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

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Bayesian Data Analysis: Section 3.4 Sampling from the joint posterior distribution Example

For reference: $$ p(\sigma^{2}|y) \propto \tau_n N(\mu_n | \mu_0, \tau_0^{2}) \text{Inv}-\chi^{2}(\nu_0, \sigma^{2}_0) \prod_{i=1}^{n} N(y_i|\mu_n,\sigma^{2}) \tag{3.14} $$ The book states: As ...
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How should I compare posterior samples of the same parameter from two Bayesian models?

I have run 2 Bayesian regression models and would like to compare the posterior samples of a parameter that is common to both models. For example, if model A is $y=\alpha + \beta_1x_1$ and model ...
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1answer
17 views

Bayes Rule for Random Samples?

Suppose I have $M$ samples from unknown distributions $F(X)$ and $F(Z|X)$. Is there a way from these two vectors to get samples of $F(X|Z)$? I understand Bayes rule, but I only know how to apply it ...
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1answer
23 views

Different results for Bayesian ANOVA in JASP and R (BayesFactor)

I computed Bayes Factors for a repeated measures ANOVA in JASP and the R package BayesFactor. There were two between-subjects groups (factor "group") with multiple measurements. However, the results ...
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30 views

Can we use MLE estimates as hyperparameters of bayesian linear regression?

Given a linear regression \begin{align} y_i = \mathbf{x}_i^T \mathbf{b} \qquad i = 1,..,N \end{align} or in matricial form: \begin{align} \mathbf{y} = \mathbf{X}^T \mathbf{b} \end{align} MLE ...
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Hamiltonion monte carlo

Can someone explain the main idea behind Hamiltonion Monte Carlo methods and in which cases they will yield better results than Markov Chain Monte Carlo methods ?
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1answer
27 views

Bayesian errors-in-variables model definition in JAGS and symbolically

I'm fairly new to probability theory and am attempting to understand and implement an errors-in-variables simple linear regression model. I am assuming a model of the form $$ Y=\theta X_a+\epsilon_Y ...
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Posterior pointwise uncertainty of multivariate normal-Wishart (variational GMM)

Given a variational mixture of Gaussians (as per, e.g., Chapter 10 of Bishop, 2006), we can compute the posterior predictive pdf: $$ \left\langle p(x|\alpha,\beta,\nu,\mu,V) \right\rangle $$ where ...
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17 views

How is Bayesian Probability different than “normal” probability? [duplicate]

I understand how Bayes' Theorem works and it seems logical to me. How is "normal" probability any different? Is there some alternative way to use the same data to compute a different probability?
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1answer
12 views

inferential structure determination

I'm trying to get my head around Bayesian inference and the difference between the posterior and likelihood. Going off the back of these answers, I'm under the impression that the posterior is ...
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1answer
42 views

Laplace's law of succession using different priors

Laplace's law of succession is a well-known rule, relying on Bayes' theorem. A possible proof of the rule of succession can be found on Wikipedia. Note that for this proof we use a uniform ...
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More variability than Beta posterior suggests - A/B testing

I'm using a Beta-Bernoulli to model conversion rates between two groups, A and B. For each group, my prior is $Beta(a,b)$ and likelihood is $Bernoulli(p)$. My posterior distribution is therefore a ...
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18 views

applying Posterior predictive distribution on the data from which the coefficient of regression were estimated [on hold]

I am new to linear regression modelling. For the given linear model $Y_{current} = \mu+ \beta X_{current} + \varepsilon \sim N(0|\sigma^2)$ Generally, in regression analysis we estimate, coefficient ...
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2answers
348 views

How do ABC and MCMC differ in their applications?

To my understanding Approximate Bayesian Computation (ABC) and Markov Chain Monte Carlo (MCMC) have very similar aims. Below I describe my understanding of these methods and how I perceive the ...
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2answers
64 views

What to take in consideration when we use Bayesian Methods on Big Data problems?

I was reading the book Bayesian Methods for Hackers by Cameron Davidson-Pilon. He use PyMC for examples. As an experiment, I created a ...
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2answers
95 views

Optimize starting parameters for Bayesian Linear Regression?

