Questions tagged [bayesian]

Bayesian inference is a method of statistical inference that relies on treating the model parameters as random variables and applying Bayes' theorem to deduce subjective probability statements about the parameters or hypotheses, conditional on the observed dataset.

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1answer
55 views

Why is a Gelman-Rubin diagnostic of < 1.1 considered acceptable?

In multiple sources a Gelman-Rubin MCMC convergence diagnostic of less than 1.1 is considered evidence that chains have converged. For example in this thread: https://stackoverflow.com/questions/...
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30 views

Calculations in a Bayes Network

I am working through a text book (Probabilistic Graphical Models, Principles and Techniques) to learn BNs, but I am confused as to the accuracy of the example. The text references the figure above. We ...
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What is the difference between Bayesian Network and Dynamic Bayesian Network?

I just got the sentences below from a web site while studying Bayesian Network: "​A dynamic Bayesian network (DBN) is a Bayesian network extended with additional mechanisms that are capable of ...
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1answer
32 views

How to calculate the mean of standard deviation when data are drawn from a Gaussian population?

This is a pure algebraic question. Drawing a sample of size $n$ from a Gaussian population $N(0, \sigma^2)$, the posterior probability of $\sigma$ is proportional to $\frac{1}{\sigma^{n+1}} e^{-(s/\...
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Observed deterministic variables in MCMC

I need to model a measurement of an "exponential decay" i.e. I have a histogram of counts $Y$ over an array of (time-) intervalls. I want to use MCMC to infer parameters ($A_1,\lambda_1,A_2,\lambda_2,\...
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1answer
39 views

Why “sum of squared Pearson residuals” is around “number of dependent variables” in Binomial distribution?

A Pearson residual is defined as: $r_{i}(\theta)=\frac{y_{i}-E(y_{i}|\theta)}{\sqrt{Var(y_{i}|\theta)}}\tag{1}$ Sum of squared standard residuals $X^{2}$ is: $X^{2}=\sum_ir_i^2\tag{2}$ where $y_{i}$ ...
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1answer
60 views

What is the posterior mean of $\mu$ given a randomly stopped i.i.d. observations from a Normal

Let's imagine I have a machine giving me an independent random number from a normal distribution $N(\mu,1)$ whenever I push a button. I have a stopping rule to decide how many samples to collect - I ...
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1answer
37 views

How to infer a missing observation in a state space model?

I read here that "structural time series models handle missing values naturally, following the rules of conditional probability. Posterior inference can be used to impute missing values, with ...
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1answer
38 views

How can Bayes avoid Cromwell?

I'm studying widgets and their failures. Generally a widget will run for many years without trouble, but 1-2% of widgets will fail in a given year. I have a table which lists widget manufacturers (A, ...
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27 views

Computing Local Evidence for Bayesian Networks

I am reading through Kevin Murphy's "Machine Learning: A Probabilistic Perspective" book. I'm interested in understanding how to do exact bayesian inference over a tree structure, as discussed in ...
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2answers
60 views

a general question about posterior probability

I know $P(B|A) = \frac{P(A|B)P(B)}{P(A)}$, where $P(B)$ is known as a prior probability, $P(B|A)$ is a posterior probability and $P(A|B)$ is called the likelihood. We can interpreting $B$ as the model ...
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What is the meaning of calculating Maximum Likelihood from complete data for Bayesian Nework paremeter learning?

I am taking a subject on Bayesian Network on Youtube. Somehow, I am struggling from understand the meaning of calculating Maximum Likelihood estimates from complete data for a for bayesian nework ...
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Data normalization (reward/penalization) in a learning paradigm

wondering if folks can offer guidance on this data normalization problem. When trying to determine retention of information after a time delay, the retention rate needs to be adjusted for # of items ...
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74 views

What is Bayesians' equivalence of sample size determination? [duplicate]

Under the frequentist framework, one can use power analysis to determine sample size for a given effect size, significance level, and power. How are similar questions answered under Bayesian framework?...
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Are “improper uniform priors” in Bayesian analysis equivalent to maximum likelihood estimations?

