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Questions tagged [beta-distribution]

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

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Uniform conjugate prior for a Beta distribution

Given $$ \pi \sim \text{Beta}(\alpha, \beta) $$ I'd like to place a prior on $\alpha$ and $\beta$. The "trick" mentioned in this post and this post seems to be to recognize that since $$ \begin{...
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Likert Scale data analysis [closed]

I have a dataset where judges give singers scores on Likert Scale on various attributes. Now we are to find what their ranks should be. Now usually normal distribution is assumed, but my teacher asked ...
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Evaluating model fit for glmmTMB with beta distribution

I am looking for a way to evaluate model fit (e.g. R2, %deviance explained) for the following model: m1<-glmmTMB(AUC_indep~woody+dispersal+log_densitys+log_seedwts+I(log_seedwts^2)+(1|modeltype), ...
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Proving transformations of two independent chi-squared random variables is equivalent to a Beta distribution

I came across the following in some old class notes of mine: if $\chi_{v_{1}}^{2}$ is independent of $\chi_{v_{2}}^{2}$ then $\frac{\chi_{v_{1}}^{2}}{\chi_{v_{1}}^{2}+\chi_{v_{2}}^{2}}\backsim ...
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1answer
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Moment Generating Function of Beta ( Taylor series)

Suppose X is a random variable with a Beta ( a =$\frac{1}{2}$ , b=1) distribution and x in (0,1) Then the moment generating function is calculated as below $ M_X(t) $ = $\mathbb{E}[e^{tX}]$ =$ \...
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Logistic regression - binary to continuous - how to interpret?

Given data with a binary outcome, i.e.: $0$ = no event, $1$ = event which can be modeled with logistic regression, how then do we understand the following logic: ...
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R package for meta-analysis of beta-distributed data

I am conducting a meta-analysis, and my effect sizes are proportions, with a credible interval for weighing. I think I should assume a beta-distribution, but I do not know which package to use. This ...
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Expectation of Sufficient Statistic for Beta Distribution

I am looking at question 1b of the following notes: http://www.gatsby.ucl.ac.uk/teaching/courses/ml1/asst1.pdf In 1a, I have shown that the Beta distribution has a density that can be written in the ...
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Uber's Pyro less accurate than expected on toy example

Trying to understand the pyro example here: https://pyro.ai/examples/svi_part_i.html which starts with a Beta(10,10) prior, adds 10 Bernoulli likelihood datapoints with a 6,4 split. The analytic ...
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MLE for Beta distribution, with $\beta$ = 3

I'm trying to calculate the Maximum-Likelihood Estimator for $\alpha$, using the beta distribution with $\beta = 3$. I'm kind of stuck at the last bit. Perhaps I've made a mistake somewhere, or this ...
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1answer
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If $X\sim\text{Beta}(\theta,1)$, obtain the confidence interval of $100(1-\alpha)\%$ based on the asymptotic distribution of the score function

Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample whose distribution is given by $\text{Beta}(\theta,1)$. Obtain the approximate confidence interval of $100(1-\alpha)\%$ based on the asymptotic ...
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Is fair to compare Dirichlet to a Multivariate Beta regression?

I am conducting some analysis on my data I found a strange behavior and would greatly appreciate some guidance or suggestions. I am trying to investigate the effect of a categorical variable (cl) to ...
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1answer
26 views

Valid highly informative prior for proportion

I am trying to find a prior distribution for a proportion $\theta$ that is highly informative i.e. it is almost point mass at $\theta$ but I am not able to find such distribution that is valid for a ...
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Squared internally studentized residual over $n-p$ is Beta distributed

Assume a regression model $y = X \beta + \varepsilon$ with $n$ observations and $p$ parameters. Let $r_i$ be the $i$-th internally studentized residual: $$r_i = \frac{e_i}{\sqrt{\hat{\sigma} (1 -h_{ii}...
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Model reputation with beta distribution

