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

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What are the parameters of a Wishart-Wishart posterior?

When infering the precision matrix $\boldsymbol{\Lambda}$ of a normal distribution used to generate $N$ D-dimensional vectors $\mathbf{x_1},..,\mathbf{x_N}$ \begin{align} \mathbf{x_i} &\sim ...
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7 views

marginal posterior distribution in linear regression

Let's assume our posterior distribution looks like the bayesian linear regression posterior, \begin{equation} p(\mathbf{w}|D) = \mathcal{N}(\mu, \sigma) \end{equation} where \begin{aligned} \mu ...
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1answer
64 views

Why do we lose conjugacy when assuming unknown $\mu$ and unknown $\sigma^2$ in a normal distribution?

Model: The following model corresponds to samples drawn from a Gaussian distribution with unknown mean and unknown variance: \begin{align} x | \mu, \sigma^2 &\sim \mathcal{N}(\mu, \sigma^2 )\\ ...
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2answers
166 views

How to calculate the Bayesian posterior analytically and by simulation?

I am working with this model: Prior: $P(\lambda)$~ N(0, 1), only the positive part likelihood: $P(x) = 1 - e^{-\lambda x}$ or $P(\vec{x}|\lambda)=\prod(1-e^{-\lambda x})$ Posterior: ...
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13 views

Incorporating Risk Aversion in Bayesian Expected Loss functions

In Berger's Statistical Decision Theory and Bayesian Analysis, he presents the following expected loss function for decision theory: $\rho(\pi^*,a)=\int_\Theta L(\theta,a)d\pi^*(\theta)$ Where ...
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1answer
34 views

Posterior probabilities with decision trees or decision forests

Is there a way to get posterior probabilities $P(C | \vec{x})$ (probability that a data item $\vec{x}$ belong to one of the given classes) in a multiclass classification problem using decision trees ...
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1answer
35 views

Clarification on a paper regarding estimating N from a Binomial Distribution

I was wondering if someone could clarify the following for me. In the paper "Inference for the binomial $N$ parameter" by Adrian Raftery, his first example outlines the posterior of $N$ given $x$ as ...
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1answer
45 views

Skewed posterior distribution on constrained parameter space for Bayesian inference of MCMC. Advice on what to do?

I am running a fully Bayesian MCMC procedure to estimate some time series models, and my model has a lot of parameter estimates. In particular, one of these parameters, $\phi$, is $\in [-1,1]$. The ...
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17 views

find posterior distribution parameters by simulation

I've calculated the posterior distribution parameters of a variable X analytically and by simulation. But doesn't mach. X ~ Normal(mu,s=6).And the prior distribution of X is a Normal(mu=100,s=20). As ...
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1answer
140 views

Unrealistically high significance when marginalizing over large number of parameters

The setup I've tried to reduce the problem to a self-contained subset for this question, but it still ended up being pretty long. Sorry about that! I have a set of observations of a set of objects ...
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1answer
47 views

General questions on MCMC

This is a continuation of the following question. The previous link was related to rejection sampling. This is related to MCMC. General questions on rejection sampling 1a. As far as I understand, ...
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6 views

How do I calculate posterior probabilties from HMM?

The title may be misleading but I don't know if a better one exists. Consider the following figure: Basically, I want to replicate that. Its a bit easier for me because my HMM only has 12 states. ...
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43 views

Implementation of Bayes posterior predictive check

I have a question concerning the implementation of a bayes posterior predictive check. Let us assume i have this model (implementation is in R and jags): ...
5
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1answer
187 views

Posterior predictive check following ABC inference for multiple parameters

I am relatively new to Bayesian statistics so please be gentle. I have just performed Approximate Bayesian Computation (ABC) for the inference of a multi-parameter model. Now I am looking to perform ...
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0answers
16 views

Meta analysis, joint posterior distribution of study effect

Meta analysis (with common study variation $\sigma$) often assumes that: $$ X_{i,j} | \theta_i \overset{ind}{\sim} N(\theta_i,\sigma)\\ \theta_i \overset{i.i.d.}{\sim} N(\mu,\tau) $$ where ...
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1answer
23 views

Posterior mode estimator unchanged under coordinate transformation?

