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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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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37 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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105 views

In a Beta-Binomial 'Bayesian' A/B test, is it possible to add a third, fourth, etc. recipe?

For context: How to define prior for beta-binomial A/B test For P(A > B), you can draw samples from A's posterior and B's posterior and then count the number of times the sample from A is greater ...
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94 views

Inferential statistics for vector of percentages

I'm getting confused by this and was wondering if someone can enlighten me: I have a random sample consisting of 50 percentages. Each percentage can take on any value between 0% and 100% inclusive ...
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199 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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132 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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206 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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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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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, ...
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155 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-...
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224 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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209 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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401 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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232 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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140 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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80 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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119 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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136 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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585 views

log-likelihood for a beta-binomial in R

Although I may ask a question that is already solved (but I found none that would explicitly refer to this), I would like to know, if (and I am no statistician) I am right with programming the log-...
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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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70 views

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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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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20 views

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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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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118 views

Intuition of Beta-Binomial Distribution

I am given $$Q \sim\mathrm{Beta} (2,3)$$ $$[X|Q=q]\sim\mathrm{Bin} (3,q)$$ I am asked to find (1) $E[X]$, (2) $\operatorname{Var}[X]$, (3) $E[Q|X]$, (4) $\operatorname{Var}[Q|X]$, (5) $v(q|X=x)$. I ...
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254 views

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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96 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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229 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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15 views

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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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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67 views

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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152 views

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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45 views

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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75 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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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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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-...
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67 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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465 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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107 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 ...