54
votes
Choosing between uninformative beta priors
First of all, there is no such a thing as uninformative prior. Below you can see posterior distributions resulting from five different "uninformative" priors (described below the plot) given different ...
Tim♦
- 136k
31
votes
Accepted
Is the Gaussian distribution a specific case of the Beta Distribution?
They are both symmetric and more or less bell shaped, but the symmetric beta (whether at 4,4 or at any other specific value) is not actually Gaussian. You can tell this even without looking at the ...
- 277k
28
votes
How to implement a mixed model using betareg function in R?
The package glmmTMB may be helpful for anyone with a similar question. For example, if you wanted to include pond from the above question as a random effect, the following code would do the trick:
<...
- 406
21
votes
Accepted
Beta distribution on flipping a coin
The quotation is a "logical sleight-of-hand" (great expression!), as noted by @whuber in comments to the OP. The only thing we can really say after seeing that the coin has an head and a tail, is that ...
- 17.4k
21
votes
Accepted
Why exactly can't beta regression deal with 0s and 1s in the response variable?
Because the loglikelihood contains both $\log(x)$ and $\log(1-x)$, which are unbounded when $x=0$ or $x=1$. See equation (4) of Smithson & Verkuilen, "A Better Lemon Squeezer? Maximum-Likelihood ...
- 559
21
votes
Whence the beta distribution?
Thomas Bayes (1763) derived the Beta distribution [without using this name] as the very first example of posterior distribution, predating Leonhard Euler (1766) work on the Beta integral pointed out ...
- 103k
20
votes
Accepted
What is the relationship between the Beta distribution and the logistic regression model?
Beta is a distribution of values in $(0,1)$ range that is very flexible in it's shape, so for almost any unimodal empirical distribution of values in $(0,1)$ you can easily find parameters of such ...
Tim♦
- 136k
16
votes
Accepted
Efficiently sampling a thresholded Beta distribution
The simplest way, and most general way, that applies to any truncated distribution (it can be also generalized to truncation on both sides), is to use inverse transform sampling. If $F$ is the ...
Tim♦
- 136k
16
votes
Accepted
Limit of $n$ times Beta$(1,n)$ variables when $n$ goes to infinity
First, let's get a sense why this should be true. The density of a Beta$(1,n)$ variable which has been multiplied by $n$ should be proportional to
$$\left(\frac{x}{n}\right)^{1-1}\left(1 - \frac{x}{n}...
- 316k
15
votes
Accepted
Pdf of $y = - \log(X)$ when $X$ is beta distributed The expected value of $Y$
Let $Y=-\ln X,\quad X \sim Beta(\alpha,\beta)$ then
$$
F_Y(y)=P(Y<y)=P(-\ln X < y)=P(X > e^{-y})=\int_{e^{-y}}^1\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\...
- 316
15
votes
Accepted
How to implement a mixed model using betareg function in R?
The current capabilities of betareg do not include random/mixed effects. In betareg() you can only include fixed effect, e.g., ...
- 15.2k
15
votes
Uniform vs Beta(1,1) prior
They both are equivalent.
$P(\theta) = { \Gamma(\alpha + \beta) \over \Gamma(\alpha)\Gamma(\beta)} \theta^{\alpha-1}(1-\theta)^{\beta-1}$
if $\alpha = \beta = 1$
$P(\theta) = { \Gamma(\alpha + \...
- 191
15
votes
Why is the Beta Distribution Called the Beta Distribution?
Florian Cajori, in History of Mathematical Notations Vol. II (1928), wrote
... in the same paper of 1730 Euler gave what we now call the "beta function." ... About a century after Euler's first ...
- 316k
15
votes
Accepted
Square root of a Beta(1,1) random variable
If $X^{2}\sim\operatorname{Beta}(1,1)$ (which is a uniform distribution), then $X^p\sim\operatorname{Kumaraswamy}(1/p, 1)$ (see the Wikipedia page). The PDF of the resulting Kumaraswamy distribution ...
- 29.4k
15
votes
Accepted
Distribution of argmax of beta-distributed random variables
When the $x_i$ are independent for $1\le i \le d$ with distribution functions $F_i$ and density functions $f_i,$ respectively, the chance that $x_j$ is the largest is (by the very definition of the ...
- 316k
14
votes
How to calculate the PDF of the 'difference' between two Beta distributions?