I'm looking at a data set where the posterior mode has noticeably less "bias" than the posterior mean and posterior median, and somewhat less error. However, the posterior mode is not invariant under ...
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9 views

Do the marginalised posterior and likelihood function converge in the limit of a large number of observations

Short question Do the likelihood function evaluated at the ML estimate and the marginalised posterior converge in the limit of a large number of observations? Long question I expect the two ...
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36 views

Combining Posterior Distributions of Separate Models

I am running Bayesian models to estimate the number of fruits on a plant, given the presence/absence of herbivores. I get a posterior distribution on each mean. I then run a separate model to estimate ...
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161 views

How to apply Bayes' theorem to the search for a fisherman lost at sea

The article The Odds, Continually Updated mentions the story of a Long Island fisherman who literally owes his life to Bayesian Statistics. Here's the short version: There are two fishermen on a ...
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1answer
61 views

Calculation of the expectation of a posterior distribution using numerical integration methods

I want to calculate the expectation of the following posterior distribution: $$E( \theta \mid {\bf u} ) = \int\limits_{ - \infty }^\infty \theta \cdot g(\theta \mid {\bf u} )\,d\theta $$ and if ...
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20 views

Verifying propriety of MCMC

I have a posterior I'd like to sample: $p(\theta\mid Y)\propto L(Y\mid \theta) p(\theta)$ where $p(\theta)$ is proper, so the posterior is proper. I can write $L(Y\mid\theta) = \int f(Y, Z\mid ...
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1answer
37 views

Is there an article/book reviewing different methods for constructing posterior point/interval estimates?

Given a one-dimensional posterior distribution it is often the case that you want to calculate a point estimate and a credible interval for the corresponding parameter. There are, of course, many ways ...
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1answer
46 views

R Bayesian prediction of a Gaussian process

I have a Gaussian model with mean zero, variance is arbitrary constant, and correlation function $e^{-\theta(x-x')^2}$ where $\theta$ is again an arbitrary constant. I've plotted some realizations of ...
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26 views

Simulation m values from posterior distribution with WinBUGS

I try to simulate from posterior distribution with WinBUGS. My data came from Multinomial distribution, i.e. : $y_i~\text{Multi}(n;p_1,p_2,p_3)$. A common prior for multinomial is Dirichlet, i.e.: ...
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28 views

Simulation from Dirichlet distribution with WinBUGS

I have a question. Now I am learning WinBUGS, doing bayesian statistics. How, can I simulate a Dirichlet distribution (which is the posterior, for my model ...
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19 views

Are pooled results from multiple imputation equivalent to a posterior mean?

I am fairly new to multiple imputation and trying to be sure I understand the approach. Say I have a data set with missing values, so I create 5 imputed data sets using multiple imputation by ...
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1answer
29 views

How to visualize a set of many optimizations of posterior simulations of an objective function?

I started by fitting a model: $y = f(X) + \epsilon$. The model includes random effects and coefficients -- there is a lot of heterogeneity in the population (and the data is longitudinal). I then ...
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1answer
39 views

Posterior parameter distribution

I am considering the following non-linear state space model: $X_t=\frac{X_{t-1}}{2}+25\frac{X_{t-1}}{1+X_{t-1}^2}+8\cos{1.2t}+\epsilon_t, \epsilon_t\sim N(0,\sigma_x^2 ) $ ...
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1answer
46 views

Posterior Predictive Checks

I understand what the posterior predictive distribution is, and I have been reading about posterior predictive checks, although it isn't clear to me what it does yet. What exactly is the posterior ...
3
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1answer
245 views

Deriving the posterior density for a lognormal likelihood and Jeffreys's prior

The likelihood function of a lognormal distribution is: $f(x; \mu, \sigma) \propto \prod_{i_1}^n \frac{1}{\sigma x_i} \exp \left ( - \frac{(\ln{x_i} - \mu)^2}{2 \sigma^2} \right ) $ and Jeffreys's ...
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1answer
126 views

Plotting a “posterior median surface”

As part of reproducing a model I described partially in this question on Stack Overflow, I want to obtain a plot of a posterior distribution. The (spatial) model describes the selling price of some ...
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43 views

Parameter Estimation for Naive Bayes - Maximum a posteriori and Maximum Likelihood

I am wondering if I understand those terms correctly. To summarize my thoughts: In naive Bayes, our decision rule is basically the Maximum a posteriori (MAP) estimate of our hypothesis. We assign an ...
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18 views

