Questions tagged [beta-distribution]

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

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Source for KL-divergence of Beta distribution?

This post explains how to derive the Kullback-Leibler divergence between two beta distributions. https://math.stackexchange.com/questions/257821/kullback-liebler-divergence#comment564291_257821 I ...
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How would we set the initial parameters of beta distribution of Thompson sampling if we want to start the model with the existing data?

This was one of the business-related questions from my technical interview last week for a data science position in a recommender system team at a search engine company focusing on advertisement ...
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Beta distribution equivalence with two redondant parameters [duplicate]

context In Factor graphs on discrete variables, the parameters are contained in factors associated each with a subset of the random variables in the system. Each factor provides a different positive ...
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How can one confidently choose a p and q for a beta distribution?

Is there a methodology or R tool that allows you to best attribute a p and q value for a Beta distribution? I am currently building a simulation that involves stochastic processes, and from the little ...
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Bayesian Test in hypothesis testing with Bernoulli random variables

Let $X_1,\ldots , X_n$ be iid Bernoulli$(p)$ $(0<p<1)$. Our sample is $x_1=\ldots x_n=1$. Suppose that the prior distribution of $p$ is Uniform$(0,1)$. Consider a Bayesian test for $H_0: p\geq 0....
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Beta coefficient related to percent change?

I am trying to determine the factors that impact changes in bat activity across the midwest. I am using package MuMIn to rank the following factors: habitat, park, years since white-nose syndrome ...
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Mixed Model with beta response and glmmTMB

I conducted an experiment in which I am trying to model the relationship between my response weed_coverage [%] and the predictors soil moisture [%] + treatment + distance. Weed_coverage and ...
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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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Can beta regression be used with continuous numerical data?

I am trying to fit a generalized linear model with Gamma distribution in R, but when I examine the residuals they are not normally distributed. I have a continuous numerical response variable, ...
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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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How to derive the distribution of 1 - X [duplicate]

Suppose X is a random variable from the Beta(0.5, 1) density. I want to derive the distribution of Y = 1 − X My attempt: $F_{1-X}(\alpha)= P(1-X \leq \alpha) = P(1-\alpha \leq X) = P(X\geq 1-\alpha) =...
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Is it appropriate to approximate an unknown probability distribution with a beta distribution?

I am creating a model that utilize the output probability distribution of another model, as an input. The model outputs a probability of an event occurring per year, the distribution is a probability ...
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How to derive the distribution of 1 - X if X is Beta(0.5, 1)

I've been reading about uniform distributions and I'm wondering how statisticians derive these distributions. Lets say we have X, a random variable from the Beta(0.5, 1) density. How could you derive ...
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How can you calculate the confidence interval for a parameter that's beta distributed when you only have the mean and the standard error? [closed]

i have the mean and the standard error of a parameter that has a beta distribution but unsure how to calculate the 95% confidence interval. Thank you!
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How to simulate a calibrated prediction model given prevalence and auc

I want to be able to simulate a prediction model given some prevalence of the event and the AUC of the model. I followed the method proposed here but, although this works for giving AUC and predicted ...
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How to improve the fit of a beta zero-inflated regression model (GAMLSS)?

I'm working with a response variable with values between 0.0 and 1.0. I have a lot of zero. Thus, I'm using beta zero-inflated regression model. Specifically, I'm using the function gamlss from the ...
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How to prove $Y=X^{2}$ is Beta$\left(0.5,\ 1\right)$ if $X$ is Uniform$(0,\ 1)$

I've been reading about uniform distributions but I can't see how $Y=X^{2}$ is Beta$\left(0.5,\ 1\right)$ if $X$ is Uniform$(0,\ 1)$. Is there a way to prove this using the cumulative distribution ...
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How to use the exponential distribution to generate samples from the chi-square and beta distributions?

I am supposed to use rexp() in R to draw from an exponential distribution with mean 1, and then use those draws to generate 1000 draws from each of the following: ...
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Beta distributed transition probability in CEA

I am reading a paper on cost-effective analysis and trying to replicate their results. (paper: https://ascopubs.org/doi/full/10.1200/GO.20.00288) The probabilistic model in the paper assumes that the ...
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Using Beta Distribution for probabilities over varying time periods

I'm running into an issue with my math not working out as expected and wondering what in my approach is causing it. I'm doing probabilistic analysis using the Beta distribution, with alpha being the &...
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Prediction of a variable that lies within the interval $[0,1]$ with masses at the ends

I have a data set on kilometers travelled by households and the associated means of transport and now want to predict a means of transport's share in households' total kilometres travelled based on ...
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How do I write out the components of my beta regression, and how do I interpret them afterwards?

I am estimating a beta-regression using the logit-link function with a continuous dependent variable bound between 0 and 1 and 5 continuous predictor variables. In R the equation is given by: betareg(...
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Creating a generalized linear model on percent/proportion data that is non-normal, heteroscedastic, and arguably zero-inflated

My objective here is to evaluate how pre-germination treatments (Pre.G) and light treatments (Light) affect percent germination (...
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What distribution would make a good hyper-prior for a Beta distribution parameterized by mean and sample size?

I have a model which includes a Beta distribution and I am looking for guidance on how to parameterize a hyper-prior for it. For example, this post uses a Beta parameterized with a mean and ...
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Replace parameter with estimate for confidence interval. Case Beta distribution

I'm trying to get a confidence interval for the mean of a beta distribution $B(\theta,1)$, using $[\hat\theta - z_{1-\alpha/2}\hat\sigma_{\hat\theta};\hat\theta + z_{1-\alpha/2}\hat\sigma_{\hat\theta}]...
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Simulating Correlated Continuous Variable Given 2 Existing Binary Variables

I am looking to draw samples from a Beta distribution (let's say α = 3 and β = 2) conditional on two existing binary variables in a correlated manner. Let's call the variable distributed as a Beta as <...
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Is there a "beta distribution" over the entire real line?

