Questions tagged [posterior]

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

Filter by
Sorted by
Tagged with
2
votes
0answers
29 views

Does a conjugate prior always exists? [duplicate]

Are there distributions where no conjugate prior exists? Is there a necessary and/or sufficient condition which guarantees the existence of a conjugate prior? Edit: Why has this question been closed? ...
4
votes
1answer
48 views

Can the posterior mean always be expressed as a weighted sum of the maximum likelihood estimate and the prior mean?

See this question. Is this always true? Can the posterior mean always be expressed as a weighted sum of the maximum likelihood estimate and the prior mean (after choosing some appropriate prior)?
0
votes
1answer
28 views

poisson posterior = gamma function x uniform prior?

I was reading the section 3.2 of this paper. Above equation (3.2), the authors say "The posterior on the background estimate is then conservatively taken to be the Poisson posterior using a ...
4
votes
2answers
64 views

Under what circumstances can an improper prior be used in bayesian analysis?

I am attempting to gain some intuition about the use of priors in bayesian analysis. I have read in some instances that an improper prior can be used when no information is known. However here is my ...
0
votes
0answers
19 views

Question on working out a pdf from a posterior distribution

Need help with part (a) of this exercise. The exact step I'm concerned about is calculating the pdf from the relationship given in the exercise. I will appreciate any explanations on how should I ...
0
votes
1answer
29 views

Uncertainty and distribution of a percentile

In a Bayesian analysis (Normal case), it is possible to obtain a posterior distribution of the mean and variance. We can also obtain quantiles, median,... of these distributions. My question now is: ...
2
votes
1answer
45 views

What is the correct procedure for posterior inference about the cumulative distribution $F(y|\theta)$?

Let $y$ follow $f(y|\theta,x)$ with $\theta$ parameters with prior $\pi(\theta)$ and $x$ covariates with distribution $p(x)$. Let $p(\theta| D)$ be the posterior distribution of the parameters, where ...
2
votes
1answer
58 views

How can I practically “marginalize away a nuisance parameter”?

I generate my data with this model: $y=ax+b+\nu$ where $\nu$ is a random value from a random variable which follows a normal distribution with mean equal to zero and standard deviation equal to $\...
0
votes
0answers
21 views

Posterior distribution for a real sample

I'm supposed to write a function in R that evaluates the posterior density, up to a multiplicative constant, at $\boldsymbol{\pi}$ for a logistic regression given a dataset $y$ and $X$. Where $y$ ...
1
vote
0answers
17 views

Calculating the conditional expectation of an exponential family [closed]

If we have $X$ with a density depending on the scalar parameter $\theta$, where the density is from of the exponential family: $f(x;\theta) = \exp(\theta x−\phi(\theta))h(x)$. Also we have that $\...
1
vote
1answer
30 views

Posterior density for normal distribution

I have that $Y_i$ has distribution $N(\beta x_i^2, 1)$, $i=1 \dotsc n$ where $\beta$ has a prior distribution $N(0, \sigma^2)$. I need to calculate the posterior density of $\beta$, find its mean and ...
0
votes
0answers
12 views

Is there a way to measure the degree of correlation between two posterior distributions?

I have the following two distributions of MCMC samples from a Bayesian analysis and want to know the extent to which values from the first distribution are correlated with those from the second. Is ...
1
vote
0answers
34 views

Do most scientific discoveries commit the conditional probability fallacy?

Usually, in an experimental scientific paper where we have observations $X$ taken in laboratory conditions, we either reject some model $\mathcal{M}$ under the basis that $P(X \mid \mathcal{M}) \ll 1$,...
0
votes
0answers
42 views

Posterior of a Normal distribution

I have a problem obtaining a posterior of a normal multivariate distribution. The problem is as follows: Assuming $ \mathbf{X} \sim N_p (\boldsymbol{\mu}, \boldsymbol{\Sigma}) $, known $\boldsymbol{\...
3
votes
1answer
72 views

Linear combination of conjugate prior

Let's say we want to find the posterior distribution for $\Theta$, where the likelihood model $X|\Theta$ ~ $Binom(8000, \Theta)$. Suppose instead of one distribution for the prior, we use a linear ...
1
vote
1answer
65 views

What is the posterior in-sample vs the posterior out-of-sample?

I'm watching this video on Bayesian modelling for the stock market by Thomas Wiecki, Thomas has a slide with two posterior distribution over the mean parameter in his stock return model. Around 18:26 ...
3
votes
1answer
80 views

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 ...
1
vote
1answer
43 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 ...
0
votes
0answers
11 views

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,...
1
vote
1answer
29 views

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 ...
-1
votes
1answer
27 views

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 ...
2
votes
1answer
142 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 $...
1
vote
1answer
24 views

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 $\mathbb{P}(P \leq s, X = 1)$. ...
1
vote
2answers
36 views

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 ...
0
votes
0answers
46 views

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 ...
1
vote
0answers
20 views

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&...
0
votes
0answers
6 views

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 ...
0
votes
0answers
9 views

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 ...
3
votes
1answer
109 views

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 ...
-1
votes
1answer
39 views

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+\...
2
votes
2answers
64 views

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, ...
0
votes
0answers
7 views

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?
0
votes
0answers
36 views

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 ...
-1
votes
1answer
41 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 ...
0
votes
1answer
62 views

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 ...
0
votes
1answer
42 views

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 ...
2
votes
2answers
130 views

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, ...
2
votes
2answers
127 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 ...
0
votes
0answers
43 views

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 $...
1
vote
1answer
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,...
0
votes
2answers
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, ...
0
votes
1answer
59 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 ...
1
vote
0answers
18 views

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 ...
0
votes
0answers
18 views

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 ...
0
votes
0answers
47 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 ...
2
votes
1answer
88 views

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 ...
1
vote
0answers
23 views

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. ...
0
votes
1answer
18 views

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 ...
4
votes
2answers
196 views

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 ...
0
votes
1answer
17 views

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)...

1
2 3 4 5
16