# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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: 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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}} \...