I am trying to develop a reputation score in which the reputation is calculated by the ratings that buyers give to sellers. What i am trying is to replicate a paper very well known http://faculty.neu....
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1answer
36 views

Efficiently Computing The Beta CDF [duplicate]

I am using numba to JIT compile some looped python functions as part of a larger application. Ideally, everything will run in numba's "no python" mode, such that the loop can be parallelised. One of ...
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1answer
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Understanding the Bayesian question [closed]

I have a Bayesian question here: In a drug experiment, patients with a chronic condition are asked to choose between two drugs, C (control), and T (new treatment). (You may assume for the purpose ...
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What is support of beta distribution? [duplicate]

I do know that the probability density function of beta distribution is $$ \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\displaystyle \mathrm {B}(\alpha,\beta)}\! $$ where $$ {\displaystyle \mathrm {B} (\...
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Can't understand the Beta distribution as described in this paper

I'm reading through the following paper (1). On page 2, section 2.1, there's a description of the selection of a beta distribution used: Since both the choice of s and the strategy for handling the ...
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Distribution of dot products of two random independent unit vectors in $D$ dimensions

Duplicate of the stats stack exchange question here; however, I need some help with some of the steps in the accepted answer. A uniform distribution on the unit sphere $\mathbb{S}^{D-1}$ is ...
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Proportion/Rate data and zero-inflation (two counts)

I have an experimental dataset which makes use of two counts. Using vague terms, we are studying animals as they become behaviourally inactive and then apply a stimulus once an hour. One of the counts ...
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How to relate beta CDF to student-t CDF? [duplicate]

We can relate the student-t and beta distributions as such: If $X$ has a Student's t-distribution with degree of freedom $\nu$ then one can obtain a Beta distribution: $$\frac{\nu}{\nu + X^2} \...
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2answers
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Interpretation very small standardized coefficient beta

In one of my studies I have results similar to the below: β=-0.0007 (95% CI: -0.0009, -0.0002), p=0.01 Since β is so small (but also the Confidence Interval (CI)), is this result still meaningful? ...
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In a manuscript, is an incomplete beta function a useful simplification of a sum, if the argument ends up negative?

I'm working on a manuscript which I intend to submit to a statistics journal. I have a sum which I "simplify" to the regularized incomplete beta function $I_{-t}(a,b)$, where $t$ is a positive real ...
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Can I improve an estimate of a coin-flip probability from a single trial using an imperfect oracle?

I have the following generative model: I have a unknown random variable $S\in[0,1]$ and samples $s_i \sim S$. I do not observe $s_i$ directly, but instead an imperfect oracle $q_i$, which might or ...
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Appropriate statistical test for continuos bounded r.v

I have a number of continuous random variables bounded in the domain [0,b] and I need to perform a statistical test to check whether their average is different from ...
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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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1answer
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Why p-values are not significant even though AIC values improved a lot in model selection using GAM mix modelling and beta regression

Dear StatExchange community, I am studying disease progression in plant leaves and I am trying to estimate differences between a wild-type and a mutant plant. To achieve this I am using the ...
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0answers
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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 =...
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1answer
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Simulate from a mixture of two beta distribution [closed]

I have this distribution : $X \sim 0.75\mathcal{B}e(\alpha_{X_1}=1,\beta_{X_1}=3)+0.25\mathcal{B}e(\alpha_{X_2}=5,\beta_{X_2}=1.75)$ where the density function is given from this function: ...
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1answer
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Mean of a truncated non-standard beta distribution

I have a non-standard beta distribution in the interval [-0.02 , 0.005] (as opposed to [0,1]). I know its mean and variance (and thus α and β). I want to calculate the mean of its truncation to [0 , ...
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2answers
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All positive Beta exponents in a binomial logistic regression

I'm running a regression in which the predictor is a categorical measure of living arrangement (1=two parent household, 2 = single parent, etc). There are 8 total categories in this variable. I'm ...
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Question regarding FamaMcBeth regression

