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

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Closed form posteriors for a simple bivariate Bayesian regression

I'm analyzing a simple linear regression $Y_{i}$~$a+b*X_{i}+e_{i}$, with $e$ being normally distributed with known variance and where I have normal priors on $a$ and $b$. I'm trying to piece together ...
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Estimating total number of people from an observed sample

The well known "German tank problem" shows how to answer the question: "If I have tanks which have an increasing serial number, and I see a sample of tanks and record their serial numbers, what is the ...
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52 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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60 views

Basic problem in Bayesian inference

I have questions with the following Bayesian inference problem I found in the book by Bertsekas & Tsitsiklis (Introduction to Probability 2nd ed.). Problem is as follows (P.445, Problem 2): ...
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Bayesian stats: trick to accept the null?

There's a lot to be said and read about this, but I haven't found a clear answer to this question: Bayesian statistics are said to 'penalize' vague hypotheses with weak priors, by giving more support ...
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83 views

Estimating abundance using non-normal count data

I have sample counts of $n=20$ or $n=7$ taken from right-skewed and zero-inflated populations. The challenge in each case is to use the sample to estimate the total count in that population. Each of ...
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Is there a Bayesian approach to density estimation

I am interested to estimate the density of a continuous random variable $X$. One way of doing this that I learnt is the use of Kernel Density Estimation. But now I am interested in a Bayesian ...
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1answer
27 views

Obtaining posteriors for exclusive and exhaustive hypotheses

I'm trying to solve a Bayesian problem where I have two mutually exclusive and exhaustive hypotheses: $H_1$ and $H_2$. Given Baye's formula: $$P(H|D) = \frac{P(H)P(D|H)}{P(D)}$$ (where $D$ is my ...
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53 views

How to determine whether an indirect effect is statistically significant using Bayesian statistics?

I've used bayesian estimation to test the indirect effects within a model and identified 95% credible intervals. I'm typically used to using the Sobel's z test to identify significant mediation, what ...
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6 views

Comparing IRFs derived from Bayesian VARs with other extraneous information

I was wondering if anyone here could help me with the following: I estimate a standard Bayesian VAR with Normal-Inverse Wishart priors. I identify some policy shock in it, and then derive the IRF for ...
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26 views

Bayesian method for computing credibility interval for correlated time series

I'm studying a stochastic process generated by simulation using two different methods. In the first, the waiting time between events can be shown to be exponentially distributed. To model the ...
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24 views

Applying the Akaike Information Criterion to Data

I have some variables that I would like to run regressions on, to create a model, but I am unsure about how to actually AIC (or the BIC). Unfortunately I have not yet taken a mathematical statistics ...
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49 views

Mixture of probits: understanding truncated-based likelihoods

I am trying to implement a mixture model of probits to infer the best decision boundary for every latent subpopulation. When doing Gibbs sampling, we eventually have to compute $P(y^* | w_c)$ where ...
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Subjectivity in Frequentist Statistics

I often hear the claim that Bayesian statistics can be highly subjective. The main argument being that inference depends on the choice of a prior (even though one could use the principle of ...
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38 views

Given the data set is the Bayesian estimation the best solution for solving the expected value? [closed]

I am very new to this. I have several measurements from different instrument about a parameter A. Each of the measurements comes with an estimated error. I know that the observation error is biased, ...
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74 views

Bayes decision theory: Classification error probability

In Bayesian decision theory: Given $\omega_1$ and $\omega_2$ as two classes for classification, $P\left( \omega_1 \right)$ and $P\left( \omega_2\right)$ their prior probabilities, $x$ the feature ...
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1answer
64 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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48 views

Confusion related to online bounds for bayesian algorithms

I was reading this paper Online Bounds for Bayesian algorithms and they had some derivations. I didn't get how they arrived to the conclusion I didn't get how equation 3 was derived any ...
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31 views

Definitions of Prediction vs. Predictor

I am writing an article which includes discussion of the MMSE estimator of the posterior predictive distribution. Since I use this term quite frequently, I was considering referring to this estimator ...
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31 views

Book for Bayesian regression modeling [duplicate]

I am new in regression modeling. Please suggest Book for Bayesian regression modeling. Thanks in advance.
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Bayesian Perceptron - how can I generate many different perceptrons?

I am going to implement the Bayesian version of a perceptron that I read in the Statistical Mechanics of learning, by Engel-Van Den Broeck. The idea to improve the performance is to use many Gibbs ...
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136 views

Combining data from different sources

I want to combine data from different sources. Let's say I want to estimate a chemical property (e.g. a partitioning coefficient): I have some empirical data, varying due to measurement error around ...
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53 views

What would be an effective strategy to correlate congressional attendance and precipitation?

