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

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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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12 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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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
66 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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52 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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35 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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80 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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153 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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501 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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53 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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18 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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20 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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37 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
45 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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65 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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330 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 ...
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80 views

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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56 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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46 views

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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116 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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50 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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23 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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87 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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88 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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35 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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24 views

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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49 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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63 views

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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28 views

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 ...
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26 views

Calculating posterior and prior odds

Question: Now, I'm confused about assigning probabilities here. I find $P(A^c|E) = (.001)(.99) = .00099$ and $P(E|A) = .99$, but what about the first two sentences? Does that mean that $P(E) = .001$ ...
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Pitfalls of posterior simulation when analysis didn't begin as Bayesian

I've got a situation where I'd like to evaluate a function of a fitted model, and account for the uncertainty in the fitted model. For example, say I want to calculate the minimum of the function ...
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79 views

Can likelihood be changed when the prior changes?

I have a data which follows gamma distribution and want to know the uncertainty of the parameters of this data. $\text{Data} \sim \text{Gamma} (\alpha, \beta)$ Parameters $\alpha \sim \text{Gamma} ...
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197 views

Bayesian inferencing: how iterative parameter updates work?

I have been struggling with this for a while. A typical optimisation problem can be viewed as optimising some cost function which is a combination of a data term and a penalty term which encourages ...
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19 views

Simple Maximum a Posteriori for Matching Points in Two Sets

I have been studying about Maximum a Posteriori and I tried to apply this concept to the problem of matching points, i.e. given two point sets $X$ and $Y$, I would like to find the most likely ...
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26 views

Given a series of Posterior probability scores, how can FDR be estimated?

I have a series of events which I manage to compute their significance using an specific software. Given that the software it outputs as a result the significant events followed by a posterior ...
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40 views

What is the interval that relates to the mean as the equal tailed interval relates to the median and the highest density interval relates to the mode?

When summarizing a one dimensional continuous distribution (e.g. a posterior distribution) it is common to use either an equal tailed interval (aka quantile-based) or a highest density interval. The ...
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61 views

performing parameter updates on Gamma distribution

I have the following form for a joint distribution $$ P(w, \lambda, \phi \vert y) = P(\phi) \times P(w \vert \lambda) \times P(\lambda) \times \prod_{i=1}^{N}P(y_i \vert w_i, \phi, \lambda) $$ I ...
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28 views

updating posterior parameters when involving conditioning

I have a setup where the joint posterior is written as: $$ P(w, \lambda, \phi \vert y) = P(\phi) \times P(w \vert \lambda) \times P(\lambda) \times \prod_{i=1}^{N}P(y_i \vert w_i, \phi, \lambda) $$ ...
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38 views

Pricing Model for desktop computers

I am trying to implement a pricing model to automatic price desktops based on some attributes(say memory, disk and brand for now). I have collected some sold history data from the internet and they ...
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99 views

Best method to estimate the mean of a normal distribution?

Let $X = ( x_1, ..., x_n ) $ be $n$ samples from a normal distribution with unknown mean. What is the best estimator for this mean? I can think of at least 2 unbiased estimators: The empirical mean ...
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Does the Bayesian posterior need to be a proper distribution?

I know that priors need not be proper and that the likelihood function does not integrate to 1 either. But does the posterior need to be a proper distribution? What are the implications if it is/is ...
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On the tractability of posterior distributions

I am trying to understand what makes estimating the posterior distribution such a hard problem. So, imagine I need to estimate the posterior distribution over a set of parameters given the data y, so ...
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Why not use Beta(1,1) as boundary avoiding prior on a transformed correlation parameter?

In Bayesian Data Analysis, chapter 13, page 317, second full paragraph, in the modal and distributional approximations, Gelman et al. write: If the plan is to summarize inference by the posterior ...
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210 views

Posterior very different to prior and likelihood

If the prior and the likelihood are very different from each other, then sometimes a situation occurs where the posterior is similar to neither of them. See for example this picture, which uses normal ...