The improper uniform distribution for parameter $\theta$ is : $p(\theta)=1,\ for -\infty<\theta<\infty$. It is called "improper" since it does not integrate to 1. Because Bayesian theorem is ...
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Bayesian fitting with very noisy data

I am trying to fit a response curve through noisy data. The curve is supposed to model a saturating return, which takes the analytical form: $$ x \to y(x) = \alpha( 1- e^{-\frac{x}{\beta}})$$ where $...
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1answer
63 views

Integrate out (covariance) matrix in Normal-Wishart distribution

In Gelman's Bayesian Data Analysis Chapter 3.6, he introduces the multivariate normal with unknown mean and variance, with the priors $\Sigma\sim \text{Inv-Wishart}_{\nu_0}(\Lambda_0^{-1})$ $\mu\...
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67 views

Bayesian: learning about a normal mean and variance

OK, question edited in response to comment from @user158565 for those not in possession of the Jackman book. I tried to abstract the relevant information below. I'm struggling with Simon Jackman's ...
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In bayesian updating, how to keep precision matrix suceptible to data?

In bayesian updating, I want precision matrix, or standard errors to keep susceptible, if much data comes. I am considering bayesian updating of simple regresion, using conjugate prior of Normal-...
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2answers
54 views

Understanding bayesian inference for parameter estimation

I am reading this article an am a bit confused about one of the terms. My understanding is we are trying to estimate a gaussian distribution Θ from which a certain value we are interested in is drawn....
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Results of grid approximation and analytic determination of posterior probability

I am learning Bayes methods and cannot reconcile my grid approximation estimate of the posterior with an analytic solution. My prior has 5 events from 129 trials. My observed data has 13 events from ...
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3answers
117 views

If the Bayesian probability is not a belief, what is it?

In this blog by William Briggs, who seems to be a prolific lecturer/writer on pop-statistics, he condemns the "Bayesian Metaphor" which is essentially referring to Bayesian probability as belief. ...
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What is the idea behind Bayes By Backprop?

Having looked through the internet and the paper, I find Bayes by Backprop very unaccesible for my intermediate understanding of variational inference. Most online guides also lack some explaining ...
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23 views

Interpretation of Bayes' Theorem in Different Scenarios

I am currently working on an image-classification problem with an unbalanced dataset, namely one with very different number of training examples within each class. I am trying to understand why such ...
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1answer
36 views

Quasi/Complete separation due to huge and infinite values

(R statistics) My question is regarding this warning. My data contains patients and healthy subjects. Exponential decay is my outcome measure. I have a example dataset here I managed to run ...
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2answers
65 views

Why the distribution $Y/\sigma$ does not depend on $\sigma$?

This book describes the scale parameter as below: Suppose $\sigma$ is a scale parameter, in the sense that $p(y|\sigma)=\sigma^{-1}f(y/\sigma)$ for some function $f$, so that the distribution ...
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Bayesian Hyperparameter Optimization. What makes it “bayesian”?

I'm using some bayesian hyperparameter optimization. I know how they works . They always calculate the next values of the hyperparameter dependent on the result of former evaluations. But what makes ...
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What are other ways to say “cutting off 2.5% from the tails”? [closed]

I was reading a text that state Cutting off $2.5\%$ from the tails of these simulated distributions gives us Bayesian interval estimates of $\mu$ and $\theta$. I was wondering what are other ways ...
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1answer
53 views

Can Park & Casella's Bayesian LASSO be applied to generalized linear models?

In Park & Casella's Bayesian LASSO model the LASSO is estimated through a scale mixture of normals: ...
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2answers
184 views

Why not to use Bayes theorem in the form $p(\theta | x) = \frac{L(\theta | x) p(\theta)}{p(x)}$?

There are a lot of questions (like this) about some ambiguity with Bayesian formula in continuous case. $$p(\theta | x) = \frac{p(x | \theta) \cdot p(\theta)}{p(x)}$$ Oftentimes, confusion arises ...
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1answer
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Rearranging Conditional Probability Equation to Show Dependencies

Given the random variables $X$, $Y$, and $Z$, with joint pdf given by $p(x,y,z)=kf(x,z)g(y,z)h(z)$ for some constant $k$, my task is to show that $p(x|y,z)$ is a function of $x$ and $z$. My work is as ...
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Why bayesian needs prior and neural net does not? [closed]

Bayesian requires prior distribution that should be magically taken from somewhere. Neural nets does not require that magical foreknowledge. Why Bayesian requires it, why can't it work without it ...
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1answer
19 views

What is the effect of “bounding the range” for a prior distribution?

