Questions tagged [beta-binomial-distribution]
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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Posterior of binomial and mixed prior
I'm currently studying posterior distribution with likelihood $y|\theta \sim B(n,\theta)$ and mixture of prior distribution $\theta \sim \pi Beta(\alpha_1, \beta_1) + (1-\pi)Beta(\alpha_2, \beta_2)$. ...
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Inference of Beta-Bernoulli Distribution
Assume $x_1, x_2, \cdots, x_n$ follows a $Bern(\pi_0)$, Let $y_{ik}$ follows $Beta(\alpha,\beta)$, $i\in \{1,\cdots, n\}$, and $k\in \{1,\cdots, K\}$. Let $z_k$ follows a Bernoulli Distribution with a ...
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Interpreting predict() with gamlss() and beta-binomial family=BB
Note after answer is posted: The issue here was actually about formatting the data of the dependent variables in the formula when using (beta-)binomial families. Not about ...
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Laplace's law of succession for ordinal variables
this is just a little curiosity of mine, but has anyone heard of some simple model like Laplace's law of succession, however for ordinal RVs instead of a binomial? Specifically, I'm thinking about ...
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Significance of outliers with multiple weighted coins
tl;dr: I have N weighted coins, each of which has been flipped some number of times (n_i), and for which I've measured the ...
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Express infinite serie in terms of a shape parameters of beta distribution
Player A and B participate in a match where the probability that A will win each point is $p$, for B its $1-p$ and a player wins when he reach $11$ points by a margin of $\ge2$
The outcome of the ...
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Variations on Uniform Priors with Binomial Signals?
the standard result is that with a uniform prior on p (from 0 to 1) and binomial signals (h H signals and (n-h) L signals from n draws, each with probability p), the posterior mean is (h+1)/(n+2) and ...
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RJAGS - Bayesian Beta binomial syntax
I am working on a classic problem when the posterior probability distribution of a proportion must be obtained. This parameters is assumed to followed a beta distribution, therefore the number of ...
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For a binary outcome, is the distribution necessarily Bernoulli? Could, for instance, "beta-Bernoulli be in play?
When a variable is binary, it sure seems like its distribution is totally characterized by the probability of being in one group: the variable takes one value with probability $p$ and the other value ...
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Parameterization of the beta-binomial family Glmmtmb
I have trouble understanding the documentation for the glmmtmb package (https://cran.r-project.org/web/packages/glmmTMB/glmmTMB.pdf)
On page 26, in the details on the beta-binomial distribution, it ...
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How to get posterior number of trials using Beta-Binomial model?
Let's say $y$ is known, is there any way to compute the number of trials N such that $P(\theta_N<\theta_0|y)=0.95$ ?
For the sake of the example, let's say #successes $y=0$ and prior probability $\...
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Beta Binomial Distirbution - Updating beta with 0 occurence [closed]
I am dealing with different problem where a count data has to be modelled either with binomial distribution or hypergeometric. I have done a extensive literature read and it seems that occurence equal ...
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Posterior predictive distribution for Bernoulli (and categorical)
I'm trying to confirm something I've tried to figure out about the posterior predictive distribution for Bernoulli vs. Binomial (and categorical vs. multinomial) random variables after a Bayesian ...
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Use of the Beta-Binomial Distribution in Capture-Recapture Sampling
During capture-recapture sampling, we aim to estimate a population size (e.g. of organisms) by capturing a sample of size $ n_1 $, marking them, releasing them, then re-sampling (assuming they have ...
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Can you create Kalman filter (or a recurssive state estimator) with Beta and Binomial distributions?
I have to infer the probability of a system failing from observations. Since probabilities are bounded between 0 and 1, they are sometimes modeled using Beta distribution. While the traditional Kalman ...
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Multiple testing adjustments for Bayes Factors
I manage an platform that has succesfully gotten a Bayesian approach to experimentation in production.
One new feature we want to implement is to do inference for multiple metrics (e.g. conversion ...
