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 GLMM dispersion statistic issue

I am analyzing proportion data with GLMM in which the number of occurrence a behaviour of interest has been displayed and has not been displayed are concatenated and fitted as a single response ...
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Maximum Likelihood Estimation on Zero-Inflated data using Constraint

I have written a function that evaluates the log-likelihood of Zero-inflated Beta Binomial data: ...
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41 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 ...
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Generating data from a Beta-Binomial distribution by inverting CDF in R [closed]

I am trying to generate data from a beta-binomial distribution by inverting its cdf in R. The code I have written to calculate the cdf seems to be working fine for most cases, but gives me values ...
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19 views

deriving shrinkage factors for beta binomial distributions

I am confused by the derivation of shrinkage factors currently online on sites like Wikipedia and ProbWiki. To be on point, I am confused how we get quickly from $\theta_i \sim Beta(\mu,M)$ to $E(\...
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36 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-...
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125 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 ...
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1answer
27 views

How do you compute the beta-binomial probability of “at least one”?

When working with the binomial distribution, it's sometimes useful to compute the probability of "at least one", which is 1 - P(none) or, after setting x to 0 in the formula for the PDF of the ...
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38 views

How do I compute the estimated values of x for a beta-binomial distribution?

I understand how to set up a binomial probability distribution. I'm trying to extend my understanding to the beta-binomial. On Wikipedia, there is a beta-binomial example given at https://en....
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61 views

Estimating beta parameters in truncated beta-binomial distribution

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 of $N_i$ possibilities. However, I am only ...
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2answers
98 views

How to define prior for beta-binomial A/B test

I would like to run an A/B test using a Bayesian beta-binomial model whereby I would state probabilities such as P(pB>pA) in place of using a traditional T-test. I've read that the prior should be ...
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Beta prior on (a,b)

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 ...
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1answer
125 views

Convert probabilities of “binomial events” into beta distribution for binomial parameter (theta)

I have three events that essentially arise from a binomial distribution with k=2, i.e., X~Binomial(theta,n=2). Given some model and data I obtained a conditional (pos distribution for a particular X_i,...
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1answer
104 views

Probability distribution for a binomial proportion 'derived' from serially dependent data

Consider the following type of data: This is data from a single-case experiment: an experiment in which one entity (i.e. one person) is observed repeatedly over time (cf. measurement times 1 to 20)...
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1answer
92 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 ...
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60 views

Estimate betabinomial distribution parameters from weighted observations

Suppose we observe $N$ independent Bernoulli sequences of different lengths. Let $k_i$ be the number of trials in each observation, and $o_i$ be the number of "successes": $$\{(k_1,o_1),(k_2,o_2),...,...
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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 ...
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259 views

Hyperprior Noninformative Beta Binomial Model

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 ...
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35 views

Impact of conjugate priors on mutual information for Naive Bayes

I am currently thinking about the following problem. Suppose you have a simple Naive Bayes model for binary classification based on binary random variables. For example, suppose you want to predict ...
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111 views

Beta-binomial logistic regression model for binomial data with small samples

I have fitted a nonlinear beta-binomial logistic regression model on data y_i: y_i ~ beta-binom(n_i,mu_i,\Phi) where mu_i = exp(\eta_i)/(1+exp(\eta_i)) , and \eta_i=\beta_0+\beta_1/(1+exp(\beta_2x_i ...
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1answer
107 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 ...
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491 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 ...
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609 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 + ...
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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 ...
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79 views

Beta binomial mean different from actual mean

I have data that I am fitting a beta binomial distribution to. The VGAM package for R is being used to do this. However, the the mean of the fitting beta binomial distribution is vastly different from ...
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1answer
240 views

Beta distribution vs beta binomial distribution: alpha and beta

I have been attempting to estimate alpha and beta from a beta binomial distribution given my data. There are R packages like VGAM to do this. I am wondering if there is a difference between estimating ...
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44 views

Is it possible to validate very small p-values via bootstrap?

Suppose I want to see whether the p-values generated by a certain parametric method are close to the “true” p-values. To avoid relying on any parametric assumptions, the latter are produced via ...
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120 views

Calculate expected value from discrete beta distribution

I have a lookup table in 2 variables, $Z_l$ and $T_l$. So, $Z_l$ and $T_l$ are vectors with same length where $Z_l$ goes from 0 to 1 and $T_l$ varies between 300 and 2000. If you are curious, $Z_l$ ...
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53 views

Recursive Bayesian Estimation, $p(C_k|\mathbf{x})$ as (discrete) likelihood

I''ve been struggeling with this problem for the last couple of days. The main goal is to use the probabilistic classification output $p(C_k|\mathbf{x})$, from for example a logistic regression, to ...
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67 views

Difference test for beta–binomial variables

Let $X \sim \textrm{Beta–Binomial}(\alpha_1, \beta_1, n)$ and $Y \sim \textrm{Beta–Binomial}(\alpha_2, \beta_2, n)$ and write $\mu_1 = \frac{\alpha_1}{\alpha_1 + \beta_1}$, $\phi_1 = \alpha_1 +\beta_1$...
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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 ...
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202 views

Sum of Beta-Bernoulli variables

Assume you have $x_i \sim \operatorname{Bernoulli}(p_i)$ with $p_i \sim \operatorname{Beta}(\alpha,\beta)$. I am exploring $Z=X_1+ \dots +X_n$ According to this page, it is $Z \sim \operatorname{...
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1answer
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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 ...
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138 views

Beta Distribution with Random Shape Parameters (Binomial or Poisson Binomial)

I've run into an interesting model into some of my research and I was hoping somebody with more mileage can point me to any resources or other applications. I'm familiar with the Beta-Binomial, but ...
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171 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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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: ...
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1k views

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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1answer
174 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?
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1answer
172 views

How to know dispersion if $\mu$ is close to or below 0 (chance-corrected beta-binomial model)

Background In sensory science, "replication" means having a panelist in a taste panel do multiple rounds of the same test. You cannot just count those additional rounds as additional panelists, ...
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262 views

Beta distribution and beta binomial distribution

What is the difference or relation between beta distribution, beta binomial distribution and binomial distribution?
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1answer
808 views

Distribution of the sum of two independent Beta-Binomial variables

Consider two independent discrete random variables $y_1$ and $y_2$, both distributed with a Beta-Binomial distribution, with different number of successes $n_1$ and $n_2$ but the same parameters $a$ ...
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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 ...
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1k 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 ...
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521 views

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 ...
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3answers
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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 ...
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Estimate beta binomial distribution

I have a dataset as follows: ni-xi 76 48 38 19 20 14 26 29 39 45 38 36 32 34 26 21 23 12 24 18 14 61 35 21 32 20 89 30 34 35 xi 0 0 16 26 1 8 23 19 24 16 16 19 0 0 0 6 8 3 5 9 5 0 34 ...
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281 views

prior distribution to the binomial distribution probability distributions urn model

I have an infinite population with unknown mean of successes and failures. I'm drawing 400 times from the population and get 400 successes. Now I want to generate random estimates for the true mean of ...
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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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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 ...
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3answers
555 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 ...