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

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Create a threshold for binary classification problem based on distribution of criterion

I created a criterion for my financial data-set for classify data to two classes for other processing (like neural network binary classificatio). After calculating ...
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3answers
377 views

When is beta distribution bell-shaped or concave?

Is there some restriction to parameters $( \alpha , \beta)$ that makes the beta distribution concave down? Bell-shaped like e.g. a normal? For example, the cases in purple and black, but not the red ...
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65 views

Interpretation of Zero-One inflated Beta Regression with R (GAMLSS)

I am not that familiar with the interpretation of a beta regression with the r-package GAMLSS. Papers and package manuals didn`t help me. I modeled a Zero-One inflated Beta Regression. The ...
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1answer
63 views

Determining beta distribution parameters $\alpha$ and $\beta$ from two arbitrary points

Suppose I have two points $(p_1,x_1)$ and $(p_2,x_2)$ where $p_i$ is a probability on the beta CDF and $x_i$ is a value on that same CDF. How would I go about determining the beta distribution shape ...
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1answer
30 views

PDF not matching histogram of synthetic ratios of independent beta

The PDF of the ratios of independent beta variables is described in http://www.tandfonline.com/doi/abs/10.1080/03610920008832632#.U9J02vldUcC To explore the implications, i created an implementation ...
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1answer
66 views

sum and product rules of probability

I am reading Bishop's Pattern Recognition and Machine Learning. In page 73, chapter 2.1. I can't understand the formula 2.19 : $$p(x=1|\mathcal{D})=\int_0^1 p(x=1|\mu)p(\mu|\mathcal{D})\text{d}\mu ...
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1answer
35 views

Sum of Beta-Bernoulli variables

Assume you have $x_i \sim \operatorname{Bernoulli}(p_i)$ with $p_i \sim \operatorname{Beta}(\alpha,\beta)$. I am exploring $Z=X_1+ \dots +X_n$ According to this page, it is $Z \sim ...
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1answer
261 views

Is the beta distribution really better than the normal distribution for testing the difference of two proportions?

I'm working at an online agency, where we run a lot of AB testing in order to test differences in proportion between two groups (test vs. control). Standard practice in the industry for testing ...
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55 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 ...
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1answer
45 views

$\alpha,\beta\ge 1$ in a Beta distribution. What does it imply for the mean and variance?

The Beta distribution has the PDF: $$f\left(x\right)=\frac{x^{\alpha-1}\left(1-x\right)^{\beta-1}}{\mathrm{B}\left(\alpha,\beta\right)}$$ for $0<x<1$, and $f(x)=0$ otherwise. The parameters ...
5
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1answer
95 views

Product of Gamma by Beta rv

If $X$ has a beta distribution $ \beta(\alpha,b)$, $Y$ has a gamma distribution $\Gamma (K,\theta)$ and $X$ is independent of $Y$. What is the distribution of the product $P=XY$ . Thanks!
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41 views

Drawing a random Sample on a Probability Distribution

Say I have multiple normal or beta distributions. So I have two questions. How do I Draw a random sample from a distribution? How do I compare determine which of my distributions has the largest ...
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1answer
61 views

Find parameters $\alpha$ and $\beta$ of a beta distribution, if I have one quantile and the mean [duplicate]

Suppose I'm given the mean and one quantile (e.g. the 95% quantile) of a random variable $x$, and I want to find the parameters $\alpha$ and $\beta$ of a Beta distribution that has the same mean and ...
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0answers
24 views

modeling error when estimating the parameters of the beta distribution

After reading Basketball Beta and Bayes, I started to think about parameter estimation with observational error. If you are counting the number of successful and failed free-throws how can you ...
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0answers
27 views

Issue with Beta distribution

Question 1: When we merge two beta distribution, how do we arrive at the posterior parameter values? Which is the right formula and why? In one of my earlier queries, Henry mentioned Option 2 as ...
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0answers
65 views

Appropriate GLM when response variable is proportion, but not binomial

The response variable I'm dealing with is the proportion of a total area that is suitable habitat for a species of interest. So although the response variable is bounded between 0 and 1, my intuition ...
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2answers
42 views

Bayes factor for selecting between two beta-distributions

I have two beta-distributions: $H_1 = Beta(\alpha_1, \beta_1) $ and $H_2 = Beta(\alpha_2, \beta_2) $ (parameters are known), and I'd like to estimate whether a new sample $D$ rather comes from ...
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47 views

Outlier detection in beta distributions

Say I have a large sample of values in $[0,1]$. I would like to estimate the underlying $\text{Beta}(\alpha, \beta)$ distribution. The majority of the samples come from this assumed ...
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2answers
589 views

Why is this distribution uniform?