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Property of two independent Beta distribution
I have been working with beta-bernoulli posteriors recently. Is it true that if $X,Y$ are independent rvs with $X \sim Beta(a_1+1,b_1+1)$ and $Y \sim Beta(a_2+1,b_2+1)$ then $\mathbb{P}(X>Y)>0.5$...
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Overdispersion in logistic regression --- Use beta-binomial?
I have some cell counts obtained via flow cytometry - simply put, I have the amount of positive cells (Successes) from the overall number of cells (Successes + Failures). Based on the data structure, ...
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Power and sample size calculation for hypothesis test on a beta-binomial
I came across this page about choosing a conjugate prior for a hypergeometric distribution. I noticed that one of the answers described a beta-binomial as the posterior distribution posterior ...
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How do I understand the intuition behind percentile point function?
I'm really trying very hard to understand the intuition behind percentile point function. From wikipedia, It is the inverse of cdf.
From the CDF, a point on its curve indicates the percentage of ...
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Length of Clopper-Pearson CI reaches maximum when $x=N/2$?
Suppose we perform $N$ trials and find $x$ successes, then the $1-\alpha$ Clopper-Pearson CI is $[B(\frac{\alpha}{2};x,N-x+1),B(1-\frac{\alpha}{2};x+1,N-x)]\ (0<\alpha<1)$, and $B(z;a,b)$ is the ...
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Estimating variance of success probability in Poisson-binomial distribution
I am looking at a very large yet finite sequence of Bernoulli trials, each with its own probability. From the physical nature of the process, I know that the probabilities $p_i$ of each trial should ...
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Reasonable to incorporate sample size into beta-binomial?
Setup:
The relationship between the beta and binomial distributions is well known.
$$\frac{\pi^{\alpha - 1} (1 - \pi)^{\beta - 1}}{B(\alpha, \beta)} \leftrightarrow {{n}\choose{x}}\pi^{x} (1 - \pi)^{n-...
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Can I use a Prior with Simulated data?
I have a prior about some proportion that follows a Beta distribution. Unfortunately, I do not have (yet) observed data but I was offered a thousand simulated datasets.
Each dataset comes from ...
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Beta-Binomial Gibbs Sampler
I am self-studying Bayesian statistics from the book Computational Bayesian Statistics by Turkman et al, but I am stuck on Problem 6.3 from the book:
Suppose we want to consider a Binomial (unknown $\...
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Am I wanting to perform a random walk?
I am trying to determine the best method to obtain a probability distribution that describes the likelihood of flipping only heads for a number of successive coin tosses with a coin that has an ...
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Histogram of the MLE of the probability in binomial distribution and the plot of beta distributions
I have data with columns "y" and "n", which for this example can be "y" count of heads out of "n" coin flips. There are "i" rows ie "i" ...
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help computing the beta likelihood when we only observe the number of successes and failures (not the latent probability of success)
Imagine we have a biased coin that generates heads with unknown probability $\theta$ where $\theta$ is drawn from a beta distribution with known parameters $(\alpha, \beta)$.
Next imagine that we flip ...
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Betabinomial (BB) regression in the gamlss package (R)
I'd be grateful for any help with beta-binomial (BB) regression in the gamlss package: predictions
from the fitted model of "mu" values for each observation in my data seem reasonable. But ...
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Change shape parameters in a beta distribution based in each datapoint [closed]
I am new to Bayesian statistics and I have been trying to implement a Beta Binomial model from a PhD thesis in rjags. The thesis describes prior distribution for the variables but I am stuck in how to ...
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Comparing performance of Quasi-binomial model and Beta-binomial model
I read some books in biostatistics about fitting binary date with Beta-Binomial regression model and Quasi-Binomial regression model. It proposes a setting:
Setting:
Assuming we have a sequence of ...
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Marginal Distribution in Beta-Binomial Model with Overdispersion Parameters
Problem Setting:
We assume there is a sequence of binomial trials of size $N_i$, $Y_i$ is the number of events of interest, $x_i$ is the predictor associated with trial $i$, and $\pi_i$ is the ...
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Inconsistent posterior estimates in Beta-Binomial likelihood vs Binomial in Bayesian, multilevel models?
