Tagged Questions

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

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2
votes
2answers
164 views

Problem interpreting the Beta distribution

On p38 of Lee and Wagenmakers (2012) "Bayesian Cognitive Modeling: A Practical Course" the following passage appears: "One of the nice properties of using the θ ~ Beta (α,β) prior distribution ...
0
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0answers
16 views

Kernel density estimation on bounded support

I was looking for some way to deal with boundary bias of kde in case of unit interval. One example is an usage of Chen estimators (or Beta estimators; an example might be seen here: ...
1
vote
0answers
18 views

Regression Modeling with Upper and Lower Bounds on target distribution

I am trying to run a simple regression model between a couple of variables, one of which is bound between values of (.25 - .75). This will be my target variable. I understand the beta distribution can ...
2
votes
1answer
16 views

Model averaging when linear and quadratic effects are modeled in a global model

I am trying to derived estimates of model-averaged parameter effects on a fairly complicated set of models using an information-theoretic approach. I have several models that investigate continuous ...
0
votes
0answers
12 views

Estimating the loss between two Beta distributions

Suppose I have two coins, $A, B$ that each come up heads with probability $p_A, p_B$. Starting with a uniform prior on the values of $p_A, p_B$, and seeing data $s_A$ heads out of $N_A$ attempts, ...
1
vote
0answers
8 views

Bounds on binary event estimation

I would like to paint an objective picture of some binary outcome. Now I have data like this: 1085x yes, 1704x no. The percentage of the positive outcome is 40.72%, but I want to give an estimation ...
1
vote
2answers
36 views

Beta regression

I have a data set where the response variable Y is a rate between 0 and 1, where the histogram of Y is bimodal. So I feel the linear regression is not suitable.s I have been reading papers about ...
2
votes
1answer
52 views

Recursively updating the parameter of a Beta function in a bayesian way?

I ask, because it is very hard to find information regarding the beta distribution and the bayesian inference, where the beta distri is NOT the prior. My goal is to identify or to improve the two ...
0
votes
1answer
34 views

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 ...
2
votes
3answers
401 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 ...
1
vote
0answers
101 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 ...
2
votes
1answer
79 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 ...
0
votes
1answer
33 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 ...
3
votes
1answer
69 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 ...
1
vote
1answer
40 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 ...
5
votes
1answer
295 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 ...
4
votes
0answers
56 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 ...
2
votes
1answer
52 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
votes
1answer
104 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!
0
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0answers
56 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 ...
1
vote
1answer
71 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 ...
1
vote
0answers
30 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 ...
0
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0answers
29 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 ...
2
votes
0answers
84 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 ...
1
vote
2answers
53 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 ...
4
votes
0answers
53 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 ...
12
votes
2answers
600 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 ...
0
votes
1answer
52 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, ...
1
vote
1answer
68 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 ...
3
votes
1answer
53 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 ...
4
votes
1answer
145 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.: ...
1
vote
0answers
52 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 $Z = X - Y$?
1
vote
1answer
252 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 ...
0
votes
0answers
50 views

Approximation for linear combination of truncated beta variables

Let $x_1, x_2$ be random variables that follow truncated beta distributions: ...
0
votes
0answers
26 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 ...
1
vote
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 ...
2
votes
2answers
63 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, ...
0
votes
0answers
44 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 ...
3
votes
3answers
91 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 ...
4
votes
0answers
226 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 ...
1
vote
0answers
105 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
votes
2answers
84 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 ...
1
vote
0answers
85 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 ...
2
votes
0answers
55 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 $k_i$ ...
9
votes
1answer
396 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 ...
0
votes
0answers
207 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 ...
3
votes
1answer
111 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, ...
4
votes
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
votes
1answer
46 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 ...
1
vote
0answers
110 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 ...