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Questions tagged [posterior]

In Bayesian statistics, the term 'posterior' refers to the probability distribution of a parameter conditioned on the observed data.

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Basic question about deriving MAP estimator

Say we have a random process $X(t, u)$ parametrized by $t$ and $u$ that generates data $x$. We also have a prior on $u$, $p(u)$. Am I correct in stating that the expression to find the maximum a ...
DangerousTim's user avatar
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how can predictive distributions be considered as expectations?

I guess that the prior and posterior predictive distributions can be considered expectation of $p(y|\theta )$ (in case of prior predictive distribution) and $p(\widetilde{y}|\theta )$ (in case of ...
Sherlock_Hound's user avatar
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two-step gibbs sampling vs block gibbs sampling

While reading Bayesian-related technical articles, I can see algorithms such as two-step Gibbs sampling and block gibbs sampling ...
user3269's user avatar
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known variance in conjugate normal

$Posterior\ mean=\frac{1}{\frac{1}{\sigma_{0}^{2}} + \frac{n}{\sigma^{2}}}\left( \frac{\mu_{0}}{\sigma_{0}^{2}} + \frac{\sum_{i=1}^{n} x_i}{\sigma^2} \right)$ Using this updating equation with known ...
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Bayes factor for hypothesis

I am studying Bayesian hypothesis testing and I want to calculate the Bayes factor for \begin{align*} H_0: \lambda = 1 \hspace{0.2cm} vs \hspace{0.2cm} H_1:\lambda > 2 \end{align*} with $p(\...
daniel's user avatar
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How to obtain likelihood ($P(B/R)$ given the prior $P(R)$ and the posterior $P(R/B)$

I am working on a topic related to multiple-choice response. I would like to measure the efficiency of the information source (or a student’s information search) and I believe Bayesian statistics is ...
Francisco 's user avatar
2 votes
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39 views

Likelihood from posterior [closed]

This question is strange and perhaps silly but it would be very useful for my research. Is there any method to find the likelihood given a prior distribution and its corresponding posterior ...
Francisco 's user avatar
1 vote
1 answer
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How to choose what to integrate out or what to condition on for marginal distributions?

I am trying to work out the Bayesian posteriors of $\theta$, $\tau$ and the $\varepsilon$ in the following model: $$y(t) = \phi(t,\tau)\theta+v(t),$$ where $\{v(t)\}$ is an iid sequence of random ...
MJPeel's user avatar
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Normal approximation for posterior distribution

I am reading the example 4.3.3 of "The Bayesian Choice" by Christian P. Robert and I was wondering if it is possible to obtain a normal approximation in this case to estimate the posterior. ...
daniel's user avatar
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Interaction: Posterior comparison brms, difference between as_draws, and posterior_predict. Is it correct to interpret posterior_predict instead?

I have an interaction effect in my model, and I want to extract the posterior of each of my parameter in order to compare them and make inference about them. I couldn't simply use the as_draws() ...
Guillaume Pech's user avatar
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How to prove the posterior probability for multivariate case (i.e. dimension $d\ge 2$)?

Suppose there are $k$ groups, $\pi_1, \pi_2, \cdots, \pi_k$, with the probability density function for group $\pi_i$ being $f_i(\boldsymbol{x})$, where $\boldsymbol x\in R^d$. The prior probability ...
John Stone's user avatar
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How to leverage the separable functions in MCMC sampling? [closed]

I'm considering the posterior of a parametric model via the Bayesian approach. More specificity, I have a parametric model $u(p_1,p_2, p_3) = u_1(p_1) \times u_2(p_2) \times u_3(p_3)$ and I want to ...
CC Kuo's user avatar
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Overcoming posterior correlation for a model with random effects (for a Gibbs sampler)

I am trying to infer parameters for a model of case numbers of different infectious diseases in different locations over time. The model is $$ \log \left(1 + y_{ijt}\right)\sim\mathsf{Normal}\left(\mu ...
Till Hoffmann's user avatar
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Can we get probabilistic predictions evaluable by proper scoring rules from bayesian inference without evaluating the marginal likelihood?

