Questions tagged [beta-distribution]

A two-parameter family of univariate distributions defined on the interval $[0,1]$.

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11
votes
2answers
2k views

Scaling the backward variable in HMM Baum-Welch

I am just trying to implement the scaled Baum-Welch algorithm and I have run into a problem where my backward variables, after scaling, are over the value of 1. Is this normal? After all, ...
10
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0answers
2k views

How to generate 2 correlated Beta random variables

I was wondering if it might be possible to generate 2 correlated $Beta$ random variables? In other words, I want to generate two Beta random variables which can be said to have come from two Beta ...
7
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1answer
228 views

Finding the distribution of sample range for a Beta population

Let $X_1,X_2,\ldots,X_n$ be i.i.d random variables having density $$f(x)=2(1-x)\mathbf1_{0<x<1}$$ I am trying to derive the distribution of the sample range $R=X_{(n)}-X_{(1)}$. The usual way I ...
5
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0answers
309 views

Hierarchical model: question on frequentist estimation

I am interested in understanding the differences between Bayesian and Frequentist estimation in the context of hierarchical models. Consider $n$ subjects, where for subject $i$ there are $k_i$ ...
4
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0answers
74 views

Why are $\mathbb{E}( \ln(x))$ and $\mathbb{E} ( \ln(1 - x))$ reasonable descriptions of knowledge about a beta distribution?

The max entropy philosophy states that given some constraints on the prior, we should choose the prior that is maximum entropy subject to those constraints. I know that the Beta($\alpha, \beta$) is ...
4
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0answers
523 views

Modeling a Correlated Bivariate Beta Distributions in PyMC3

My goal is to perform a bayesian A/B test of probabilities of success in two groups considering a hypothesis about non-zero covariance between those probabilities. Bivariate beta distribution I am ...
4
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0answers
89 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
75 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, ...
4
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1answer
398 views

What is the likelihood function of this random variable (beta distribution parameterizing a Bernoulli distribution)?

This is related to an earlier self-study question of mine. The setup is that there are $N$ individuals, indexed by $i$, and two time periods. Individuals choose whether to "invent" something in the ...
4
votes
1answer
474 views

How to fit newer cohorts using Pareto/NBD or Beta/Geo for CLTV

I am trying to fit the popular Pareto/NBD or Beta/Geometric models for non-contractual, continuous customer data. On top of that I then fit the Gamma/Gamma model for monetary value (using the very ...
4
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0answers
1k views

Overall p-value for zero-inflated beta regression mixed model

I am analysing vegetation percentage cover data from grazed and ungrazed plots in R using a zero-inflated beta regression in package gamlss. Here are some example ...
4
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0answers
203 views

Meaning of the Bayesian Cramer-Rao bound for a coin flip

Consider a coin $X\sim\operatorname{Bernoulli}(p)$. Its Fisher information is given by $J=\frac{1}{p(1-p)}$. Now suppose we are in a Bayesian setting and our prior on $p$ is $\pi=\operatorname{Beta}(...
4
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0answers
1k views

Beta Distribution Fitting with constrained location and scale

I'm fitting a set of beta distributions. My data is constrained to live on $[0,1]$, both theoretically and empirically. A typical output of scipy.stats.beta.fit ...
4
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0answers
273 views

Calculate expected value of CDF of a different Beta variable

Let $$ X_1 \sim Beta(\alpha_1,\beta_1) \\ X_2 \sim Beta(\alpha_2,\beta_2). $$ Let $F_X(x) = P( X \le x )$ be the CDF of $X$ and $\mathbb{E}_{X}(\cdot)$ be expectation with respect to $X$. How to ...
4
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0answers
2k views

Weibull Distribution v/s Beta Distribution

I've recently fallen in love with the Weibull Distribution and have gotten a reason to see if there's a mapping of this distribution to an interval (0,1). After plotting the Beta Distribution against ...
4
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0answers
482 views

How to integrate products of beta distributions with large $\alpha$, $\beta$?

