computation with R it's mentioned that, in the context of Normal Linear Regression, the posterior joint density is: $$g(\beta, \sigma^2 | y) =g(\beta|y, \sigma^2)...
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Squared Error Loss for Bayesian estimator of Normal distribution

I'm following Brad Efron and Trevor Hastie book "Computer Age Statistical Inference" (link). In chapter 7, they begin to debate the James-Stein estimator by calculating the Bayes rule, or ...
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The Bayes' Theorem Components of the Probability Output of a Classifier

Let's give a simple setup. I have $500$ photos of dogs and $500$ photos of cats, all labeled. From these, I want to build a classifier of photos. For each photo, the classifier outputs a probability ...
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Posterior mean of $\mu$ in Bayesian Hierarchical model (Poisson-Gamma)

Chapter 7 of Jim Albert's book considers the case of using a hierarchical model, to estimate heart-transplant mortality rates ($\lambda_i$) from 94 hospitals, each with it's own exposure (# of ...
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Laplace approximation in high-dimensions

Obviously computing the inverse Hessian is hard when a probability distribution is fitted on high-dimensional datapoints. One idea to reduce computational cost would be to approximate the distribution ...
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109 views

Bayes Estimator for Bernoulli Variance

I have the following question: Let $X_1,\dots,X_n$ be independent, identically distributed random variables with $$ P(X_i=1)=\theta = 1-P(X_i=0) $$ where $\theta$ is an unknown parameter, $0<\theta&...
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What is the posterior distribution of two parameter Weibull likelihood and a uniform prior?

I've seen examples of Jeffreys' prior with a Weibull likelihood. I was wondering if $p(\alpha) = 1$, then what is the posterior distribution of alpha for two parameter Weibull in that case.
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How are artificially balanced datasets corrected for?

I came across the following in Pattern Recognition and Machine Learning by Christopher Bishop - A balanced data set in which we have selected equal numbers of examples from each of the classes would ...
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Metropolis-Hastings exercise with Cauchy and normal distributions [self-study]

I have the following exercise to solve and would appreciate some help. Consider a linear regression model $y = X\beta + \varepsilon$, where $y = (y_1,...,y_T)'$, $X = (x_1,...,x_T)$, $x_t$ is a single ...
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72 views

Bayesian inference with an arbitrary prior

A classical problem in Bayesian inference arises when we wish to learn about (say) the fraction $\theta$ of balls in an urn that are white; and do so by sampling from the urn with replacement. In such ...
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1answer
36 views

Beta-Binomial conjugate proof

Can someone explain this proof to me? I get stuck on the transition from the third line to the last line. Namely: Is the integral being evaluated or not? How does the entire expression reduce to a ...
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What's the score employed by Platt scaling to compute SVM posterior probabilities?

I have read about the Platt scaling approach to compute posterior probabilities for the SVM classifier $P(y=1|x)$. In Scikit-learn's SVC (SVM) implementation this is the approach used to produce ...

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