Marginal Likelihood Latent Variable Model

I am trying to apply the method proposed by Chib in Marginal Likelihood from the Metropolis Hastings output to calculate the marginal likelihood of a logit model the includes latent variables. ...
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19 views

Transforming/collapsing bivariate distribution to a univariate distribution

I have a joint probability density function f(x,y) numerically in R. X is the probability males get a disease; Y the probability females get the disease. I want to extract from this bivariate data the ...
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67 views

In confusion with a Bayesian statistical problem

I was learning some probability basics. I am stuck with a problem, that I need your help with in solving. An $e$-fair coin is a coin with probability of head $(\theta)$ in interval ...
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1answer
75 views

Posterior distribution of a random variable

Im not understanding the following; suppose $y \sim N (\mu,\sigma^2)$ and we have a prior $\mu \sim N (\mu_0, \sigma^2_1)$ Then we can figure out the posterior distribution. What i dont understand ...
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1answer
86 views

burn in for Metropolis Hastings MCMC

I was wondering if there is a principled way to figure out how many samples to discard during the MH-MCMC burn-in stage. So, as I understand it, the initial samples can introduce bias in the ...
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40 views

Covariance update from Jacobian of transition function

In this paper on particle filtering with gradient descent, the authors sample Xk+1 through gradient descent, then update the covariance matrix P associated with Xk+1 as follows: Pi+1(k + 1|k + 1) = ...
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1answer
112 views

How do MCMC methods allow the estimation of the posterior distribution in this example?

I am reading a book example (diagram from p10) in which a person scores 9/10 on which we assumed a uniform prior. The posterior distribution could be easily worked out analytically, but the book gives ...
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181 views

What does it mean to integrate over the posterior?

I have been reading a book that cites an example where a uniform distribution is the initial prior, and then a person scores 9/10 on a test. Then the resulting posterior becomes the prior ...
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1answer
521 views

What's wrong with this illustration of posterior distribution?

I have the following image which I've been told is an illustration of how the posterior probability distribution is a combination of the prior and likelihood distributions. I've been told that ...
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16 views

Constructing a Gaussian from the posterior distribution

I have a Linear Dynamical System as in the following graphical model: Every random variable is a scalar; we know $B$ and $x_1$ a priori. $A$ comes from a Gaussian, the variances $R$ and $Q$ come ...
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141 views

How to calculate the posterior distribution given Inverse Gamma conjugate prior?

I have a state-space model (actually belonging to a Kalman Filter) as in the given graphical model: This is a typical 1 dimensional Linear Dynamic System model. The variances $Q$ and $R$ have ...
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29 views

How to use posterior function within Matlab

I would like to create a GMM within Matlab and then input an observation x to get the probability of this x. I know the equation is the following: $$p(x) = \sum_{k}p(C_{k})p(x|C_{k})$$ I would like ...
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31 views

a question on posterior distribution

Suppose that $E[T]=E[\gamma_i|X_1]+E[e_2|X_1]$ (1) where $\gamma_i$ and $e_2$ are distributed uniformly on the interval [0,1]. $X_1= \gamma_i + e_1$ So the background is that im trying to find ...
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27 views

help with this gradient computation in Expectation Propagation

I am trying to use Expectation propagation (EP) for approximating a posterior distribution in the Gaussian family. In this case, it is done by finding the Gaussian distribution with the same first and ...
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43 views

How can I calculate joint posterior distribution for a von bertalanffy growth function

I have recently begun to look into Bayesian Inference for fisheries. I have some difficulties in playing around with distributions. This is my model; \begin{equation} L_t = ...
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1answer
50 views

Dispersion measure - probability density function

I am wondering whether someone has a tip on this potentially very basic question. i have done some grid-based bayesian analysis and ended up with a non-standard discrete posterior density function ...
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113 views

Posterior distribution of precision for multivariate normal with normal-wishart prior

I'm trying to derive the posterior distribution for the precision matrix for the multivariate normal with normal-wishart prior. According to wikipedia and other sources the answer is as follows: ...
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3answers
752 views

How can an improper prior lead to a proper posterior distribution?

We know that in the case of a proper prior distribution, $P(\theta \mid X) = \dfrac{P(X \mid \theta)P(\theta)}{P(X)}$ $ \propto P(X \mid \theta)P(\theta)$. The usual justification for this step is ...