We are investigating Bayesian statistical testing, and come across an odd (to me atleast) phenomenon. Consider the following case: we are interested in measuring which population, A or B, has a ...
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1answer
51 views

Non-binomial posteriors for a binomial prior?

Let's assume we have a discrete binary random variable K (K=0 or K=1) for which the prior distribution is binomial. My understanding of Bayesian statistics tells me that regardless of the likelihood, ...
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1answer
66 views

Beta function approximation of delta function

I have modified the original question. Does beta distribution function $$f(x,\alpha) = \frac{[x^a(1-x)^b]^\alpha}{B(a\alpha+1,b\alpha+1)}$$ where $B$ is the beta function, approach delta function ...
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1answer
51 views

Univariate priors for the parameters of a Beta distribution

I need a rather a prior on the parameters of a Beta distribution (i.e. $\alpha$ and $\beta$). I have an external constraint that requires me to use univariate priors, one for $\alpha$ and one ...
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1answer
121 views

Remove effect of a factor on continuous proportion data using regression in R

I have a data set of continuous proportions which depend on a fixed-effect factor, e.g.: ...
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0answers
35 views

Mode of the difference of two beta distributions

Given two beta distributions $X \sim \beta(m_1, n_1)$ and $Y \sim \beta(m_2, n_2)$, how to compute the mode of the distribution of $Z = X - Y$?
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1answer
247 views

Expected value of a random variable

Random variable $X$ has the probability density function \begin{equation*} f\left( x\right) =\left\{ \begin{array}{ccc} n\left( \frac{x}{\theta }\right) ^{n-1} & , & 0<x\leqslant \theta ...
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36 views

Approximation for linear combination of truncated beta variables

Let $x_1, x_2$ be random variables that follow truncated beta distributions: ...
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20 views

How to measure whether the YES/NO (Bernoulli trials) answers have converged about their estimated mean for a specified error bound?

I have a questionnaire with questions that require a Yes or a No answer (1 or 0 input) from participants. From the set of answers I want to use the mean, $\mu$, estimated from the data for the ...
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0answers
40 views

estimating product of 2 independent variables

I have 2 independent variables with distributions: $X \sim \text{beta}(\alpha_1,\beta_1)$ and $Y \sim \text{beta}(\alpha_2,\beta_2)$ I would like to estimate $P(X * Y >= \text{val})$, and I am ...
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2answers
60 views

What is the distribution of a sum of a subset of probabilities, with each probability having the same distribution?

Suppose I have $k$ outcomes with probabilities, $p_i$, with $p_1+p_2+\dots+p_k=1$. Each probability has the same distribution. What would the distribution of a sum of probabilities be? For example, ...
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0answers
40 views

Hypothesis testing for cost-per-click metrics

I have two ads, and I want to figure out which is better. If they both have identical cost, my null hypothesis is that the probability of success in the first is equal to the second, and I can use ...
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3answers
83 views

What probability distribution is to the discrete uniform distribution as the beta distribution is to uniform distribution over $[0,1]$?

A beta distribution with its parameters $\alpha = \beta = 1$ is the uniform $[0, 1]$ distribution. What distribution is to the discrete uniform distribution (the sample space is left undecided), as ...
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176 views

Distribution of the ratio of dependent chi-square random variables

Assume that $ X = X_1 + X_2,...+ X_n $, where $\; X_i \sim N(0,\sigma^2)$ and independent. My question is, what distribution $$ Z = \frac{X^2}{X_1^2 + X_2^2 + ... + X_n^2} \;\;\; (1)$$ follows. I ...
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85 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, ...
2
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2answers
77 views

Beta distribution on discrete data

Suppose that my data $y \in \{0,0.1,\ldots,1\}$. What are the consequences of modeling that data as continuous, i.e., as if $y \in [0,1]$, by using the beta distribution? Is there a version of the ...
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0answers
73 views

Multivariate Beta distribution (no Dirichlet!)