Let's say we have a vector of inputs, $X=[x_0,\dots, x_{n-1}]$, and a vector of outputs, $Y=[y_0, \dots, y_{n-1}]$. We would like to predict the distribution of a new output ,$\hat{y}$, given a new ...
QMath's user avatar
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Mutual Information decay

Consider $m$ channels indexed by $i$ with $1 \leq i \leq m$. The input alphabets are from the same finite set $\mathcal{X}$. Let $\pi$ denote a probability distribution on $\mathcal{X}$. Define the ...
Sushant Vijayan's user avatar
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Confidence interval over point estimate given regression parameters

In Bayesian analysis, the posterior distribution is often sampled when the PDF is not intractable (often). If the samples are of length $n$, then every index in $range(1,n)$ corresponds to a valid ...
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Computing Bayesian model averaged posteriors

The Bayesian model averaged posterior predictive distribution for new data $\tilde{y}$ given training data $y$, across a set of $M$ models $\mathcal{D} = \{D_{1}, ..., D_{M}\}$, is defined as: \begin{...
user_15's user avatar
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1 answer
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How to decompose the conditional posterior prob? [closed]

I am learning bayesian inference now. A problem I encountered a lot of time is, when I need to calculate or simplify the posterior prob., I don't know how should I begin, according to what I have. For ...
littletennis's user avatar
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How to derive conditional posterior predictive distribution from definition of posterior predictive distribution in bayesian regression?

In my situation, I have a set of data points: $$ z_{0:n} = \\{ (x_0, y_0),\dots ,(x_{n-1}, y_{n-1}) \\} $$ I am trying to figure out how to derive the fully expanded form for the conditional posterior ...
QMath's user avatar
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MCMC for correlated Posterior

I'm simulating the posterior of a (as it seems) highly correlated Posterior distribution using MCMC (DREAM Algorithm). My setting is that I have 7 parameters where x1/x3 and x2/x4 is highly correlated,...
Sobol's user avatar
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Turning a list of cost into categorical probability mass distribution

Background Given a noisy dataset $D$, I have to solve a classification problem where the possible anserwer is $i\in\{1,\dots,N\}$. So far I can get pretty decent result with an algorithm that, based ...
matteogost's user avatar
1 vote
1 answer
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Simple example of Log-Sum-Exp trick for continuous case

I am trying to confirm my understanding of how to apply the [Log-Sum-Exp trick to recover a posterior distribution from a log-posterior distribution. I want to consider a simple example from a model I ...
John Doe's user avatar
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Recovering normalized posterior distribution from log-posterior

For a Bayesian estimation problem that I am working on, where I update the log-posterior (many times based on data) instead of the posterior itself using Bayes rule. I find the following (rather ...
John Doe's user avatar
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Understanding distribution of distributions

I was studying Bayesian plausibility but was stuck on some basic concepts. Bayesian plausibility condition is an equation that says $$\int_{\Delta(\Omega)}pd\mu(p) = P$$ Where $P$ is the prior ...
trueWarrior09's user avatar
1 vote
2 answers
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Interpretation of posterior predictive distributions

QUESTION UPDATE due to the comments I have received so far. The data, the example and the results below are fictitious, as I am interested with the correct interpretation of these results. Suppose I ...
Rustam's user avatar
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Posterior distribution of shape & rate parameter in Poisson-Gamma Mixture

Currently I'm struggling to handle the following question. Suppose $x_i,(i=1,2,\dots,n)$ follows Poisson distribution: $$p(x_i|\theta) = \frac{\theta^{x_i}e^{-\theta}}{x_i!}, \quad x_i\in\mathbb N,\...
jason 1's user avatar
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0 answers
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Predictive Posterior Distribution of a Bivariate Normal Distribution with Unknown Mean and Variance