I have data sets of the attendance of congresspeople, and of the weather in Washington, DC. I would like to examine whether and to what degree precipitation is correlated with the likelihood of ...
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Accuracy of Laplace approximation of posterior density

I consider approximating the following integral: $$ \int p(t|\alpha,\beta)p(\alpha,\beta)d \alpha d\beta $$ Where $t|\alpha,\beta$ follows a multivariate normal distribution and $p(\alpha,\beta) = ...
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Relation between variational Bayes and EM

I read somewhere that Variational Bayes method is a generalization of the EM algorithm. Indeed, the iterative parts of the algorithms are very similar. In order to test whether the EM algorithm is a ...
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46 views

Stochastic Programming (e.g. LP) with MCMC

I have just started learning about MCMC (using PyMC), and it seems to be a hammer that can be used to solve a large class of inference and optimization problems. While I understand that there are ...
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Comparing regression outputs for different response variables

I have inherited some legacy code designed to solve the following problem - we are given a set of observable binary variables $Y_1, \dots , Y_N$ that we believe are related to a fixed set of input ...
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Problem constructing model matrix to generate predicted values to plot Bayesian glmmBUGS output

My goal is to plot the predicted values generated from a Bayesian model using glmmBUGS run through R. I believe my problem stems from a lack of understanding in 1) how to properly construct a model ...
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26 views

Updating a Dirichlet distribution with partial data

I've got some categorical data where each observation has multiple attributes, and I want to make a probabilistic model of this using Dirichlet distributions. For example, in the two dimensional case ...
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39 views

What is the Probability Distribution of NLTK Naive Bayes?

As I know Naïve Bayes has various distributions, as said in Sci-kit learn manual “The different naive Bayes classifiers differ mainly by the assumptions they make regarding the distribution of P(x_i ...
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39 views

latent variables versus model parameters

I am quite confused with the distinction between a latent variable and model parameters. So say I have two observed variables $x$ and $y$ and they have some unknown relationship between them i.e. $y ...
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21 views

recursive feature elemination in R with caret

i work with R caret software package to select the most important features from some set of data. My response is a factor of multiple classes (e.g. nominal ...
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1answer
39 views

What is the difference between Binary Clasification and Multiclass classification?

Apology for posting almost one question daily. I am trying to learn some aspects of Statistical Machine learning, so every day many questions coming and if I am not finding answer in my offline peer ...
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26 views

What machine learning tool is best suited for taking time series data as well as descriptive data and making a binomial classification

I have an interesting task of utilizing log data from computer servers in a server farm and predicting if a particular server is likely to fail in the next 24 hours. My data set will be comprised of ...
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49 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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Gibbs sampling versus general MH-MCMC

I have just been doing some reading on Gibbs sampling and Metropolis Hastings algorithm and have a couple of questions. As I understand it, in the case of Gibbs sampling, if we have a large ...
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Acronyms to use for Bayesian posterior predictive distribution estimators

I am considering writing an article that discusses the Bayesian MMSE and MAP of the posterior predictive distribution. I was wondering if there are acronyms that have been used so that instead of ...
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Why isn't bayesian statistics more popular for statistical process control?

My understanding of the bayesian vs frequentist debate is that frequentist statistics: is (or claims to be) objective or at least unbiased so different researchers, using different assumptions can ...
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Incorporating function of random variable in bayesian network.

Let A be a random variable hence a node in my bayesian network. I wish to apply some function on A and then connect it to some other node in the network. So how to do that?
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42 views

Estimation in Naive Bayes

I have a very silly question. In Multinomial Naive Bayes Classifier, which parameter estimation do we use, is it Maximum Likelihood or Maximum A Posteriori? If any one of the esteemed members may ...
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1answer
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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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Library for using plates in bayesian networks

In my Bayesian network there are plenty of repetitive variables leading to the use of plates(http://en.wikipedia.org/wiki/Plate_notation). I do not want the exponential space complexity in ...
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147 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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How to Implement an Empirical Bayes Analysis in BUGS/JAGS/Stan

My data is a set of $N$ observations $y_i$. I would like to estimate $\mu$ and $\sigma$ in the following model: $y_i \sim \mathrm{Normal}(\theta, \sigma)$ $\theta \sim \mathrm{Normal}(\mu, ...
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Bayesian bandits with delayed rewards

I looked into the topic of bayesian bandits in order to create a simple testing tool for headline optimizations. UCB1 seemed easy enough until I discovered that there is probably a problem with the ...
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43 views

Taking into account Bayesian model uncertainty

I recently received a review of a paper from a Bayesian Statistics Journal. The Associate Editor wrote this mini-review (quoted below in full). The paper is talking about Bayesian modeling of DNA ...
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103 views

Why likelihood is not always a density function? [duplicate]

I try to self-learn Bayesian machine learning (mostly by studying Bishop and Kevin Murphy's books). While working with formulas I was puzzled by the quote that "Note that the likelihood function is ...
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56 views

Bayesian model with unknown mean and variance with lognormal prior

For $i=1, \ldots, K$ and $j=1, \ldots,n$, assume the following model. \begin{align} X_{ij} \mid \mu_i, \sigma^2 & \stackrel{_\text{iid}}{\sim} N(\mu_i, \sigma^2) \nonumber \\ \mu_i & ...
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Books for learning non parametric Bayesian model

Having studied parametric Bayesian statistics during the two last years, I plan to begin to self-study non parametric Bayesian model during this summer and look for recommendations. I would like the ...
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Conditional distribution in a Bayes net

I have this Bayesian network where variable C is dependent on variables A and B. I want to know how come $ P(B|C) = \frac{P(B)}{P(B)+(1-P(B))P(A)} $