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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431
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10answers
165k views

What is the intuition behind beta distribution?

Disclaimer: I'm not a statistician but a software engineer. Most of my knowledge in statistics comes from self-education, thus I still have many gaps in understanding concepts that may seem trivial ...
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5answers
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Relationship between Binomial and Beta distributions

I'm more of a programmer than a statistician, so I hope this question isn't too naive. It happens in sampling program executions at random times. If I take N=10 random-time samples of the program's ...
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3answers
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What is the appropriate model for underdispersed count data?

I am trying to model count data in R that is apparently underdispersed (Dispersion Parameter ~ .40). This is probably why a glm with ...
16
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2answers
306 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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3answers
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When to terminate the Bayesian A/B test?

I'm trying to do A/B testing the Bayesian way, as in Probabilistic Programming for Hackers and Bayesian A/B tests. Both articles assume that the decision maker decides which of the variants is better ...
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0answers
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Hyperprior Noninformative Beta Binomial Model [closed]

I've been working through Gelman's Bayesian Data Analysis 3 text and have been trying to understand one of the hierarchical models revolving around rat tumors (Chapter 5). He uses a binomial model ...
9
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1answer
617 views

Prediction interval for a future proportion of successes under Binomial setting

Suppose I fit a Binomial regression and obtain the point estimates and variance-covariance matrix of the regression coefficients. That will allow me to get a CI for the expected proportion of ...
8
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1answer
201 views

Which distributions on [0,1] other than the beta distribution form nice compounds with the binomial distribution?

For which distributions x, other than beta, is the x-binomial distribution nice? The beta and binomial distributions are famously conjugate but I am curious if other non-conjugate distributions will ...
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1answer
1k views

Taking into account the uncertainty of p when estimating the mean of a binomial distribution

I have a binomial distribution with parameters $N$ and $p$, and the estimate for the mean of my distribution is N$\times p$. The values of $N$ and $p$ are such that we can use the Gaussian ...
7
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1answer
472 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/(...
7
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3answers
3k views

Estimating beta-binomial distribution

Suppose that I culture cancer cells in n different dishes g₁, g₂, … , gn and observe the number of cells ni in each dish that look different than normal. The total number of cells in dish gi is ti. ...
7
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1answer
6k views

Minimizing symmetric mean absolute percentage error (SMAPE)

I am working on a forecasting application in which forecast errors are measured using the symmetric mean absolute percentage error: $$ SMAPE = \frac{1}{n} \sum\limits_{t=1}^n{\frac{|F_t - A_t|}{F_t + ...
7
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1answer
477 views

How to implement generalized hypergeometric function to use in beta-binomial cdf, sf, ppf?

I'm writing a subclass of scipy.stats._distn_infrastructure.rv_discrete for the beta binomial distribution whose PMF is $$P(X=k \mid N, \alpha, \beta){N \choose k} ...
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2answers
4k views

How to specify a Bayesian binomial model with shrinkage to the population?

Problem I’m currently working on a problem where I have count data for $n$ items in the following form: ...
6
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1answer
10k views

Fitting a beta-binomial model in the case of overdispersion in R

I'm estimating some count data. I have counts for say $m=100$ individuals. Unfortunately when using the Poisson regression overdispersion occurs. So I was thinking to fit a negbin model. But this is ...
6
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3answers
779 views

Fast integration of a posterior distribution

I wish to infer the posterior distribution on the probability of success $\theta$ in some binomial process, the twist being that I know that $\theta$ lies in the interval [0.5, 1]. The trouble is ...
6
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1answer
288 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 ...
6
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2answers
394 views

Should one use the same overdispersion parameter when comparing Binomial models?

McCullagh & Nelder, 2nd edition, p 91 claim that to make comparisons "fair", it's best to use a single estimate of overdispersion parameter, usually derived from the most complex model. I noticed ...
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1answer
5k views

Properly interpret the alpha / beta parameters in the Beta Distribution

For quite a while I believed that the proper interpretation of a Beta distribution with $\alpha$ and $\beta$ is: "what is the most likely $P$ given $\alpha -1 $ success (heads), and $\beta -1 $ of ...
5
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4answers
756 views

Beta binomial Bayesian updating over many iterations

I'm using a beta binomial updating model for a piece of code that I am writing. The software is real time updating - meaning that data is continually being gathered and after N data points are ...
5
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2answers
800 views

A continuous generalization of the binary bandit

There is plenty of reading out there about Bayesian (beta-binomial) multiarm bandits for 0/1 data, but I would like to extend this slightly. To give some context, suppose I have two webpages, A and ...
5
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1answer
377 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 $\rho$? $$Y = \sum_{i=1}^n X_i$$ $$...
5
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1answer
90 views

Multiple binomial variables: probability they are all 1

We have observed four phone booths by recording, at a given time point, whether each phone booth is occupied (0 or 1). We have approximately 50 such observations (i.e. at 50 different time points, so ...
4
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1answer
547 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
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 ...
4
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2answers
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Update rule for beta distribution with fixed K/confidence/sample size

