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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3answers
310 views

Does order of events matter in Bayesian update?

I'm wondering whether the order of events can lead to different Bayesian update. For example, consider a coin-tossing problem with unknown $p$, the probability of Head. Initially, $p$ is known to ...
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61 views

Conditional probability of tossing coins with uncertain head probability

Suppose there are two coins A and B. When tossing a coin $i$, "head" happens with probability $p_i$. The problem is that $p_i$ itself is a random variable. Say that the associated probability ...
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Bayesian inference for a conditional probability

I'm simplifying my research question and want to know whether the question can be properly modeled or not. Suppose we have two coins $X_1,X_2$ and assume that the outcomes are possibly correlated. ...
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1answer
53 views

Calculating the parameters of a Beta-Binomial distribution using the mean and variance

I'm trying to do the same thing that was done in this question: Calculating the parameters of a Beta distribution using the mean and variance for the Beta-Binomial distribution for which the mean is ...
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50 views

Multi-armed bandit where you must pull N pulls in T timesteps

Multi-armed bandit where you must pull N pulls in T timesteps Consider the beta-Bernoulli multi-armed bandit, with the following wrinkle: We have a total of $T$ time steps. We must pull $N$ pulls ...
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Bayesian estimation of weighted proportion

Having bayesian estimates of a proportion is relatively easy. You model that proportion as a binomial variable, you choose a beta-binomial prior and by using the likelihood you obtain a beta-binomial ...
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2answers
103 views

How do I carry out a significance test with Tarone's Z-statistic?

Context In this blog the author suggests using Tarone's Z-statistic to test for overdispersion in a binomial model to determine whether or not it is necessary to use a beta-binomial model instead. In ...
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Can (log-)likelihood be used to compare a binomial model to its beta-binomial equivalent?

In this article the author talks about fitting beta-binomial models to data when the there data is over-dispersed relative to the assumptions of a model with binomial errors. Near the end they present ...
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How to fit a Beta-Binomial Distribution to a dataset [duplicate]

I have a data set which is defined over positive integers and I have reasons to believe it follows a beta-binomial distribution. I am aware there is the ...
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1answer
27 views

Binomial distribution for randomly drawn probabilities

Setting Probability theory can be a weird place sometimes. Here I was, confident in my insane math skills, trying to solve the following problem: Let $N, \alpha$ and $\beta$ be given. ...
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49 views

Beta-binomial Model with missing values

I have read http://www.sumsar.net/blog/2018/12/visualizing-the-beta-binomial/ this simple explanation of how the posterior is changing while more data are added: in this visualization there are six ...
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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 randomly drawn from a beta distribution results in the so called beta-binomial distribution. One can calculate ...
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1answer
141 views

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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164 views

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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2answers
101 views

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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174 views

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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441 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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1answer
151 views

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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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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1answer
221 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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1answer
198 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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497 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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1answer
571 views

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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1answer
199 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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1answer
138 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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1answer
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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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2answers
86 views

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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1answer
225 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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0answers
257 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
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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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1answer
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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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0answers
152 views

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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1answer
238 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 [closed]

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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1answer
98 views

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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1answer
316 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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2answers
382 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
43 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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0answers
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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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1answer
2k 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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0answers
78 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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3answers
369 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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1answer
604 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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227 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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1answer
703 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 ...