I'm using PyMC3 in Python 3 and I'm not sure exactly how to optimize my starting parameters. The example uses the regression ...
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16 views

How to update latent discrete variables in MCMC?

Most of the discussion on Bayesian model with latent variables that I've seen fall into two classes: continuous latent variable underlying the observed discrete outcome (e.g. probit model (Albert ...
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13 views

DIfferent zero-inflated poisson model outputs of Frequentist and Bayesian

I modeled zero-inflated poisson regression using crime data. There are 10034 obs and let assume spatial autocorrelation is not significant. For the Frequentist approach, R was used and code is ...
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7 views

Estimating days of usage for a limited time frame

I would like to estimate how many individual days of a week user have been using a feature in my app, but I'm kinda lost as to do that? Basically the set up is an A/B test, where some users get one ...
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1answer
23 views

How to proper evaluate the PDF of a Beta Distribution?

On page 40 of "Think Bayes - Bayesian Statistics Made Simple", Allen evaluates the PDF of the Beta distribution as ...
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Modelling a Static Bayes Net versus Dynamic Bayes Net

I have a Bayes Net with 20 variables, but I found out that one of the Parent variables is dependent on the previous value of its Child as: C(t-1)->P(t)->C(t) C and P are binary (True or False). All ...
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1answer
14 views

How to interpret marginal likelihood definition?

Say we have a Beta-Bernoulli model where $X_i$ are i.i.d. Bernoulli variables, $p(X_i=1)=\theta$, and $\theta\sim\operatorname{Beta}(\alpha,\beta)$. The marginal likelihood is defined as $$ ...
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20 views

Credible Interval for Likert-Scale data

I am begining at Bayesian statistics. I'm currently trying to build a credible Interval for ordinals data. I have 16 observations from a likert scale. Each of my 16 participants has been asked to rate ...
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1answer
15 views

Update the posterior after sampling from a mixture

Suppose I put 10 red balls and 40 blue balls into a bucket. Then I put another 50 balls in that I drew at random from a set of 100 red balls and 100 blue balls. At this point I believe the ...
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1answer
28 views
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Intraclass correlation coefficient in Bayesian statistics

I need some references about intraclass correlation coefficient in Bayesian statistics and hypothesis testing. I already take a look in A. Gelman, J.B. Carlin, H.S. Stern and D.B. Rubin, Bayesian ...
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Bayesian inference via approximate data likelihood

Suppose that we have a very large i.i.d. sample $x_1,...,x_n$ and a data likelihood defined by $$p(x | \theta,\beta) = \prod_ip(x_i | \theta,\beta)$$. Further suppose that $\theta$ is the parameter ...
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24 views

How can we prove this equation using marginalization and conditioning? [closed]

I want to prove $$P(A|C) = \sum_{B} P(AB|C) $$ How can we prove this using marginalization and conditioning?
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17 views

fdjoint multivariate longitudinal and survival modelling code in any one of statistical software [closed]

I have two continuous response varaible (systolic and diastolic blood pressure) over time which is bivaraite longitudinal.Then i need to model those bi-variate responses with a single time to event ...
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2answers
65 views

Rule of Succession for Unfair Coin

Given the first n flip results from an unfair coin, we wish to estimate the probability that the next flip is a heads. I can take 2 approaches to this: ...
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1answer
30 views

Using Bayesian statistics to analyze food production [closed]

I am in the process of learning about Bayesian statistics with the help of R, and I would like to know the kind of analysis questions I should pose. Say for instance, with this dataset of food ...
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1answer
30 views

Bayesian test with exponential density function

I need to do a bayesian test for a simple random sample with exponential distribution and N(2,3) as a prior distribution conditioned by: Null hypothesis=> $\theta$ less or equal to 1 ; Alternative ...
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3answers
258 views

Monty hall problem, getting different probabilities using different formulas?

In my Monty hall problem, I am computing what is the probability that P(H=1|D=3) i.e. price is behind door 1 and the 3rd door is opened. $P(H=1|D=3) = p(H=1) * \frac{p(D=3|H=1)} {p(D=3)} = 1/3 * ...
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Composite-likelihood ratio test

I am interested in a number of articles (such as Kim and Stephan 2002 and follow-up articles) that use composite likelihood ratio tests to infer selection pressure on linked (phased) genetic data. I ...
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1answer
46 views

Why does Bayesian p-value involve the parameters in addition to the data?

