Questions tagged [beta-distribution]

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

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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 $$\frac{X_1}{X_1+X_2}\perp \!\!\!\perp \frac{X_3}{1-X_1-X_2}~ and$$ $$\...
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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 ...
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Probability of multiple events leading to overall probability [closed]

I am struggling with a rather intuitive problem. Let's give an example of what I am trying to achieve. I have $n$ buckets with $m_{i}$ balls in them. I would like to randomly draw a number of balls ($...
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How can I understand if my Beta Distribution is converging?

I am evaluating a Bayes AB Test on 2 variants, A and B. I then plotted a graph which shows the Probability of B is better than A on a daily basis. My worry comes in on the topic of 'peeking'. Let's ...
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Gamma to Beta: Successes and Failure Proportions?

I have a question after reading this article on the relation between Gamma and Beta distributions and this article on the intuitive understanding of the Beta Distribution. Question As the Gamma ...
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1answer
48 views

How to draw u-shape distribution? [closed]

I want to build a distribution as U-shape, with the x-axis of values between [1,5] continuous, and the y-axis is probability [0,1]. I am thinking of beta distribution of alpha=beta=0.5 but couldn't ...
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43 views

PyMC's treatment of shape versus deterministic data, when a random variable's parameter is vector-valued

I'm working on a problem with PyMC3 that makes me think I need to better understand how it deals with random variables whose parameters are vector-valued. Data description and problem setup I have $...
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Deriving the binomial-beta conjugate model

I think I am correctly deriving the binomial-beta conjugate model, but my answer differs slightly from what's on Wikipedia's page on conjugacy. My solution Assume that $$ X_t \sim \text{Binomial}(m,...
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Beta regression or logistic regression with an offset term?

I have a dependent variable (DV) that is a proportion that is bounded by [0,1). Initially I was considering using a beta regression to model the relationship between this proportion and two other ...
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Beta rectangular distribution

I need to sample data from a beta rectangular distribution. As far as I know, this distribution is a mixture of the beta and uniform distribution. I am using Python and in particular SciPy. There is a ...
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Beta-Binomial regression or Poisson-Gamma model to account for uncertainty in (empricial Bayesian) prior? Explained in simple terms?

I have a dataset of $m$ individuals. For each individual $m$ I have $n_m$ (binomial ) observations with $s_m$ corresponding to the number of 'successes'. I use this data to fit a beta-binomial ...
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would it be possible to pick a different likelihood model achieve the same posterior estimation?

Take the coin flipping example. When we decide to use the Bernoulli distribution to model a coin flip, of course with and without a conjugate prior would make some difference for estimation. Would it ...
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Modeling 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 ...
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Is it possible to express the density of Beta distribution with its mode?

the density of beta distribution can be express in term of mean and variance as $$ f(x;\mu,\phi) = \frac{\Gamma(\phi)}{\Gamma{(\mu\phi)}\Gamma((1-\mu)\phi)}x^{\mu\phi-1}(1-x)^{(1-\mu)\phi - 1}$$ ...
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Infinite discounted sum of betas

Let $0 \leq \gamma < 1$, $X_i \sim \text{Beta}(\alpha, \beta)$, and $$Y \sim \sum_{i = 0}^\infty \gamma^i X_i$$ What is the distribution of $Y$? Does it have a closed form? Can it be sampled ...
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Find the value of $\nu$ so that $n^\nu (1-X_{(n)})$ converges in distribution

Let $X_1, X_2, \cdots$ be iid. If $X_i \sim Beta(1,\beta)$, find the value of $\nu$ so that $n^\nu (1-X_{(n)})$ converges in distribution. My thoughts: Since $X_{(n)} \to 1$ in probability, I was ...
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Beta-like tail bounds for ratios of sums of i.i.d generalized gammas?

I am trying to derive a tail bound for a random variable $Z = \frac{\sum_{i=1}^a X_i}{\sum_{i=1}^a X_i + \sum_{i=1}^b Y_i}$, where $X_i, Y_i$ are i.i.d. $GGamma(1, k, 1+\epsilon)$ random variables ...
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Regression betas of X on Y and Y on X are both less than one? [duplicate]

Intuitively, I can't really wrap my head around this. If I regress y on x and the beta is less that one, shouldn't the beta from a regression of x on y be greater than one. Mathematically, I know the ...
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How to calculate the PDF of the 'difference' between two Beta distributions?

