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Questions tagged [beta-binomial]

The beta-binomial is a discrete distribution on 0, 1, ..., *n* where the probability of success in a binomial distribution (*p*) is itself drawn from a beta distribution.

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Beta-binomial distribution for scaled and translated Beta

Recall, that a binomial distribution in which the probability of success at each trial is fixed but randomly drawn from a beta distribution results in the so called beta-binomial distribution. One can ...
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Correct usage/understanding of Bayes Factor when comparing two proportions

I'm just starting to learn R and explore Bayesian statistics, but I keep getting tripped on using Bayes Factor and (honestly), I'd love a little confirmation if my process is correct in interpreting ...
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Poisson-binomial vs. Beta-binomial

I have N distinct bernoulli trials with a distinct probability for each trial given by, P=(p1, p2, ..., pN). I want to know the distribution of the number of successes. Given that I know P, I can ...
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Bayesian hypothesis testing with multiple beta-binomials

I want to test questions relating to whether individual ants of a certain species have personal food preferences, using a Bayesian model built up of multiple beta-binomial distributions. My problems ...
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Hypothesis testing with beta binomial. Dealing with overdispersion

To make the question more understandable I will use a reproducible example. I have count data, how many connections different groups share with a unique group. In my case I have an upper bound of <...
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How to infer a prior belief after observing a behavior

My participant goes through a maze made of 32 T intersections. At each intersection he must choose whether to go either to the left or to the right: one option will lead to another T intersection, ...
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How to convert the parameters in a binomial distribution to those in a beta distribution?

I know that the beta distribution is the generalized continuous case of the discrete binomial distribution. Let's say I have a binomial distribution, $B(N,p)$. I would like to know the corresponding $\...
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Sample from multiple urns - when to stop?

I have a problem that hasn't yet been addressed, although follows similar lines of reasoning here and here. My problem is as follows: I have $N$ urns, each with a different number of black and white ...
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Given an observed sample from a binomial distribution and a known prior, how can I update the probability distribution of possible 'p' values?

Let's say that I have an 'unfair' coin, for which I'm interested in estimating the 'heads' likelihood or 'p' value. I've been told that the 'heads' likelihood for my 'unfair' coin is normally ...
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Relation between mean probability and overdispersion

I have proportion data for N observations x P variables. That is to say for each variable, I have proportions that vary between observations. A proportion is the number of successes divided by the ...
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197 views

What is the distribution of a sum of identically distributed Bernoulli random varibles if each pair has the same correlation?

What is the distribution of a sum of $n$ Bernoulli random variables, each having success probability $p$, where each pair is correlated with correlation coefficient $\rho$? $$Y = \sum_{i=1}^n X_i$$ $$...
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beta binomial hierarchical model with two groups, inference on the group hyperparameters

The problem I want to solve: Lets imagine that I have two factories A and B, where each factory produces coins. What I suspect is that the probability of tails (denoted as $\theta$) varies ...
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Events/Trials Syntax vs. Single-Trial Mixed Model - Beta-Binomial Model

I'm a touch confused about modeling an events/trials outcome when the Bernoulli trials are not independent, such as when a series of Bernoulli trials are observed from the same person. I cannot find a ...
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Analysis of Paired, Longitudinal Ordinal Data

I am doing an analysis of paired survey responses. The responses are on a 5pt Likert scale and the same 100 respondents are asked a question at year 1 and year 2. I would like to do some pairwise ...
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Binomial-logit-normal

I'm watching a pool match and at the start of the first game the person next to me says "I think that the chance of player A winning the first game is $.700$. If he wins that I think the chance of ...
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Entropy of the beta-binomial compound distribution

I have a generative process as follows: $$ x \mid \alpha \sim \textsf{Beta}\left (\alpha,\beta \right) \\ y \mid x \sim \textsf{Bernoulli}(x). $$ How does one go about calculating the Entropy of ...
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Prevalence of class c in random subsets

Suppose we have a set S of N objects, K of them belonging to class c and (N-K) not belonging to c. What is the probability that, if I pick a subset at random among the (2^N-1) nonempty subsets of S, ...
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108 views

Number of parameters mixture model

In order to do a LRT between two mixture models with different numbers of components, I need to know the number of parameters. I would like to know the answer both for: a) Gaussian mixture model b) ...
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What correlation coefficient is appropriate for two beta-binomial variables?