On page 146 of Gelman's Bayesian Data Analysis, Gelman discusses Bayesian p-value as a way to check the fit of the model. The idea is to compare the observed data ($y$) with data that could have been ...
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20 views

Expected value of inverse-Wishart prior in JAGS

I am trying to estimate a covariance matrix in JAGS, using a prior centered around the expected variances. Thus, I want to assign a prior with expected variances that are about equal to the observed ...
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1answer
21 views

Determining coin biases, given some information

I have a real-life problem which essentialy boils down to the following: Given n biased coins, given to you ordered from highest to low bias, determine the last coin (by index) whose bias is still ...
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Is it possible to estimate the posterior distribution of the difference of two means using published t statistics?

I am interested in using summary statistics from published papers to obtain posterior distributions for the difference of two means. In the setting of the classic t-test, we can imagine measurements ...
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1answer
38 views

Conjugate prior for parameter W when the likelihood is normal with mean and variance both functions of W

Suppose that $x$ is an observable scalar variable and $$ x \thicksim N(W\mu_0,W^2\sigma_0^2) $$ Where $W$ is a parameter that must be estimated from data, and $\mu_0$ , $\sigma_0$ are known ...
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23 views

X/Y/Z axis in a bayesian system?

So I've come up with catagories and the probabilities are fine, except, I also need to represent the order in which they fall to properly represent the data. ...
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1answer
30 views

Variance of Level 2 random effect depending on value of variable in highest level (Level 3) in Hierarchical model

I was reading material on hierarchical models, especially 3 level hierarchical models. One of the examples I found was having the hierarchy Hospitals-Wards-Nurses. Only 1 measurement per Nurse. So ...
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1answer
117 views

allocating degree belief to forecast value

We have a number of providers for a forecast of wind power generation per country per date. Values are forecast up to one week ahead. Forecasts may be compared with actual values of reported wind ...
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1answer
68 views

On the Bayesian setup in inference

I've been trying to get into the chapter 4 in Lehmann's Theory of point estimation, but I can't seem to understand his presentation of the Bayesian setup. He starts of by the introduction below and ...
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1answer
129 views

PyMC3; create simple Linear Regression model with real-world datasets

The Linear Model I understand the concepts of Bayesian Inference in that the observed data, $x$, is fixed, and the parameters, $\theta$, are the random variables that follow a particular ...
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64 views

Overestimated MCMC standard deviation (SD)

I did a simulation study with 100 replications for a complex model with rstan, and I found the standard deviation (SD) of posterior samples is larger than the true ...
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0answers
9 views

How to Calculate the CI of the LTV given the CI of the Expected Revenue

I am calculating the LTV (lifetime value) of some users. What I have so far is a table of months, expected revenue and the credible interval (lower and upper bounds) of that expected revenue. A ...
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2answers
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Simple example of how “Bayesian Model Averaging” actually works

I'm trying to follow this tutorial on Bayesian Model Averaging by putting it in context of machine-learning and the notations that it generally uses (i.e.): ...
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1answer
40 views

What is causing autocorrelation in MCMC sampler?

When running a Bayesian analysis, one thing to check is the autocorrelation of the MCMC samples. But I don't understand what is causing this autocorrelation. Here, they are saying that High ...
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29 views

Probabilistic Graphical Model - Modelling a Bayesian Network on Real Life Data

Recently, I have started studying about Probabilistic Graphical Models (PGMs). While the examples provided in the textbook essentially convey the message of what and how things are happening, I am ...
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46 views

About the Jeffreys prior for multivariate regression

In certain cases, the Jeffreys prior for a full multidimensional model is clearly inadequate, this is for example the case in: $$ y_i=\mu + \epsilon_i $$ where $\epsilon \sim N(0,\sigma^2)$, $\mu$ and ...
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13 views

Causal Impact and using multiple control series with their regressors

Hi all I am analyzing several DMA's for campaign effectiveness using the CausalImpact package by Kay Brodersen. I have data for participants and non-participants INCLUDING their contemporaneous ...