I start with two Beta distributions: $$\mathrm{Beta_A}(p; \alpha_A, \beta_A) = \frac{p^{\alpha_A-1}\,(1-p)^{\beta_A-1}}{\mathrm{B}(\alpha_A, \beta_A)}$$ $$\mathrm{Beta_B}(p; \alpha_B, \beta_B) = \...
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Difference between two Beta distributions - is there an exact joint probability distribution (that avoids having to use MC)?

please excuse dodgy terminology - I have tried to use correct terms so I hope everything makes sense Background I am investigating whether there is a difference in conversion rate between two cells. ...
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Can distributions that are in the exponential family, but not the natural exponential family, be formed as GLM?

The lognormal and beta distributions are in the exponential family but not the natural exponential family. Generalized Linear Models are often advertised as being models for response variables that ...
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Bayesian updating via priors

Currently learning about using the Beta distribution and the Beta-Binomial distribution in Bayesian inference. I am confused regarding the following statement: $f(p | X=k)=\frac{P(X=k|p)f(p)}{P(X=k)}...
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Parameter Estimation for the Generalized Beta Distribution

I am interested to determine if there are explicit formulas for the parameters of the Generalized Beta (GB) distribution. McDonald [1] eludes to this but does not provide solution. I was wondering if ...
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Aggregation with an overlap: Dirichlet distribution

Suppose that we have $$(p_1,p_2,p_3,p_4)\sim Dirichlet(a_1,a_2,a_3,a_4),$$ where $p_4=1-p_1-p_2-p_3.$ When we add random variables for example, $p_1+p_2$ and $p_3+p_4$, the resulting distributions ...
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Comparing two coin tossing experiments using Kullback-Leibler divergence

In a coin-tossing, suppose two people have conducted two separate experiments to find out the probability of head $p_H$ of the same coin. They both are Bayesians and have started from the prior $$...
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Extended regular variation

In extreme-value theory, an important property of distributions in the Frechet/power-law domain of attraction distribution is "regular variation". My question: What is the corresponding property (if ...
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279 views

How reparameterize Beta distribution?

Consider $X \sim N(\mu,\sigma)$; I can reparameterize it by $X = \epsilon\mu + \sigma; \epsilon \sim N(0,I) $ But given Beta distribution $X \sim Beta(\alpha,\beta)$; is there easy way (closed form ...
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Uniqueness of change of variable function

Let $X$ and $Y$ be continuous random variables with probability density function as $p_x(X)$ and $p_y(Y)$. If $X$ and $Y$ are related by an invertible function $f$ as $f(X)=Y$, then using change of ...
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Can adjusted r-squared (or r-squared) be used to compare the strength of independent variables in linear multiple regression?

Relative newbie with quantitative analysis here, so forgive me if the question is naive or ill-specified. I have argued in a manuscript that along with using Beta values in linear multiple regression ...
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truncated predictions for continuous proportion response variable using either beta regression or quasi-likelihood methods

I have tried two approaches for predicting a continuous proportion response variable in both SAS (proc glimmix) and R (glmx): beta regression and a quasi-likelihood approach. I have tried both logit ...
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Issue with proof in Statistical Theory related to Beta and Binomial distributions [duplicate]

Assume $X_n$ is distributed $\text{Beta}(1/n, 1/n)$ and $X$ is distributed as $\text{Binom}(1,1/2)$. Show that $X_n$ converges to $X$ in distribution. I'm having some issue with this question. I ...
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Sum of elements in a Dirichlet random vector conditional on one element is greater than another

I'm working on a research project in machine learning but I was stuck at some point while developing a theory background of the project. Let $(X_1,X_2,X_3,X_4)\sim Dir(\alpha_1,\alpha_2,\alpha_3,\...
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Tail bound of beta distribution when $\alpha$ is sufficiently close to zero while $\beta$ greater than 1

I am interested in finding sharp enough bound for tail probability $\Pr(X\gt t)$ given $X\sim \operatorname{Beta}(a,b)$ when $a$ is very close to 0 while $b$ is fixed value greater than 1. Numeric ...
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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 $...
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Bayesian update for Beta distribution