I am working on a homework assignment and have been stuck on it for a while and would like to seek for some clarification. Given a likelihood function $y_i|\theta,\sigma^2 \sim N(\theta,\sigma^2)$ for ...
ak_mng's user avatar
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MCMC seems very sensible to the evidence

currently starting to study bayesian ML, and specifically MCMC, in order to compute the posterior: $$ P(\theta|D) = \frac{P(D|\theta)P(\theta)}{P(D)} $$ Now, I see how the acceptance ratio makes sense ...
Alberto's user avatar
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3 votes
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How should you determine the probability returned by a flat uniform prior function

I am currently doing an analysis that involves fitting a model to a 1D graph. Following the example on the emcee documentation, I started with Maximum likelihood estimation and am now looking at using ...
shram's user avatar
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1 vote
0 answers
38 views

Using bootstrap for accurate posterior in Variational Bayes

A common well-known issue in Variational Bayes is the variance underestimation of the posterior. Some methods using "sandwich" variance have already been proposed but provide frequentist ...
Mangnier Loïc's user avatar
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9 views

Choosing Distortion Measures for Decision Rules with Logarithmic Posteriors

I've been delving into Bayesian decision theory and specifically looking at scenarios where we work with the logarithm of the posterior distribution (log-posterior). My understanding is that in such ...
Alireza's user avatar
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0 answers
35 views

How to derive conditional destribution of MVN variable

I am working with following model specifications (Regression_ Modelle, Methoden und Anwendungen-Springer-Verlag Berlin Heidelberg (2009), p. 147): $$Y \sim MVN(X\beta, \sigma^2I)$$ $$\beta|\sigma^2 \...
BlankerHans's user avatar
1 vote
1 answer
66 views

Full conditional posteriors

so up to now I dealt with posteriors in the form of: $$p(\theta|x) \propto p(x|\theta) p(\theta)$$ No we started to model a linear regression with the bayesian approach: $$Y \sim MVN(X\beta, \sigma^2I)...
BlankerHans's user avatar
3 votes
0 answers
42 views

Does the mode of MCMC samples equal the MAP of the posterior?

If I had millions of MCMC samples from a posterior, should the most frequent value among those samples (i.e., the peak of a histogram of those samples) at least in principle always equal the maximum-a-...
Durden's user avatar
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2 votes
0 answers
57 views

Credible intervals with parameter near boundary

When doing Bayesian inference on a parameter that is bounded, often we use priors that approach 0 as the parameter approaches the boundary. For example, when estimating $(\mu, \sigma^2)$ for normal ...
half-pass's user avatar
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2 votes
1 answer
195 views

Given N observations - Bayesian Posterior for Unknown Variance of a Normal Distribution with a Known Mean?

So, starting from no information besides N trials from a Gaussian with $\mu = 0$, I'd like to know the best Bayesian posterior for the unknown variance, $\sigma^2$. My approach so far as been to ...
SSD's user avatar
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0 answers
41 views

Gaussian posterior/posterior predictive mean [duplicate]

I'm a little confused regarding the difference between the posterior and the posterior predictive distribution. I know that the posterior is a distribution over the parameter, while the posterior ...
Jannik's user avatar
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Why is the noise included in the posterior predictive distribution in bayesian regression?

Assume the following model: $y = b_0 + b_1 * x$ where we set some priors to $b_0, b_1$. Let $I$ denote our historical data and $x^*$ denote future inputs. Let $p(b_0, b_1|I)$ denote our posteriors. We ...
karl henriksson's user avatar
7 votes
4 answers
287 views

A seeming paradox regarding estimation of the number of buttons

There is a computer with $N$ buttons in a secret room. We do not have access to the computer and we do not know $N$. But we know that $N\leq 100$ and we have a ever so slightly larger prior for ...
Feri's user avatar
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2 votes
2 answers
72 views

How can I get the probability of a predicted outcome conditional on the posterior in bayesian regression?