In this Google Colab, I've simulated Binomial count data and compared the performance of Binomial-likelihood and Beta-Binomial-likelihood models. Both models have the same Beta prior on theta, the ...
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Beta-Binomial mixture vs Beta-Binomial multilevel model?
I first read about the Beta PDF in the context that it was conjugate to the Binomial distribution; a Beta prior with a Binomial likelihood returns a Beta posterior. So this sounds to me like a ...
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Conditional beta posterior for uniform priors
I have three different random variables $\theta_1, \theta_2, \theta_3$ . These random variables are actually parameters of binomial likelihood
Assume that I have prior distribution of $\theta_2 \sim ...
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Calculate the posterior probability in two groups, Pr(p1>p2 | Data)?
Assume the prior distribution of p1 and p2:
p1~beta(1,1)
p2~beta(2,3)
Assume the data in group1 and group2 follows bernouli distribution:
y1~Binom(10,0.3)
y2~Binom(10,0.6)
How can I calculate the ...
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Calculating ICC for a beta-binomial GLMM
I understand that ICC in binomial GLMMs with a logit link can be calculated via R, where the residual deviance is (pi ^ 2) / 3. However, this is assuming that the ...
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Beta distribution for uncertain binary trials
I have a larger problem but have presented what I believe is a minimal example.
Imagine that you are trying to determine the true probability of a potentially-biased coin landing on heads, and want to ...
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Can I use posterior beta parameters from a previous experiment to use as priors for my current experiment?
I am doing a Bayesian comparison between two proportions, H0 being Proportion(Protein)> Proportion(Mixed). Here the Proportion is of no. of times a free-ranging dog(s) ate from a box(Protein, Mixed)...
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How does Pfizer justify it's Beta-Binomial model? [closed]
This is a bit of a hodge podge of different clinical trial related questions, but starting with some basics. It seems like they're defining vaccine efficacy differently depending on where you look. At ...
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Beta-binomial vs updating a prior beta distribution
Bear with me, as I've just recently been learning about conjugate priors, prior and posterior distributions, and such material. My understanding of the beta-binomial distribution is that it basically ...
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How to estimate the effects of vaccines with Beta- Bernoulli inference
The original Tutorial comes from toward data science
Given the following description:
"Moderna: The vaccine is being tested in 30,000 people. Half received two doses of the vaccine, and half ...
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Beta-Binomial parameter estimation
The MLE or method of moments estimation of parameters of a beta-binomial distribution makes use of (c, y) -- total number and positive counts. However, if we only have one such pair, then $\frac{\...
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How to estimate beta distribution parameters using a beta binomial with empirical bayes
I would like to estimate parameters for a beta distribution using a maximum likelihood approach in python (as mentioned here). I can do this for a beta:
...
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Discrete Probability Density that is Monotonically Decreasing as K Increases and is 0 at K=N+1
My knowledge of distributions is limited, so I apologize beforehand for what may be a silly question.
I am looking for a discrete probability distribution with domain $\{1,2,...,N\}$ that satisfies ...
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Calculating the tail bounds for a beta-binomial regression
I have a beta-binomial regression model that depends on a probability $p$ and a given over-dispersion $\beta$ and is used to parametrise the distribution of $Y$ in the following way
$$ Y(x) \sim ...
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Compound beta-binomial and beta distribution
I have a process that is modelled by a beta-binomial, parametrised by mean $\mu$ and correlation $\rho = 1/(\alpha+\beta+1)$ (as per dbetabinom in the R VGAM package). I know $\rho$, but the mean $\...
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Selecting between a zero-inflated binomial, OLRE and beta-binomial model
I need some help in deciding which of the following models fits best the data that I have.
This was a survey where participants reported proportions of successes (defined as n/m) in condition A and B. ...
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Significant dispersion test
I used DHARMa for my residual diagnostics. For two models, the dispersion test is significant even though the rest of the diagnostic output looks good. I am wondering if both my models are correct ...
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Discrepancy between binomial and beta in R?
I'm getting a result I cannot explain when using beta distribution.
I've got a result which came from a binomial distribution: 2 successes in 6 trials.
I would think the maximum likelihood estimator ...
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