I'm wondering how to find a posterior of a beta distribution when the "new information" is not an outcome of a binomial trial. Let $p$ be the probability of Head of a (biased) coin toss. As usual in ...
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Incoherent post hoc pairwise Tukey test using beta binomial distribution and betareg()

I am fitting a beta binomial regression to my data, which is proportion data (lifetime reproductive success ratio) ranging [0,1]. I transformed the 0s and 1s following Smithson and Verkuilen 2006 (y.(...
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Outlier Detection via Beta Distribution

Suppose I have a continuous random variable which is bound between 0 and 1. The distribution is left skewed like the picture below: My goal is to identify outliers that are small or farther away from ...
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Beta-distribution: how to generate a peak at certain mean value with a control on variance in extrems

Following Distribution that has a range from 0 to 1 and with peak between them?, I generated a beta distribution that has a peak between 0 and 1 at the mean value. When the mean value is 0.5, I can ...
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Is $f(t,\beta ,\mu , \theta )= e^{\theta(t-\mu)^{\beta}}$ a well known distribution on the unit interval?

$\newcommand{\erf}{\operatorname{erf}}$ I have tried out to define a new distribution lie in [0,1] using my special function which is montioned in this paper, The new special function is defined as: $...
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Are the beta distribution and binomial distribution related? [closed]

I've seen questions like this and this, but it hasn't quite answered my question. How intertwined are the Beta and Binomial distributions? A quick sidenote: the Poisson distribution and the ...
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1answer
73 views

Can I plot a beta distributed model with percentage rather than proportion? [closed]

I am analyzing a response variable (herbivory) which is a proportion, so I am using betareg function from the betareg package. The structure is similar to the GasolineYield dataset, so using it as ...
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1answer
45 views

Geometric distribution with a capped number of trials - finding expectation and prior predictive distribution

So I am modeling a random variable which follows a geometric distribution with probability $\theta$ except that the total number of trials is capped at some value $n$. I.e., the probability mass ...
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How do I use a beta distribution to model this bimodal data?

I am trying to model data that looks like this: The data reflects frequencies of items at 0, 25, 50, 75, and 100. The majority of the time, values are concentrated at the 0, and 100 points. I believe ...
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Deriving conditional probability of bivariate bernoulli by using Dirichlet

While I was working on my research project, I found it difficult to derive a conditional probability from Dirichlet dist. Consider two Bernoulli trials that are possibly correlated with each other. ...
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1answer
132 views

Why does the mean for a beta distribution when input into the scipy beta cdf not return 0.5? [duplicate]

Very likely I am missing some fundamentals here, or my python coding is not up to scratch. When I use the standard formula to calculate the mean for a beta distribution ~ $B(\alpha,\beta)$ $$ \mu = \...
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1answer
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Variance of beta distribution (fastest way)

Suppose a Random variable $X \sim \mathrm{Beta}(a,b)$ Find the $\mathrm{Var}( \frac{X}{1-X} ) $ My initial approach is to calculate $\mathrm{E}( \frac{X}{1-X} ) $ and $\mathrm{E}( [\frac{X}{1-X}]^...
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Bayesian inference for Beta distribution after an uncertain outcome

Normally, when we have $$p\sim Beta(a,b)$$ and an observation of $x=1$ (''success'') from a Bernoulli trial with ''success'' probability $p$, the Bayesian inference on the parameter value $p$ is $$p|x\...
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Is there a equivalence test for beta coefficients in regression analysis?

There are established ways to rule out medium/high effects like TOST for two-groups. But is there a way to rule out medium/high effects in one multiple regression? Maybe using eta-squared? What ...
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Tuning the “strength” of updates to a posterior distribution for conjugates

I'm asking this question as a sanity check- I am not a statistician or research scientist, and just am doing a gut check on a model I am building. I want to quantify uncertainty of a specific metric ...
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Multi-armed bandit where you must pull N pulls in T timesteps

Multi-armed bandit where you must pull N pulls in T timesteps Consider the beta-Bernoulli multi-armed bandit, with the following wrinkle: We have a total of $T$ time steps. We must pull $N$ pulls ...

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