I have a product of Beta distributions, like $$\frac{1}{Z}\prod_{i}\frac{\left(x-A_{i}\right)^{\alpha_{i}-1}\left(B_{i}-x\right)^{\beta_{i}-1}}{\left(B_{i}-A_{i}\right)^{\alpha_{i}+\beta_{i}-1}\...
4
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0answers
114 views

Approximate a product of Beta PDFs with another Beta PDF

The PDF of the generalized Beta distribution in the interval $[A,B]$ is defined as: $$f(x) = \frac{(x-A)^{\alpha-1}(B-x)^{\beta-1}}{(B-A)^{\alpha+\beta-1}\mathrm{B}(\alpha,\beta)}$$ for $A<x<B$...
3
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0answers
98 views

Improving fit of underdispered beta regression model in glmmtmb

I have survey data where the outcome is the proportion of a research budget interviewees wished to assign to one of three different "types" of research into solutions for various issues. I ...
3
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0answers
477 views

Fitting a Beta distribution only using coin flips from the biased coins it generates

I have a Beta distribution $D$ with unknown parameters $\alpha$ and $\beta$ which I wish to estimate. If I was given samples $p_1, \ldots, p_n$ from $D$, then it's relatively straightforward to fit $...
3
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0answers
620 views

Evaluating model fit for glmmTMB with beta distribution

I am looking for a way to evaluate model fit (e.g. $R^2$, %deviance explained) for the following model: ...
3
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0answers
126 views

In a bivariate normal sample, why is the squared sample correlation Beta distributed?

If $(X_i,Y_i), i = 1, \dots, n$ are independently bivariate normal distributed, with mean $(\mu_x , \mu_y)$ and variances $(\sigma_x^2, \sigma^2_y)$ and correlation coefficient $\rho = 0$. Denote $T =...
3
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0answers
511 views

Measure of relationship between two variables that are percentages containing many zeros

I am working with various different data sets (in the context of forest reclamation on industrial disturbed landscapes) that contain percent cover values of desired (planted) and undesired plant ...
3
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0answers
51 views

How to find the distribution of the ratio of random vectors having two other known distributions?

I have the following problem. Consider a random vector distributed as a multivariate $t$-distribution, $\mathbf{v} \sim t_\nu(\mathbf{0}, \boldsymbol{\Omega})$. Consider further another random vector $...
3
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0answers
216 views

Weird beta inflated distribution qq plot (easy reproducible code included)

First, I do this library(gamlss) simnorm <- rnorm(1000, 0.5,0.1) simzero <- rep(0,1000) x <- sample(c(simnorm,simzero),1000, replace = TRUE) which ...
3
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0answers
379 views

Calculate parameters of Beta distribution from the mean and variance of the log-transformed variable

If my $X$ follows the Beta distribution with shapes $a,b$, and I only know the mean and variance of $lnX$, is there any way to figure out $a$ and $b$ ? I've read in the wikipedia article about the ...
3
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0answers
117 views

Is the Gaussian distribution the only statistical distribution fully determined by the mean and variance?

I've read that the Gaussian marginal is fully determined by the mean and variance. What does this mean in reality? If we consider a Gaussian marginal PDF is given by $$ \pi_G(\xi|\mu,\sigma) = {1\...
3
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0answers
217 views

Distribution of the ratio of two shifted generalized gamma random variable

$X \sim \mathrm{GG}\left(p,d,\theta_{1},\mu\right)$ where $p$ is power, $d$ is shape, $\theta_1$ is scale and $\mu$ is location parameter. Also Consider $Y \sim \mathrm{GG}\left(p,d,\theta_{2},\mu\...
3
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0answers
293 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
270 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 ...
3
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0answers
1k views

Zero-and-one inflated beta regression vs. binomial GLMM?

I appreciate some help with deciding whether I should (and how to) construct a zero-and-one-inflated beta regression model. I want to use R to test the hypothesis that there is a ...
3
votes
0answers
89 views

Incorporating population priors into MLE fits with few/limited samples

I am fitting Beta distributions to data resulting from each of many experiments using maximum likelihood. My goal is for each experiment, given iid data $y_{1:k}$, fit a Beta distribution, and then ...
3
votes
0answers
4k views

Proportion data: - beta regression or logit transformed OLS regression?

I have proportion data (percentage viewership of TV programs) that i'd like to model as a function of various demographics (age, sex etc.) and time (year). After surveying options for appropriate ...
3
votes
0answers
1k views

What do I do if my predicted values are out of the dependent variable range?

