Bayesian inference is a method of statistical inference which uses Bayes' theorem to find probability estimates of parameters or hypotheses.

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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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Intelligence Squared Scoring and Winner Determination

There is an NPR podcast called Intelligence Squared. Each episode is a broadcasting of a live debate on some contentious statement such as "The 2nd amendment is no longer relevant" or "Affirmative ...
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20 views

Bayesian mean square error

Given a i.i.d sample $X_{1},..,X_{n}$ of bernoulli random variables test 2 hypotheses $H_{0}:p=2/3$ and $H_{1}:p=1/3$. Bayesian prior is $\pi(2/3)=1/3$ and $\pi(1/3)=2/3$. Find the bayesian criterion ...
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number of stochastic nodes in bayesian multivariate distribution?

I'm doing some bayesian modeling using BUGS - JAGS to be specific. I find it hard to infer how many stochastic (i.e. non-deterministic) nodes there really are when I use multivariate distributions. ...
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Filter algorithm for a system

I have a system with the following structure: $X_{t+1} = X_t + E_{t+1}$ $E_{t+1} \sim N(0, \Sigma)$ $Y_{t+1} = f(X_{t+1})$ $Y_{t+1} \sim {\rm Uniform}(a_{t+1}, b_{t+1})$ So $X$ is a vector of ...
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Inferring parameters for a regression with features of both multivariate probit and ordinal regression?

I am dealing with data which is generated by a complex process, which I elaborate below; I am trying to answer one or more of the following questions- a) what is the right literature to look for ...
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Bayesian mixture model for univariate continuous random variable

I'm quite new to the mixture models and I hope you'll help me to understand how they work. Suppose I have a univariate continuous random variable x which represents time of a visit, and suppose that ...
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33 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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1answer
40 views

Understanding factor potentials in PyMC

I'm trying to understand factor potentials from the PyMC documentation, but need some help on the implementation piece--or it may turn out that I am misunderstanding how potentials work altogether. ...
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34 views

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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2answers
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Ignorability in Rubin's theory of missing data mechanisms

I am trying to understand Rubin's theory of bayesian inference with missing data, specifically how the missing data mechanism affects the inference on a superpopulation parameter. The theory is ...
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How tho choose the number of components in a Bayesian Hidden markov model

I'm implementing a bayesian Hidden markov model. I now face the problem of how to choose the number of components. I have two problems: 1) which index is better to use? 2) suppose i decide to use the ...
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36 views

stick breaking model of Dirichlet process

I have a question regarding sticking-breaking model of Dirichlet process, which is defined as follows: There are further statements that I am not clear that how to derive equation 1 from that ...
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13 views

Is it possible to combine bayesian SEM with PLS SEM?

I have already read some books about both two structural equation models. It seems both SEMs are suitable to the situation with small observations and large variables. I assume to use combine both ...
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18 views

Floor effects in Bayesian estimate, can I reparameterize?

I'm replicating an old study and I have two sets of existing estimates which measure a similar effect, namely the presence of a studied item in memory over time: ...
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22 views

Fitting multiple logistic functions at once with pymc

I have read a similar post here Fitting logistic function with pymc but it seems there is a rule that I shouldn't ask question in someone else's post. My approach is to fit 10 logistic functions to ...
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1answer
36 views

How does the Bayes' theorem equation generalize all sorts of regression/classification models?

I have been reading “Pattern Recognition & Machine Learning” written by Christopher M. Bishop for some time, but I am still a beginner. I wish to get a bigger view that summarizes regression and ...
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Assuming training data as set of target funtions

In this 10th slide of http://www.cs.cmu.edu/afs/cs.cmu.edu/project/theo-20/www/mlbook/ch6.pdf presentation The Training data set $D$ is assumed as the set of target function . Actually $D = ...
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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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45 views

Formulas for probabilities in Bayes theorem

In continuation to this question $p(h|D) = \frac{p(D|h)p(h)}{p(D)}$ $p(h) = $prior probability of hypothesis $h$ $p(D)$ = prior probability of training data $D$ $p(h|D)$ = probability of $h$ ...
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+50

When to terminate the Bayesian A/B test?

I'm trying to do A/B testing the Bayesian way, as in Probabilistic Programming for Hackers and Bayesian A/B tests. Both articles assume that the decision maker decides which of the variants is better ...
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162 views

What is a “strictly positive distribution”?

I am reading Judea Pearl's "Causality" (second edition 2009) and in section 1.1.5 Conditional Independence and Graphoids, he states: The following is a (partial) list of properties satisfied by ...
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Parametrization of Gamma and Negative Binomial in R

I have some Poisson data {${y_1,...,y_n}$} and a Gamma prior, and I wish to construct a predictive posterior distribution. As I understand, if my Gamma hyperparameters are $\alpha$ (the prior number ...
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24 views

Probability of ongoing experiment

Suppose, I do a experiment where I have an event 'a' true 1000 times in 1000 trials. So, the probability becomes 1000/1000 = 1. If I am going to do another trial, my prediction about event 'a's ...
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1answer
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Inferring testlet structure in item response theory

