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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24 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 ...
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18 views
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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20 views
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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20 views
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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1answer
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 $...
2
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1answer
14 views
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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20 views
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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16 views
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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1answer
74 views
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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21 views
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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227 views
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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1answer
32 views
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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1answer
96 views
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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1answer
64 views
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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11 views
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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28 views
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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48 views
How to calculate the PDF of the 'difference' between two Beta distributions?
I start with two Beta distributions:
$$\mathrm{Beta_A}(p; \alpha_A, \beta_A) = \frac{p^{\alpha_A-1}\,(1-p)^{\beta_A-1}}{\mathrm{B}(\alpha_A, \beta_A)}$$
$$\mathrm{Beta_B}(p; \alpha_B, \beta_B) = \...
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21 views
Difference between two Beta distributions - is there an exact joint probability distribution (that avoids having to use MC)?
please excuse dodgy terminology - I have tried to use correct terms so I hope everything makes sense
Background
I am investigating whether there is a difference in conversion rate between two cells. ...
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1answer
68 views
Can distributions that are in the exponential family, but not the natural exponential family, be formed as GLM?
The lognormal and beta distributions are in the exponential family but not the natural exponential family.
Generalized Linear Models are often advertised as being models for response variables that ...
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20 views
Bayesian updating via priors
Currently learning about using the Beta distribution and the Beta-Binomial distribution in Bayesian inference. I am confused regarding the following statement:
$f(p | X=k)=\frac{P(X=k|p)f(p)}{P(X=k)}...
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Parameter Estimation for the Generalized Beta Distribution
I am interested to determine if there are explicit formulas for the parameters of the Generalized Beta (GB) distribution. McDonald [1] eludes to this but does not provide solution. I was wondering if ...
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1answer
47 views
Aggregation with an overlap: Dirichlet distribution
Suppose that we have $$(p_1,p_2,p_3,p_4)\sim Dirichlet(a_1,a_2,a_3,a_4),$$
where $p_4=1-p_1-p_2-p_3.$
When we add random variables for example, $p_1+p_2$ and $p_3+p_4$, the resulting distributions ...
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13 views
Comparing two coin tossing experiments using Kullback-Leibler divergence
In a coin-tossing, suppose two people have conducted two separate experiments to find out the probability of head $p_H$ of the same coin.
They both are Bayesians and have started from the prior $$...
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19 views
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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2answers
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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28 views
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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1answer
65 views
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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23 views
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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19 views
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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2answers
90 views
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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99 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 $...
2
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2answers
110 views
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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1answer
76 views
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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65 views
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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28 views
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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2answers
173 views
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:
$...
2
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0answers
26 views
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 ...
2
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2answers
80 views
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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0answers
53 views
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 ~ $B(\alpha,\beta)$
$$ \mu = \...
3
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1answer
96 views
Variance of beta distribution (fastest way)
Suppose a Random variable $X \sim \mathrm{Beta}(a,b)$
Find the $\mathrm{Var}( \frac{X}{1-X} ) $
My initial approach is to calculate $\mathrm{E}( \frac{X}{1-X} ) $ and $\mathrm{E}( [\frac{X}{1-X}]^...
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1answer
71 views
Bayesian inference for Beta distribution after an uncertain outcome
Normally, when we have $$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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1answer
32 views
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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0answers
9 views
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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66 views
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 ...