I have built a beta regression model with log link for predicting adherence. My dependent variable's range is 0 to 1.When I used a test set to calculate the predicted values with the parameter ...
2
votes
0answers
57 views

Beta distribution with a priors as Uniform and Pareto Distribution

I am working on a bayesian programming problem which involves a Beta Posterior, which has mean (location) parameter coming from Uniform Distribution [U(0,1)] and concentration (kappa) coming from ...
2
votes
0answers
26 views

Visualizing distribution of sample proportions that takes into account of the sample proportion uncertainties

I have calculated several sample proportions in my dataset that stem from an unknown population distribution - likely a beta mixture. I would like to perform some exploratory analysis to visualize the ...
2
votes
0answers
82 views

Mean and variance of the Beta distribution using identities of exponential families

I was studying the part of exponential families from Statistical Inference (George Casella, Roger L. Berger) and they give the following definition of an exponential family: $$ f(x|\pmb{\theta}) = h(x)...
2
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0answers
52 views

Is there anything wrong with having a Bayesian logistic regression model with beta priors?

The Bayesian logistic regression model with beta priors seem to work using JAGS. I just can't find any examples of it in any literature or any tutorials. They all seem to use normal priors. Just want ...
2
votes
2answers
1k views

DHARMa diagnostics show significant deviations in KS tests for a glmm with beta distribution

I'm trying to use glmmTMB to fit a beta-distributed generalized mixed effects model with nested random effects. DHARMa residual diagnostics show a KS test with significant deviation. Is this serious ...
2
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0answers
70 views

Is a multivariate Beta Regression appropriate for modelling multiple dependent variables where each variable is a percentage?

I have a dataset with multiple predictors, some discrete and some percentages. My response variable is essentially 3 categories: percentage of high, medium and low life satisfaction, or maybe they are ...
2
votes
0answers
412 views

How to report results for generalised linear mixed model (glmmTMB) with beta distribution and logit link?

I am currently analysing the effect of canopy cover on the proportion of birds scavenging on carrion left out in nature. The data comes from several national parks, which I included as a random factor....
2
votes
0answers
24 views

Regression alternatives for proportional data that do not rely on the 0 - 1 interval

I'm currently analyzing data consisting of change in cognitive performance and how it is related with a physiological variable (enzymatic activity) in a group of people. As all people were assessed ...
2
votes
0answers
27 views

Proving independence relationship

Let $X_1,X_2,X_3$ be continuous positive random variables satisfying $X_1+X_2+X_3<1$ and the following independence relations $$\frac{X_1}{X_1+X_2}\perp \!\!\!\perp \frac{X_3}{1-X_1-X_2}~ and$$ $$\...
2
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0answers
364 views

Choosing a ‘noninformative’ hyperprior distribution

I am trying to better understand hierarchical Bayesian models. I started here: https://blog.dominodatalab.com/ab-testing-with-hierarchical-models-in-python/ And ran into the following sentence ...
2
votes
0answers
157 views

Tight bound for Binomial distribution or, equivalently, the Incomplete Beta function?

Suppose $X \sim Binomial(n,p)$ with known $n$ but unknown $p$, and let $G(p,k) = P[X \geq k)$ for $k=0, \ldots, n$. I am looking for a tight upper bound on $|G(p_1, k) - G(p_2, k)|$ for some given $k$....
2
votes
0answers
62 views

Proof help: Coincidences in higher dimensions

Background I recently watched a 2014 Talk by Geoffrey Hinton (a key researcher in Machine Learning literature) where he discusses the concepts behind the recently published Capsule Networks. In the ...
2
votes
0answers
59 views

How could one prove that b in the GB2 distribution is a scale parameter?

In the Generalized Beta distribution of the second kind (GB2), where a, p, and q are shape parameters and b is a scale parameter, the pdf is defined on $\mathbb{R}_+$ by: $$ GB2(y;a,b,p,q) = \frac{|a|...
2
votes
1answer
64 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 $C_i/I_i$ ...
2
votes
0answers
371 views

Sampling parameters from the beta distribution with a given correlation

Let a correlation matrix $\Sigma$ be given. I would like to sample from the n-dimensional multivariate beta distribution where each marginal distribution is known and the variables are correlated as ...
2
votes
0answers
641 views

Envelope distribution for rejection sampling from $(Beta/Beta)*Normal$

I want to obtain random variates from a random variable whose probability density function is $$q(log(\sigma_t)|a_1, b_1, a_2, b_2, \sigma_{t-1},\lambda) = \frac{1}{C}\frac{\mathcal{B}(a_1; \...
2
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0answers
302 views

R_How to select hyperprior distribution for Beta distribution parameter in R?

I have 2-mixture weibull distrubution. And this distribution haver the portion parameter θ whose value should lie between (0,1). Therefore, I am assuming the prior distribution of θθ to be a beta ...