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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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 ...
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22 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 ...
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9 views

Is the beta binomial with a variance equal to almost zero equal to the binomial distribution?

I was wondering whether the following holds: if I have a certain average and an almost zero variance and I use them to estimate the alpha and beta for the beta binomial distribution. Would that be ...
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25 views

Applying a beta binomial distribution and my understanding

I am having a bit of a problem in understanding the betabinomial distribution. I've only read some short introductions, but I can't really seem to get a grasp of it. What I do not understand is the ...
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12 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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19 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": ...
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129 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 ...
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68 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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17 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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59 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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69 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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73 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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192 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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2answers
115 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 ...
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1answer
47 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
115 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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35 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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84 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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46 views

Beta Binomial Distribution with a priori $\alpha$ and $\beta$ to Account for Probability Forecasts

I am trying to use a beta binomial distribution to calculate how much a single vote would count in a 2-choice election, given $n$ voters and a $p$ forecast: $ f(\lceil\frac{n}{2}\rceil;n,p) $ where $ ...
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19 views

using a value from Beta binomial to make draws from binomial

I am having trouble solving this in R. For m draws, draw a value qi from Beta(a,b), and use this to make a single draw from a binomial(n,q). So you will have q1, q2, …, qm different probabilities for ...
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260 views

Statistical Test for Two-Alternative Forced Choice Task (2AFC)

I conducted a learning experiment and assessed whether participants learned with a two-alternative forced choice (2AFC) task consisting of 100 items (25 items of category 1, 25 of category 2, items of ...
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38 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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48 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 ...
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52 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 ...
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120 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 ...
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489 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 ...
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92 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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102 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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623 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: ...
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2answers
761 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
113 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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128 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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200 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
566 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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4k views

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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1answer
946 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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337 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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1k 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 ...
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1answer
584 views

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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225 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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1answer
161 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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805 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 ...
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470 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 ...
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1answer
356 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?
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1answer
4k 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 ...
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411 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 ...
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7k views

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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175 views

Aggregation of propensity scores with varying reliability

When trying to estimate the number of sampling units with an attribute, is there a good algebraic way to aggregate over propensity scores for that attribute which each have their own error? For ...
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2answers
706 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, ...
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1k 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. ...