What is a multidimensional generalization of the Beta distribution, in compliance with the following specification? I am not looking for the Dirichlet distribution. I am looking for a generalization ...
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0answers
53 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 $k_i$ ...
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1answer
385 views

Since the beta distribution is similar in form to the binomial, why do we need the beta distribution?

It appears that the binomial distribution is very similar in form to the beta distribution and that I can re-parametrize constants on either pdf to make them look the same. So, why do we need the beta ...
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165 views

Sufficient Statistic for Beta distribution

How can I show that the geometric mean $( \prod_{i=1}^{n} X_i )^{1/n}$ of a random sample of size $n$ from a distribution with pdf $f(x;\theta)=\theta x^{\theta-1},0<x<1,$, zero elsewhere, and ...
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1answer
98 views

The Conjugate Beta Prior proof

Hello. I'm having a problem with trying to figure out this proof that shows the beta distribution is conjugate to the binomial distribution (picture attached). I understand it until the third row, ...
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2answers
2k views

Calculate the confidence interval for the mean of a beta distribution

Consider a beta distribution for a given set of ratings in [0,1]. After having calculated the mean: $$ \mu = \frac{\alpha}{\alpha+\beta} $$ Is there a way to provide a confidence interval around ...
2
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1answer
43 views

Probability distribution satisfying constraints?

Reposted from Math.SE: Continuing from this question. Given two random variables $X$ and $Y$ where $X \sim \operatorname{Beta}(a, b)$ and $Y \sim \operatorname{Beta}(c, d)$, I'm looking for a random ...
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96 views

Computing Confidence in A/B testing

First of all, I'm fairly new to Decision Theory and my question might be rather too simple for the experts. I am having some issues in computing confidence for A/B Testing. I will first explain the ...
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1answer
122 views

Beta distribution and beta binomial distribution

What is the difference or relation between beta distribution, beta binomial distribution and binomial distribution?
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1answer
191 views

UMP of a Beta($\theta,1$) distribution

I need to find the UMP of a random sample of a BETA$(\theta,1)$ distribution. I know that the pdf of this problem is $$f(x;\theta)=\theta x^{\theta-1}=\theta e^{(\theta-1)\log{x}}$$ After some ...
4
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1answer
453 views

Approximation for Beta distribution when alpha is less than 10

I know that we can approximate Beta distribution to Normal distribution when the values of alpha and beta are large numbers. In my problem alpha lies between 1 and 10, beta is always greater than ...
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96 views

Model multilevel proportion data

Suppose that one has data on a proportion $P$ measured on a continuous scale. Further suppose that this data has a grouped structure -- some proportions are clustered according to a certain variable ...
2
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1answer
402 views

Beta distribution fitting in Scipy

According to Wikipedia the beta probability distribution has two shape parameters: $\alpha$ and $\beta$. When I call scipy.stats.beta.fit(x) in Python, where ...
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4answers
1k views

Does the beta distribution have a conjugate prior?

I know that the beta distribution is conjugate to the binomial. But what is the conjugate prior of the beta? Thank you.
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1answer
106 views

Posterior probability - change in Beta hyperparameters

Can you explain, how does $\text{B}(\alpha, \beta)$ transfrom to $\text{B}(s+\alpha, f+\beta)$ in the following equation? $$ \begin{align*} p(\left. q=x \right| s,f) &= {{{s+f \choose s} ...
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1answer
657 views

How do I choose parameters for my beta prior?

Suppose today I'm going to flip a coin. I believe that 9 of 10 flips will come up heads. I flip the coin and 8 of 10 are heads. Is my distribution of belief beta(9+8, 1+2) beta(1+9+8, 1+1+2) ...