Normally you have a beta distribution with shape parameters $a$ and $b$. The mean of this distribution is $a / (a + b)$ and the sample size, or the confidence (or K) is $a + b$. Now, if you do some ...
4
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1answer
705 views

Conjugate beta / interpretation of the “continuous binomial” signal

Note: this question has significantly evolved, thanks to inspiring comments by Tim. Assume there is some "truth" $x\in[0,1]=Beta(1,1)$ that is signaled with some precision. I assume that the ...
4
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2answers
980 views

Estimate accuracy of an estimation on Poisson binomial distribution

I manage a website that charges its customers using payment cards. Some transactions area approved, others are declined. I compute the approval rate of transactions for a interval (a calendar day) as ...
4
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1answer
2k views

Types of dispersion parameter for binomial data

For a model with a binomial proportion as response variable, which is fitted with according to a binomial distribution, a dispersion parameter $\phi$ can be calculated, which is equal to the sum of ...
4
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0answers
42 views

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 ...
4
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0answers
70 views

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, ...
3
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1answer
2k views

Understanding the multilevel / random-effects beta-binomial regression model

Suppose we have an outcome variable $y_{ji}$ which is a count of behaviors performed by group $j$ in round $i$, for $j = 1,...,n$ and $i = 1,...,8$. The outcome $y_{ji}$ counts are non-independent ...
3
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1answer
258 views

Beta prior on (a,b) [duplicate]

Is it possible to rescale a beta prior to range from, say, $(0.5,1.0)$? For example, say that your likelihood function for some parameter $p$ is binomial and you know that $p \in (0.5,1.0)$ due to ...
3
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2answers
1k views

Updating a beta-binomial

Suppose I'm modeling a set of processes using a beta-binomial prior. I can build parameterized beta-binomial models that average over large groups of the processes to give reasonable, although coarse, ...
3
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1answer
2k views

Using poisson distribution to model proportions

I have recently came across a number of articles that have used Poisson GLMs when modelling proportion data. For instance, one study modelled the proportion of pig mortality for each sow using a ...
3
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1answer
1k views

VGAM fitting a betabinomial model

I have a small question, given this: fit <- vglm(cbind(R, N-R) ~ 1, betabinomial, lirat, trace=TRUE, subset=(N > 1)) Why should I do ...
3
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1answer
247 views

Bivariate distribution: beta and binomial

Consider a pair of RVs $X$ and $Y$, with the following conditional distributions: $$X | Y=y \sim Binom(L, y)$$ $$Y | X=x \sim Beta(\alpha + x, \nu)$$ where $L$, $\alpha$, and $\nu$; are all ...
3
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1answer
79 views

Generalized Binomial Models

With the standard Binomial probability distribution we consider n trials each with a probability of success p. This can be somewhat generalized to the Beta-Binomial Distribution which is effectively ...
3
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1answer
668 views

What happens with the beta-binomial distribution, when n approaches infinity?

Short question: What happens to the beta-binomial distribution, when n increases to infinity? Is there a count distribution arising like it's for the classical binomial distribution?
3
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1answer
2k views

beta-binomial distribution with R

I am studying an experiment of the kind: Let $n_{ij}$ be the number of fetuses, $X_{ij}$ the number of responses i.e. the number of fetuses with a malformation in the jth litter of the ith dose level ...
3
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3answers
3k views

Use Bayesian hierarchical model to predict new data points

I have a data set $(n_i,y_i),i=0,...,10$. I modeled it as a Bayesian hierarchical beta-binomial model. $y_i∼Binomial(n_i,p_i)$ and $p_i∼Beta(\alpha,\beta)$. I have used MCMC to estimate $\alpha$ and ...
3
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0answers
157 views

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-...
3
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0answers
229 views

Test for Equality of Parameters in Beta-Binomial Distribution

With binomially distributed data, it's straightforward to test the null hypothesis of equiprobable responses, $H_0: p=0.5$, but say you want to test the analogue in a Beta-Binomial model fit to over-...
3
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0answers
212 views

Estimating beta parameters in truncated beta-binomial distribution

$\newcommand{\Beta}{\operatorname{Beta}}$I'm sampling a bunch of probabilities, $\theta_i \sim \Beta(a,b)$, from a common beta distribution, and then using each $\theta_i$ to sample a value $x_i$ out ...
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0answers
413 views

Should I use a beta-binomial or binomial glmm?

I have several data sets on wildlife disease incidence. One of the issues with my dependent variable is that it represents only current infection status, therefore 0 (no disease) can represent either ...
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2answers
670 views

Is the beta-binomial distribution a conjugate prior for some distribution?

The beta distribution is conjugate prior for the binomial distribution. Is the beta-binomial distribution a conjugate prior for some sampling distribution?
2
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1answer
513 views

Beta distribution and beta binomial distribution

What is the difference or relation between beta distribution, beta binomial distribution and binomial distribution?
2
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2answers
94 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 ...
2
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1answer
150 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 ...
2
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1answer
590 views

How to combine two beta-binomial distributions

Say I have the following situation. I have two weighted coins: Coin 1: In the past I've seen this coin flipped 10 times, 8 of which it came up heads. So I can model the probability of $n$ heads out ...