Is it possible to infer the testlet structure in a set of items using item response theory? Specifically, I've created a lot of variations on the story recall task, each variation being scored on 25 ...
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22 views

Method of Composition to sample from a t density

I got stuck with this, I will appreciate a lot any help. I need to make an R program in order to run this algorithm (in the photo below), with simulated data. The question is to use the method of ...
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31 views

Expectation expansion query

I am trying to do a proof. Define the best Bayesian estimator by $\theta^B=E(\theta|x)$. Prove that for another estimator $\gamma$ of $\theta$, we have $MSE(\theta^B)\leq$$MSE(\gamma)$. Proof: ...
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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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16 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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43 views

Facts about the mathematical difference between Student's t-distribution and normal distribution

I'm looking for some facts (theorems or such) concerning the properties of the Student's t-distribution compared to the normal distribution. More specifically, I understand that for the normal ...
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52 views

Within-subjects model in JAGS/stan

I have a general question regarding a varying intercept / varying slope model in jags/stan: I have data from a psychophysics experiment, with one covariate, one within-subjects factor and several ...
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29 views

Non-informative prior for Poisson/gamma density

In the Albert book on Bayesian computation with R, exercise 4.8.5 (p.83), it is suggested to use $$ p(a, b) \sim (a \times b)^{-2} $$ as the non-informative prior for the Poisson/Gamma model: $$ ...
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Derive the conditional pdf of data on prior parameters

In Bayesian statistics I see this derivation often. Given the likelihood function $f(X|\theta)$ and the prior $f( \theta |a, b)$, the author will derive $f(X|a,b)$. The steps in between are ...
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Can unimodal prior and unimodal sampling distributions lead to a multimodal posterior distribution?

Can unimodal prior and unimodal sampling distributions lead to a multimodal posterior distribution? The Bayes rule tells us that $$ f(y|x) = f(x|y) f(y) / f(x). $$ which, I think, implies a unimodal ...
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49 views

What are pros and cons of empirical Bayesian methods?

The empirical Bayesian method is a new concept to me. It raises my interest, because it offers a different philosophical and methodological perspective of statistical analysis. From my limited ...
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195 views

Why are directed graphical models called Bayesian?

Why are directed graphical models called Bayesian,while those undirected not? Does that mean that Bayesian analysis is involved in the directed ones, and not in those undirected?
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Bayesian meta-analysis

A colleague is conducting a Bayesian meta-analysis of some large-scale drug studies. Although she has experience with 'conventional' meta-analysis, she in new at the Bayesian approach. We have read ...
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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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50 views

What do you call $p(Y|X_1)$?

Given random vectors $Y= [Y_1, \dots, Y_m]$ and $X = [X_1, \dots, X_n]$, $p(y|x)$ is the conditional distribution of $Y$ given $X$. We can call $p(y_1|x)$ the marginal conditional distribution of ...
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Same or different? The Bayesian way

Say I have the following model: $$\text{Poisson}(\lambda) \sim \begin{cases} \lambda_1 & \text{if } t \lt \tau \\ \lambda_2 & \text{if } t \geq \tau \end{cases} $$ And I ...
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Confidence intervals for 6-point Likert scale

I was asked to compare certain ratings between two different groups. The distribution of ratings looks something like this: ...
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1answer
50 views

Hypothesis testing multiple choice question with single answer

I have a survey questions with five options and I ask respondents to pick their single top choice. What test should I use to figure out if the top voted answer is statistically more popular than the ...
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58 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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27 views

Laplace rule of succession for a Poisson process

Suppose that I measure for an interval of time $t$ a Poisson process that has unknown average rate $\mu>0$ and I observe no event. What could be a reasonable estimate of $\mu$ from a Bayesian point ...
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56 views

Advice on sensitivity analysis for priors in Bayesian statistics

I'm not clear on how to perform sensitivity analysis on the priors. Many sites have different answers. One site indicates to perform three non-informative, weakly informative and known priors. Another ...
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2answers
97 views

$p(D)$ in Bayesian Statistics

Say I have the following obvious Bayesian computation: where $\theta$ is a model parameter that we try to infer and $D$ is observed data. I have always understood $p(D)$ to relate to knowledge ...
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Formal justification of Bayesian inference as a model for belief

I remember a proof that Bayesian probability theory is the only valid method for representing beliefs, it went something like we represent belief by some non-negative function over some domain of ...
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Marginal Likelihood Prior

I have a model with probability matrix for a distribution of $x$, $y=\{0,1\}$, $p(x,y|w)$ where $w=[w_1,w_2,w_3,w_4]$ $p(x=0,y=0)=w_1$, $p(x=0,y=1)=w_2$ $p(x=1,y=0)=w_3$, ...
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How to find conjugate prior for a given distribution?

Given a distribution $\{p(X|\theta),\theta\in\Theta\}$: how do you show the existence/non-existence of its conjugate prior? what are some general ways/principles to construct/find its conjugate ...
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1answer
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What are the definitions of semi-conjugate and conditional conjugate priors?

What are the definitions of semi-conjugate priors and of conditional conjugate priors? I found them in Gelman's Bayesian Data Analysis, but I couldn't find their definitions.