I'm new here and hope you can help me with three questions I'm having in regard to the FamaMcBeth regression methodology. What is the advantage of using firm specific returns instead of an aggregate ...
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Change point detection in beta distribution

If the outcomes of agent is modeled as a Bernoulli distribution with success probability θ ∈ [0, 1]. Where α is number of successful tasks that agent execute, and β is number of unsuccessful tasks ...
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Power Analysis with Noncentral F distribution

I'm looking to replicate a power analysis for binary data using an MRMC analysis (details found here). The data looks like this, where Score is a binary variable. ...
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1answer
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Probability that one random variable using the Beta Distribution being greater than another, bounded intervals

I am doing some practice problems to prepare for my statistics exam, and I just want to know if my reasoning is correct on one problem, and if not, I want to know how I should reason through this. The ...
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Independence of Beta ratios of Gamma variates

If $X= x_1/(x_1+x_2+x_3)$ and $Y= x_2/(x_1+x_2+x_3)$ where $x_1, x_2, x_3$ are independent $\chi^2$-distributed random variables with d.f. $-n_1,n_2, n_3$ respectively. Are $X$ and $Y$ independent? I ...
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1answer
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How to justify using Beta distribution as a prior distribution in the following problem

Let $\theta$ be the proportion of people who are ready to quit smoking within 6 months. Let's say we perform a survey in $2017$ with a $n$ volunteers who ask people this question until they obtain yes ...
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2answers
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Beta distribution and normalization [duplicate]

Here in the 4 pictures in the last answer, is the vertical axe the probability? I.e. it seems to me that it is somewhat unnormalized : it has the value 2 in the 2nd picture and 3 in the 3rd picture. ...
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Self Study - How does $\alpha$ and $\beta$ correspond to mean and variance of a beta distribution? [duplicate]

I am trying to figure out how to provide $\alpha$ and $\beta$ in terms of $\mu$ and $\sigma$ in a beta distribution. $\mu$ is given as $\mu = \frac{\alpha}{\alpha + \beta} $ $\sigma$ is given as $\...
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1answer
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Analysis in a beta simulation

I did a simulation where shape1=1 and shape2=2, for n=10, n=100 and n=1000, the following is: ...
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1answer
59 views

How to model a binary vector by the Beta-Bernoulli distribution in R or Python?

I have a set of data $x=(x_1,x_2,x_3,...)$;e.g:$x=(0.1,0.003,0.78,...)$ which follows a beta distribution. Since $x_i$ in my data $x :x_i ~ Bernoulli(p)$, where $p$ is the prior of average $x$ and ...
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Simulate a 2D-beta distribution: spatial simulations

I'm doing spatial simulations and I would like to simulate a matrix that represent a truncated 2D-bimodal distribution with U shape. In other words, I would expect high density towards the borders of ...
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1answer
330 views

How to pick starting parameters for MASS:fitdist() with the beta distribution?

I have data set of ~700k yes/no events that I want to first aggregate on various features (e.g. group by average), always resulting in a 34 length vector. From there, I want to fit a beta ...
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1answer
148 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 ...
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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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2answers
58 views

How to convert the parameters in a binomial distribution to those in a beta distribution?

I know that the beta distribution is the generalized continuous case of the discrete binomial distribution. Let's say I have a binomial distribution, $B(N,p)$. I would like to know the corresponding $\...
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calculating credible intervals for Bernoulli trials with a beta-distribution

I'm interested in finding credible intervals for Bernoulli trials with a variable probability of success. So, the process (as I understand it) is the probability of success, p, is modeled with a ...
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1answer
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how can this missing observation model be extended to include cases where sigma is a function of other variables?

Richard McElreath's blog entry Algebra and the Missing Oxen describes a simple missing observation model in RStan. At the end of the blog, he says it can be extended easily to cases in which the ...