I know this is a bit of an old question but for what it's worth there is an established closed-form solution to this problem, found by Pham-Gia, Turkkan, and Eng in 1993. It's a piecewise solution ...
- 241
14
votes
Distribution of the exponential of an exponentially distributed random variable?
First, note that the range of $\DeclareMathOperator{\P}{\mathbb{P}} Y$ is $(1, \infty)$. First find the cumulative distribution function of $Y$ in the usual way:
$$\begin{align}
F_Y(t) & = \...
- 74.6k
13
votes
Accepted
Why is there -1 in beta distribution density function?
This is a story about degrees of freedom and statistical parameters and why it is nice that the two have a direct simple connection.
Historically, the "$-1$" terms appeared in Euler's studies of the ...
- 316k
13
votes
Accepted
Jeffreys' prior for Beta distribution
As indicated in this paper by Yang and Berger (1999) that provides a list of Jeffreys priors, the Jeffreys prior associated with the Beta distribution is the determinant of a $2\times 2$ matrix that ...
- 103k
13
votes
Accepted
How to plot $x^{1700}(1-x)^{300}$?
Stephan's answer about floating point is correct. As a work-around, you could plot the data on a logarithmic scale. Instead of plotting
$$
x ^{1700} (1-x)^{300}
$$
you would plot
$$
1700\log(x) + 300\...
- 89k
13
votes
Accepted
Is there a "beta distribution" over the entire real line?
One way to make a "real-valued beta distribution" would be to transform the interval $(0,1)$ onto the real line. One way of doing that is the logistic function
$$ \text{logit}(x)= \log(\frac{...
- 74.6k
12
votes
What is the intuition behind beta distribution?
So far the preponderance of answers covered the rationale for Beta RVs being generated as the prior for a sample proportions, and one clever answer has related Beta RVs to order statistics.
Beta ...
- 60.5k
12
votes
Do two quantiles of a beta distribution determine its parameters?
The answer is yes, provided the data satisfy obvious consistency requirements. The argument is straightforward, based on a simple construction, but it requires some setting up. It comes down to an ...
- 316k
12
votes
Accepted
Can cosine kernel be understood as a case of Beta distribution?
The cosine kernel is not a beta distribution.
Note that the following things are all true of the standard cosine density:
$f(0)=1$
$f(0.5)=0.5$
The right half of this density is rotationally ...
- 277k
12
votes
Accepted
Expected value of $1/x$ when $x$ follows a Beta distribution
First note that the pdf of a Beta$(\alpha, \beta)$ distribution is only defined for $\alpha, \beta > 0$. Which means that for when $\alpha \leq 0$ or $\beta \leq 0$
$$\int_0^1 \dfrac{x^{\alpha - 1}...
- 15.1k
12
votes
Why does dbeta not sum to 1?
The relevant property of a probability density is not that it sums (for evaluation on some particular $x$ values) to one, but that it integrates to one.
If you evaluate a density $f$ at $x$ values ...
- 117k
11
votes
Accepted
Understanding the Beta conjugate prior in Bayesian inference about a frequency
The point is that we know what the posterior is proportional to and it so happens that we do not need to do the integration to get the (constant) denominator, because we recognise that a distribution ...
- 30.4k
11
votes
Square root of a Beta(1,1) random variable
You ask for a general method. Here is one.
When $X^p$ has a Beta$(\alpha,\beta)$ distribution for $p\gt 0,$ this means for all $0\lt y \lt 1$ that
$$F_X(y^{1/p}) = \Pr(X \le y^{1/p}) = \Pr(X^p \le y) ...
- 316k
11
votes
Accepted
Distribution of i.i.d. random variables $X$ and $Y$ if $XY \sim \text{Beta}(\alpha, \beta)$
In the case of $Z=XY\sim \text{Beta}(\alpha,1)$, the moment generating function (mgf) of $-\ln(XY)=-\ln X-\ln Y$ is
\begin{align}
M_{-\ln(XY)}(t)
&= E(e^{-t\ln Z})
\\ &=E(Z^{-t})
\\ &=\...
- 10.5k
10
votes
Accepted
Distribution of the ratio of dependent chi-square random variables
This post elaborates on the answers in the comments to the question.
Let $X = (X_1, X_2, \ldots, X_n)$. Fix any $\mathbf{e}_1\in\mathbb{R}^n$ of unit length. Such a vector may always be completed ...
- 316k
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