Assume that I am running the following regression: $\hat{y_t} = \beta_0 + \beta_1 \cdot x_t$ where as $\hat{y_t}$ is a continuous variable. Lets assume a gaussian likelihood and nonconjugate priors ...
karl henriksson's user avatar
1 vote
0 answers
27 views

Understanding the Binomial likelihood notation

Let $X \sim Bin(n,\pi)$. I don't understand why the binomial likelihood is then given by $f(x|\theta)=\binom{n}{x} \theta^x (1-\theta)^{n-x}$. Shouldn't it be $B(x|\pi,n)=P(X=k)=\binom{n}{k} \pi^k (1-\...
BlankerHans's user avatar
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0 answers
43 views

Posterior of Inverse Wishart distribution with a subset of data observed

Suppose: \begin{equation} x_1\in \mathbb{R}^{p_1}\\ x_2\in \mathbb{R}^{p_2} \end{equation} such that \begin{equation} x \sim \mathcal{N}( \begin{bmatrix} x_1\\ x_2 \end{bmatrix}; \begin{bmatrix} \...
Snowy Baboon's user avatar
2 votes
0 answers
23 views

Mean and covariance kernel for the posterior GP of a Hidden Markov Model

In a hidden Markov model (HMM) we have a process $X_k$ that evolves according to: $$ X_{k+1} = X_k + W_{k+1}, \quad W_{k+1} \sim N(0, \sigma_{W}^2), $$ where $\{W_k \}$ are IID and $X_0 = W_0$. We can ...
Oskar's user avatar
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2 votes
0 answers
46 views

Mean of normal follows a T distribution

Suppose: $x \sim \mathcal{N}(x; \mu, \Sigma) \;\;\;$ st. $\;\;\; \mu \sim T_{v}(\mu; k, M)$ Where $T$ is the $t$-distribution with v degrees of freedom, location $k$, and shape $M$. Then, is there a ...
Snowy Baboon's user avatar
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0 answers
36 views

When do variance of MLE and variance of posterior match?

Assuming a Gaussian likelihood, $y \mid x, w \sim \mathcal{N}(w^\top x, \sigma^2)$, the variance of the least squares estimate $\hat{w} = \mathrm{argmax}_w p(y \mid X, w)$ is $\mathbb{V}[\hat{w} \mid ...
swuk's user avatar
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0 answers
11 views

Why do we need the variance term in SWAG method?

My question is about the SWA-Gaussian paper. I do not really understand why they need the 1/2 factor for the covariance matrix (as underlined in the picture). I understand that it is needed because ...
Mikhail Petrov's user avatar
4 votes
1 answer
76 views

Bayesian ROPE (region of practical equivalence) for simultaneous comparison of multiple parameters?

I very much like the idea of the ROPE (region of practical equivalence) (e.g. see here), where you compute the posterior probability that a given parameter is in a previous range that counts as "...
Ben Bolker's user avatar
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2 votes
1 answer
71 views

Using old posterior as new prior given new data [duplicate]

Suppose I have some data, and use this data to create a posterior distribution. Now suppose I have some new data that I believe is from the same population as the data before. Can I now use my old ...
Ewan McGregor's user avatar
3 votes
1 answer
65 views

Posterior probability for $\theta$ with a discrete prior

I'm trying to find a posterior probability for this model but I can't find the solution. Help would be appreciated! Prior distribution: $\theta$ follows a discrete probability function: $\mathbb{P}(\...
Alexandre Beaudry's user avatar
1 vote
0 answers
42 views

Showing that the estimator of the log posterior used in stochastic gradient MCMC is unbiased

In the SGLD paper as well as in this paper it is claimed (paraphrasing) that the following estimator: $$\widetilde{U}(\theta) = -\dfrac{|\mathcal{S}|}{|\widetilde{\mathcal{S}}|} \sum_{{x}\in \...
Tan's user avatar
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