I'm trying to test whether answers to 4 'yes-no' questions are correlated. Every subject answered each question 16 times. I have number of positive answers for each question for each subject. If I ...
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144 views

How can we determine the parameters of a Beta-Binomial Distribution for given mean and variance?

Given a beta-binomial random variate $X$ with $N$ known, how can I choose $\alpha$, $\beta$ such that the distribution's mean matches a chosen quantity $\mu$ and its variance matches a chosen quantity ...
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146 views

how to analyze overdispersed binary data

In my substantive research, I often use dichotomous scoring (1 correct, 0 wrong) for my tests (tests with $15~yes/no$ items). My goal is often to compare the the proportion of correct answers to all ...
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403 views

Limit of beta-binomial distribution is binomial

I am trying to understand the relationship between the beta-binomial and the binomial distribution. More specifically, I am trying to show that the limit of the beta-binomial distribution, with $p=a/(...
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Hyper-parameter estimation for Beta-Binomial Empirical Bayes

I am reading a paper Illustrating empirical Bayes methods and in the paper the author uses method of moments to derive the value of an estimate. In equation 17 the author gives the following marginal ...
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133 views

Beta-Binomial Derivation

I was looking on wikipedia and seemed to be not follow how the beta-binomial is derived. I was hoping I could provide an example I found in a paper and someone could explain the steps. Given the ...
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114 views

Bayesian A/B test - using an updated prior based on collected data

I have a question about whether I would be adding bias to an A/B test by updating my prior based on combined A & B data, and then running the A/B test on that prior. My A/B test is click through ...
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Probability with Limited Data

I have a theoretical question which has been bugging me, involving prediction based on limited data. Say you go to the horse track and see that horse A has just won its race with horse B. The two ...
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Updating posterior distribution in online fashion [duplicate]

First, I am sorry if this is an obvious question, I am starting to study bayesian statistics (mainly for machine learning) and I was seeing the classic coin flip example using a Bernoulli distribution ...
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In Bayes using conjugate prior and joint probability

While using conjugate prior in bayesian networks. For example say beta-binomial. I have assigned a beta distribution as a prior after certain observations the posterior is available as betapdf(a+no. ...
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109 views

Can I multiply the densities of a beta distribution in finding the joint density?

I have a state vector consisting of 3 nodes. Using bayesian inference I found the posterior which also follows beta distribution. Now I have beta densities(as we know beta densities are out put of ...
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172 views

Beta distribution

After studying about the beta distribution I understood that the beta density function is used to find the densities for the probabilities. Densities may range from 0 to inf. That is if the density of ...
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190 views

Visually compare binomial and beta-binomial appropriateness in R

I'm trying to choose the best distribution family for generalized linear regression. My outcome is cross-sectional, over-dispersed proportion data (# of behaviors/20-22 possible behaviors). I used the ...
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1answer
34 views

Beta Binomial Encryption

For the sake of example, suppose we have a list of advertisements $\{A_i\}_{i=1}^n$, each of which have parameters $I_i$: the number of impressions, and $C_i$ the number of clicks. Then $C_i/I_i$ ...
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How do I specify a Bayesian Beta binomial model, with predictor variables, for R2jags?

