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

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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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47 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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28 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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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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23 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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21 views

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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59 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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159 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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13 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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17 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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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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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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19 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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35 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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83 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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52 views

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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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 ...
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42 views

Expected value of posterior vs. success probability

Context Suppose I have two models, $H_1$ and $H_2$ for which I know the prior probabilities $p(H_1)$ and $p(H_2)$. Furthermore, I know the class-conditional distributions $p(x|H_1)$ and $p(x|H_2)$ of ...
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how to initialise this distribution

I am reading a paper by Tom Minka and trying to follow the steps to perform the updates for the Expectation Propagation algorithm. I am having trouble getting the initialisation of the posterior ...
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Marginalization from a distribution

I have a posterior distribution over a set of parameters denoted by $\theta = \{w, \phi, \lambda\}$ and the posterior is the joint distribution given the observed data i.e. $P(\theta | D)$. Now, this ...
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50 views

Parameters with categorical and gamma distributions in posterior distribution

I'm following a very good IPython notebook (the whole list can be found here) in which some sampling techniques are explained. However, I don't understand the use of a categorical variable in a change ...
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Expectation Propagation when the likelihood is already Gaussian

What happens with EP when the likelihood terms and the priors are already Gaussian. So, if we imagine that the posterior is given by: $$ P(\theta|y) = ...
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77 views

Is Gaussian process regression a Bayesian method?

Actually I thought Gaussian Process is a kind of Bayesian method, since I read many tutorials in which GP is presented in Bayesian context, for example, in this tutorial, just pay attention to page ...
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21 views

Classification model that allows changing priors at prediction time

I would like to know if there's a way to build a classification model in R that would allow me to change the class weights at prediction time. The scenario where I would want to do this: I have a ...
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46 views

Posterior and Likelihood probabilities meaning [duplicate]

I am a computer scientist, so I have a background at maths (however limited). I am reading about posterior distribution from here http://en.wikipedia.org/wiki/Posterior_distribution . It says there: ...
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EM on product of multinomials

I have the following conditional density: $$ P(x | \theta, \pi) = \prod_{i=1}^I \prod_{j=1}^J t_{ij}! \prod_{k=1}^K \frac{1}{x_{ijk}!}(\sum_{l=1}^L \theta_{il} \pi_{jkl})^{x_{ijk}} $$ Here, $x$ is ...
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70 views

What's the point of estimating the posterior distribution $Y | X $ using a Gaussian Process?

So classic regression methods (ridge regression, LASSO) only predict the posterior mean $E[ Y | X ]$, while Gaussian Processes give you the full posterior distribution $Y | X$. It would be very ...
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39 views

the derivation of the conditional posterior for the Poisson model setting

When discussing the Poisson process with changing point (Carlin, Gelfand and Smith, 1992), the model is assumed as I am not quite clear about the derivation of conditional posterior distribution ...
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Shape of posterior distribution

I have recently been reading and trying to understand the Bayesian paradigm and looking at various methods that people have been using to estimate the posterior distribution. Now, it seems that most ...
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25 views

Interpretation of posterior variance in gaussian regression

I'm trying to understand how the posterior variance of a Gaussian regression depends on the datapoints. Suppose $y=X\beta + e$, with Gaussian priors over $\beta$ and $e$ and known variance of $e$. ...
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Multiplying multivariate Gaussian with a univariate Gaussian

Not sure if this question makes sense but I have been spending a lot of time looking at information about expectation propagation and one of the key operations in it as explained by Tom Minka is the ...
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48 views

Finding the posterior pdf

Suppose $X$ has probability density function $$f(x, \theta) = \theta e^{-\theta x}$$ when $x > 0$ and $\theta > 0$, and $0$ otherwise; given $\Theta = \theta$. Suppose the prior probability ...
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63 views

Can I say anything about the shape of the posterior distribution?

I have a model where the noise is modelled as independent and identically distributed across the various data points. The noise $e$ is modelled as a 0 mean gaussian with $\phi$ as the precision ...
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55 views

Variational Posterior Dirichlets in LDA

I am running the c code for LDA provided on David Blei's website. The code outputs several files. The output file final.gamma is supposed to include the "Variational Posterior Dirichlets". If I ...
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Trouble understanding posterior probabilities

I'm having trouble understanding what this problem is asking for. Could someone clarify how to obtain posterior probabilities using backward/forward probabilities?
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159 views

Posterior covariance of Normal-Inverse-Wishart not converging properly

I am trying to implement a simple normal-inverse-Wishart conjugate prior distribution for a multivariate normal with unknown mean and covariance in numpy/scipy such that it can take a data vector and ...
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48 views

Posterior distribution of the parameter knowing the prior

I have exponentially distributed probability of event $E$ $$P(E|a) = a \exp(-aE),$$ where $a$ is the rate parameter of the exponential distribution. Now the probability distribution for $a$ is a ...
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31 views

Mean of Posterior distribution

If $X1..Xn$ be iid $\sim N(\theta,\sigma^2)$, and let $\theta$ has double exponential distribution, $\pi(\theta) =\frac{e^{-|\theta|/a}}{2a}$, a known. Find mean of the posterior distribution. My ...
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55 views

Posterior distribution

Suppose $X1,..,X4$ be iid from pdf $f(x|\theta)=\frac{1}{\theta}$ ,for $0<x<\theta$. The prior distribution is $\pi(\theta)=\frac{2}{\theta^3}$ , for $\theta>1$ I have to obtain: ...
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28 views

Posterior distribution as a distribution for a new random variable?

So in Bayesian framework one uses observed data $X=\{x_1,...x_n\}$ to update the prior $p(\theta)$. My question is it justified to say that $p(\theta|x_1,...,x_n)$ corresponds to a new random variable ...
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123 views

Gibbs sampling from the full conditional distribution using R

I have a likelihood function of all the data $y$ $$L(\tau ,\theta |y)\propto \theta ^{\sum \delta _{i}^{C}+\sum \delta _{i}^{H}}\tau ^{\sum \delta _{i}^{H}}e^{-\theta \sum x_{i}^{C}-\tau\theta\sum ...
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30 views

Confidence intervals and central estimates for a functional of an estimated function with uncertain parameters

I've got a problem that is leading me to dip my toes into Bayesian stats, and I've got a question about confidence (or, I suppose, credible) intervals: Say you want to know how $X$ maps to $y$. You ...
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Can posterior distribution for a continuous variable be greater than one?

I already asked this question here, but I am not sure where would be better to ask it? This might sound a dumb question but I am really confused about it. According to Bayes' rule we do have the ...
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70 views

Posterior probability

Suppose that we have have scoring functions $f(\textbf{x})$ and $g(\textbf{x})$ for classifying an object as red or blue. These are based on linear discriminant analysis. So if $f(\textbf{x}) > ...
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Confusion related to intractability in topic models

I was reading this paper related to topic models. I am a bit confused why the marginal likelihood is not tractable and how converting the graphical model into the new one actually helps. First I ...
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Test data likelihood via posterior predictive

I want to compute likelihood of the test data given train data under my model: $$ p(x_{test} | x) = \int p(x_{test}, \theta | x) d\theta = \int p(\theta | x) p(x_{test} | x, \theta) ...