Below is a material about the prior disribution for the proportions. The appropriate prior distribution for the parameter $\theta$ of a Bernoulli or binomial distribution is one of the oldest ...
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1answer
47 views

Specification and Interpretation of Repeated Measures Binomial model in BRMS

I have two questions regarding specifying and interpreting Repeated Measures Binomial Models in BRMS We have a set of data in the following format: ...
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1answer
24 views

How should I treat categorical variables in Bayesian modelling

I have got a dataframe that contains three categorical predictors and one numerical response. I would like to compare their differences using posterior uncertainty intervals of MCMC draws. The reason ...
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14 views

Bayesian Statistics, MCMC methods and “Insensitive” Models [closed]

Recently, I faced an issue when I am trying to simulate the posterior. For instance, suppose if I have the following data sets $(x_i,y_i)$ and parameters $a,b$. The model I am fitting is such that: $...
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1answer
22 views

Order of observations in calculation of likelihood

I am afraid this will be quite obvious once you tell me, but I'm just confused right now. Say we have an i.i.d. RV $N\sim Poisson(\mu)$. Now, I want to calculate the likelihood $L$ of an observation ...
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26 views

Translate data into parameter coefficients - Bayesian regression

I Have a data set of accident rates from a population in which I'm attempting to identify which factors have the most effect on how many injuries occur from each accident. Since I am trying to ...
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39 views

MCMC in Gibbs: Threshold Multivariate Linear Regression [closed]

I have a problem regarding what I expected to be a rather straightforward modelling approach: I want to investigate multivariate linear regression parameters and a thresholding parameter which defines ...
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1answer
33 views

How can I normalize Bayesian Network query result?

While taking a Bayesian network tutorial on YouTube, I was watching a video explaining the Bayesian Network probability inference. Somehow, at the end of the tutorial, the lecturer did not explain how ...
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0answers
34 views

FFBS algorithm for estimating mean log-return parameter in stochastic volatility jump model

I am currently attempting to replicate this model: https://arxiv.org/pdf/1809.01501.pdf in r. My (first) problem is regarding how to sample from conditional posterior for mu, $(μ_{(j)}|Y_n, J_{(j−1)}...
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2answers
29 views

How to calculate the probability of getting 2 heads wen the coins is biased?

I can't understand how I can implement the Bayes' Rule here and the probability of getting two heads when tossing twice. Could anyone give me a hint? Suppose you have two coins. One coin has ...
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1answer
46 views

In the context of a MCMC: how to create/interpret a trace for a matrix?

In a Metropolis-Hastings algorithm, I'm drawing a matrix from a proposal. The accepted matrices should constitute draws from a posterior. If it were just a parameter, I would know how to interpret ...
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31 views

Posterior predictive distribution example

Assume there's some normally distributed population ($X$) whose parameters ($\mu$, $\sigma$) are not known. A sample ($x_1$) of size $n$ is drawn from $X$, and statistics are calculated: $\bar{x}_1$ ...
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1answer
24 views

Why no “extra coefficient K” in the formula of Laplace's law of succession?

Below is an example about the calculation of Laplace's law of succession: Suppose we observe $y$ responses out of $n$ binomial trials. Assuming the trials are indenpendent, with unknown response ...
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1answer
56 views

Nonlinear sin model with brms

I try to fit sin function with brms using next code: ...
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Multiple priors in Bayesian estimation

Typical Bayesian estimation equation is: Estimate = ( SampleSize * SampleEstimate + PrioriEstimateWeight * PrioriEstimate) / ( SampleSize + PrioriEstimateWeight ) Typically, the PrioriEstimate is ...
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30 views

Deriving Bayesian Predictive Distribution?

Given $ p(\mathbf{w} | \mathbf{t}, \alpha, \beta) = \mathcal{N}\left(\mathbf{w} | \mathbf{m}_{N}, \mathbf{S}_{N}\right)$ and $p(t | \mathbf{w}, \beta) = \mathcal{N}\left(t | \mathbf{w}^{\mathrm{T}} \...
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Using the data to obtain a prior for an analysis of that same data?

I am refereeing a physics paper in which the authors first analyze their data using a low-accuracy method, and then use the result of that method as a prior to re-analyze the data using a higher-...
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29 views

Difference between dlm and bsts

I'm working on a project which asks me to analysis the Facebook's stock price, and I have to do it the Bayesian way. This assignment doesn't have a particular goal and we are free to decide the what ...