I want to have a beta-binomial Bayesian model, with R2jags. I understand I need something like this: ...
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Aggregated Count Data-Days in the last 30- Negative Binomial vs Beta Binomial vs Binomial Regression

I need help choosing between the binomial, negative binomial, and beta-binomial distributions and understanding the rationale of why I would choose one over the other, as well as understanding how it ...
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166 views

Beta Binomial Inverse CDF

There are p groups of size $n_1, n_2, ... , n_p$ each with number of successes $x_1, x_2, ... x_p$ and number of failures $n_1 - x_1, n_2 - x_2, ... , n_p - x_p$. $X_i$ ~ $Binom( n_i, p_i)$, where $...
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Log Beta Distribution Priors

Let $X_n$ be a binomial distribution with parameter $\theta$. Empirically, after $n$ throws, my estimate is $\hat{\theta}_n=\frac{S_n}{S_n+F_n}$, where $S/F$ are the successes and failures $(F_n:=n-...
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How to model on rate (percentage) outcome, but with consideration of number of total counts?

I would like to model on a rate outcome, with a number of covariates. I also have the number of counts which generate the rate outcome. Like below: ...
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205 views

Comparing two groups with binomially distributed data

Below (in R), I have two INDEPENDENT groups of scores that are binomially distributed. These two groups of scores are known to have different probability of success (i,e., $p_1 \neq p_2$). Let's ...
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312 views

Computational shortcuts/approximations for beta-binomial and beta-negative binomial CDFs

There are several simplifications that can be done so that computing cumulative distribution functions of beta-binomial and beta-negative binomial distributions, but still computing CDF as $F(x) = \...
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Differential effect of parents quality on sons and daughters: how to test it

Suppose you have n broods followed up during x years. In each brood there are a varying number of sons and daughters that sum up to a varying number of offspring in each brood. You dispose of this ...
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1answer
39 views

Interdependence of hyperparameters of bernouilli laws [closed]

Suppose that you have $N$ people passing an exam having 2 type of questions (type A and type B) which all have correct/uncorrect outcomes. After seeing the results, I want to modelize the probability ...
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Bayesian a posteriori law of bernouilli law, from observed correlated bernouilli law

Suppose that you have $N$ people passing an exam having 2 yes/no questions. After seeing the results of $N-1$ people and the result of the first question of the $Nth$ person, I want to know what I can ...
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1k views

Update samples of a Beta with Bernoulli likelihood to the Beta posterior

So this may seem like a strange question, but I have a good reason for it. Nonetheless, I’ll risk the XY problem and describe what I want to do without explaining why I want to do it. We know that ...
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69 views

Design of a triangular test (or 2-out-of-5) when there is intra-group variability of samples

I would like to submit to an expert group a set of images from two genetically different types of plants to see if they can find a difference. I have 20 images for each of the two types. Images are ...
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263 views

Why is there -1 in beta distribution density function?

Beta distribution appears under two parametrizations (or here) $$ f(x) \propto x^{\alpha} (1-x)^{\beta} \tag{1} $$ or the one that seems to be used more commonly $$ f(x) \propto x^{\alpha-1} (1-x)^{...
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445 views

Marginal probability of beta distribution

The problem: Suppose you have 10 voters and 2 candidates. Each voter has a probability $p$ of voting for a Candidate 1, and a probability $1-p$ of voting for Candidate 2. What is the probability that ...
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159 views

Posterior of Beta Distribution

Let $X$ be the number of packets that fail to reach their destination nodes from $n$ = 100 randomly selected packets. Use the Beta-binomial formulation to specify the posterior distribution $p(\theta|...
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512 views

How does the beta prior - binomial conjugate theory hold in classifiers?

In our machine learning class, we were given an example of a naive bayes classifier. Say, you classify a day as being good/bad depending on 2 conditions (the "X" input) - weather(X1 - hot/cold) and ...
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(un)conditional density from other densties in continuous beta model

This question relates to my other question, where I actually noticed a mistake. Suppose I have a random variable $X\in[0,1]$ and a signal $S\in[0,1]$ bearing some info about $X$. I know $f_X(x)$ and $...
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207 views

Translating user-defined joint-distribution from PyMC to PyMC3

I'm attempting to set up a simple beta binomial hierarchical model with an uninformative prior in PyMC3. I've read that the uninformative prior for this model should have alpha and beta hyper-...