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

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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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121 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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32 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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18 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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34 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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44 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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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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18 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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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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28 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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37 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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15 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 ...
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184 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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123 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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27 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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14 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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17 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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64 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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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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60 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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37 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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93 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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163 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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511 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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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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111 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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21 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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22 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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42 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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46 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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91 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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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 ...
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Sampling from Bayesian regression predictive posterior

I have the following problem: I want to obtain a predictive posterior distribution for the target logistic regression variable $y$. That is to say, given a combination of explanatory variables $X$, I ...
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61 views

What can be learnt from the sampling distribution of posterior probabilities?

I have an rather open question. In Bayesian statistics you do testing based on some posterior distribution $p(\theta|D)$. E.g. you could try something like $T=P[\theta>0|D]$ and decide based on the ...
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Are discrete single value prior distributions always lost in MAP estimation?

I’d like to illustrate my problem with a little (heavily abbreviated) excercise. I think it will help a lot to stress my point. Meet Mary, Tom and Jane. They all are programmers. Mary is a decent ...
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173 views

Output probabilities of binary support vector machine classifier in Matlab R2014a

I’m using SVM for classification of my binary output problem. I want probability of belonging to every class. How can I obtain it? For instance suppose this is our structure: ...
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52 views

Non-binomial posteriors for a binomial prior?

Let's assume we have a discrete binary random variable K (K=0 or K=1) for which the prior distribution is binomial. My understanding of Bayesian statistics tells me that regardless of the likelihood, ...
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33 views

Square distance and likelihood in k-means

In k-means algorithm, the distance minimization step is equivalent to maximize likelihood: $P(X|\theta)$ or to maximize posterior distribution $P(\theta|X)$? I think it's more logical to maximize ...
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89 views

Find posterior distribution

Let $X_{1},..,X_{n}$ be a sample from a poisson$({\lambda})$ distribution. Let the prior be ${\pi}({\lambda})=1/{\sqrt{\lambda}}$. Find the posterior distribution. My work: We have ...
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109 views

Confused about why we would use expected value instead of MLE when estimating some parameter

I have a conceptual confusion about the use of the expected value of a distribution. Often, we want to estimate the most likely value of something. For example, I have X= ten observations. I know X ...
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23 views

multiplication of 2 PDFs

If I multiply the two PDFs, does the variance of the result PDF becomes narrower than the two PDFs always? In other words, if I multiply likelihood and prior to get the posterior, is the variance of ...
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37 views

How should I construct a prior distribution with a particular kind of count data

For context I will first explain the overall problem that I am working on. I am given a catalog of product names and I am also given a large text dataset that may contain mentions of these catalog ...
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Mode of Joint Posterior - Maximization Problems

I have a problem whereby I get two different answers if I try to maximize a function. let $ \begin{bmatrix} Y_{o}\\ Y_{a} \end{bmatrix}|\phi\sim N (0,\phi^{-1}A^{-1}) $ $\pi(\phi)=\frac{1}{\phi}$, ...
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56 views

MAP Estimator with Laplacian Noise

I need to calculate the MAP estimator of $ x $ in the following case: $$ \left [ \begin{matrix} {y}_{1}\\ {y}_{2} \end{matrix} \right ] = \left [ \begin{matrix} x\\ x \end{matrix} \right ] + ...
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123 views

Testing for the compatibility of inferences

Consider that I have two balances (called 1 and 2). Each of these balances gives a posterior distribution for the weight of the object of the form $m_1 \pm s_1$ (for balance 1) and $m_2 \pm s_2$ (for ...
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Posterior Density in R

I'm new to the site, and to Bayesian statistics and was hoping to get some help. I'm currently working through some study exercises and am required to compute the mean and variance of the posterior ...
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Calculating posterior of difference given posterior of two means

I am using R and MCMCpack to do a Bayesian analysis of some data that I have. I have generated posterior distributions (...
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Applying Beta distribution for calculation latent variables

I would like to find the probability distribution function for the below scenario,its similar to Computer Adaptive technique (IRT) I need to estimate the ability of a student from answered questions ...
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Obtaining and sampling from the posterior predictive of a naive Bayes classifier

I have trained a naive Bayes classifier with on a dataset with a dichotomous outcome and multinomial attributes (predictors). I managed to get a Maximum a posteriori (MAP) estimate which is good ...