The malleability of the beta distribution pleases me; it can be symmetric, asymmetric, platykurtic and so on, as the following picture shows us: I thought it would be interesting to use it to model ...
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Distribution of the exponential of an exponentially distributed random variable?

Let $X$ be an exponentially distributed random variable, that is, with density function $f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$ ($\lambda>0$), and cdf $F_X(x)=1 - e^{-\lambda x}$. What is the ...
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Expectance of predictive posterior on binomial-beta modeling

I'm struggle with figuring out how to prove: $E(\tilde{x}|x)=n\frac{a1}{a1+b1}$ where $x|\theta \sim Bin(n,\theta)$, $\theta \sim Beta(\alpha,\beta)$ and so $\theta|x\sim Beta(a_1=\sum{x_i}+\alpha,a_2=...
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Optimal number of components in a beta mixture model [duplicate]

This is a well-written blog on how we can fit a mixture of beta distributions to a dataset: http://varianceexplained.org/r/mixture-models-baseball/ However, it would have been excellent to identify ...
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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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Application of spike and slab for sampling from posterior distribution (bernoulli and beta)

I think the gamma N term in the first equation relates to the spike and prior. However, I am unsure what the rhs of the first is used for? Further, I am unsure what the pie term of the second equation ...
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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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Bayesian Discrete Survival Analysis

I've been reading over Bruce Hardie's tutorial on survival analysis of customers in a discrete-time subscription model. Two design choices he delineates early on include the Geometric PMF: $P(T=t|\...
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Beta distributed dependant variable and GLM logistic link function [duplicate]

I would like to explain a continuous probability, beta distributed (U shaped) and continuous on [0,1], through a glm model with a logistic link function. Are there any concerns I should take care of ?
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Find beta from gamma distribution by inspection

I'm doing exercises in order to prepare for an exam, and have the following question: If Y has a probability density function given by f(y) = 4$y^2$$e^-$$^2$$^y$, y > 0 and 0 elsewhere. obtain E(...
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Distribution of argmax of beta-distributed random variables

Let $x_i \sim \text{Beta}(\alpha_i, \beta_i)$ for $i \in I$. Let $j = \operatorname*{argmax}_{i \in I} x_i$ (ties broken arbitrarily). What is the distribution of $j$ in terms of $\alpha$ and $\beta$? ...
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Expected Beta distribution within a group

I have a group a baseball players and I have to calculate the expected beta distribution of each player. Groups Group 1: Hits: 30, Misses: 471 -> $Beta_1(30, 471)$ = a/(a+b) = 5.99% Group 2: Hits: ...
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Copula Probability default n >= 2 case

I have been reading about Copula distributions and how they can be used to find probability of default for correlated assets. Let's say I have two assets $J$ that both have a marginal distribution of $...
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Obtain Bayes estimator with conjugate prior

Consider n observations $ X_1, X_2,....X_n $ from $ Beta_1 ~ B(1,\theta ) $ distribution. Obtain Bayes estimator for $ \theta $ under quadratic loss function when conjugate prior is assumed for $\...
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Simulating data from a beta distribution [duplicate]

I want to simulate from a beta distribution which has the following properties: The mode (peak) of the distribution is closer to 0.6. The first quartile and the third quartile should be 0.3 and 0.9 ...
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How is a convex combination of Dirichlet-distributed variables distributed?

Let $X = (X_1, \dots, X_K) \sim \operatorname{Dir}(\alpha_1, \dots, \alpha_K)$ and define the convex combination $Y = \sum_{i=1}^{K} c_i X_i$. In the case of $K=2$, the constraint $\sum_{i=1}^{K} X_i =...
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Reverse the max-min normalization of Beta/Gamma-distributed data

I normalized my data using max-min normalization as follows $$X_{normed}=\frac{X-\min(X)}{\max(X)-\min(X)}$$ How can I find the distribution of $X$ given the distribution of $X_{normed}$, if $X_{...
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How to model Beta distribution with Uniform prior in RJAGS? [closed]

I'm having trouble modeling a Bayesian problem in RJAGS. I'm analyzing depth damage curves. In 1988, the US Army Corps of Engineers estimated that 1 foot of flooding would result in 32% mean damage ...
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Interpretation of zero-inflated beta regression

I have constructed a zero-inflated beta regression model in gamlss. I find the output of this model somewhat confusing to interpret and was hoping that someone may ...
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Bayesian (continuous) logistic regression model with Beta likelihood?

I have a problem where my target variable are continuous/float values in the range [0,1]. If my data were integers in {0,1} this would be a simple logistic regression / Bernoulli likelihood problem. ...
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Given a mean, what is the range of variance values that make for possible Beta distribution parameters

The beta distribution can have its parameter estimated via method of moments, which I will be doing. $$\hat\alpha = \bigg(\dfrac{\bar x (1-\bar x)}{var(X)} - 1\bigg)\bar{x}\\ \hat\beta = \bigg(\dfrac{\...
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Transforming F-dist or long one tail

I have been going through different kaggle datasets recently to apply different techniques I have learned. I have seen loads of articles related to preprocessing your features with normalization or ...
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Finding method of moments estimate for density function $f(x|\alpha) = \frac {\Gamma(2\alpha)} {\Gamma(\alpha)^2}[x(1-x)]^{\alpha - 1}$

Suppose that $X_1, X_2, ..., X_n$ are i.i.d random variables on the interval $[0,1]$ with the density function $$ f(x|\alpha) = \frac {\Gamma(2\alpha)} {\Gamma(\alpha)^2}[x(1-x)]^{\alpha - 1} $$ where ...
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