16
votes
Accepted
Taking into account the uncertainty of p when estimating the mean of a binomial distribution
There are several problems with your approach. First, you want to use confidence intervals for something that they were not designed for. If $p$ varies, then confidence interval will not show you how ...
- 141k
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 ...
- 334k
11
votes
What is the intuition behind beta distribution?
Most of the answers here seem to cover two approaches: Bayesian and the order statistic. I'd like to add a viewpoint from the binomial, which I think the easiest to grasp.
The intuition for a beta ...
- 879
10
votes
Accepted
Limit of beta-binomial distribution is binomial
There are at least two ways of seeing this.
The urn interpretation of the distribution can be shown to be
The beta-binomial distribution can also be motivated via an urn model for positive ...
- 4,603
8
votes
Accepted
Property of two independent Beta distribution
It does not seem to be a correct conjecture.
It seems your condition is that the mode for $X$ is greater than the mode for $Y$. Since in non-symmetric Beta distributions, the mode is not equal to the ...
- 42.1k
7
votes
Accepted
Hyper-parameter estimation for Beta-Binomial Empirical Bayes
The hierarchical model
You don't actually even need the marginal probability mass function $m()$, you actually only need the marginal moments of $Y$.
In this tutorial, Casella (1992) is assuming the ...
- 13.5k
7
votes
Accepted
What is the distribution of a sum of identically distributed Bernoulli random varibles if each pair has the same correlation?
Have you seen this paper: Kadane, 2016, Sums of Possibly Associated Bernoulli
Variables: The Conway-Maxwell-Binomial Distribution?
In this paper, you can see that the conditions assumed in your ...
- 554
6
votes
Poisson-binomial vs. Beta-binomial
My question stemmed from my own ignorance at the time, but I thought I'd post the answer in case anyone else has the same misunderstanding I did.
When I posed the question I erroneously understood ...
- 71
5
votes
Accepted
Update samples of a Beta with Bernoulli likelihood to the Beta posterior
Let $f(x;\alpha,\beta)$ denote the beta distribution parameterized by $\alpha$ and $\beta$. The aim is to express $f(x;\alpha,\beta)$ in terms of $f(x;\alpha-1,\beta)$ and $f(x;\alpha,\beta-1)$.
$$f(...
- 391
5
votes
Relationship between Binomial and Beta distributions
Summary: It is often said that Beta distribution is a distribution on distributions! But what is means?
It essentially means that you may fix $n,k$ and think of $\mathbb P[Bin(n,p)\geqslant k]$ as a ...
- 221
5
votes
Accepted
Does order of events matter in Bayesian update?
AFAIK, you cannot say that $p > \frac{1}{2}$ or even $\mathbb E[p]>\frac{1}{2}$ is an "observation" or "event", but rather a constraint on your model parameter(s). The term "observation" is ...
- 3,144
5
votes
How do I understand the intuition behind percentile point function?
For a random variable $X$ with cumulative distribution function $F(x) = P(X\leq x)$, the usual definition of the quantile function is
$$
Q(p) = \inf\{x: F(x)\geq p\},\quad p\in(0,1)
.$$
Now if $X$ is ...
- 12.1k
4
votes
Using poisson distribution to model proportions
Yes. I believe you can use Poisson to model rates by having the denominator count as an offset on the right hand side. If you are modeling the rate $\frac{Y}{q}$, piglets per sow, then
$$\log\left(\...
- 41
4
votes
What is the appropriate model for underdispersed count data?
It seems that the solution provided by Joseph Hilbe within the vgam package is no longer available. From the manual of the package: The genpoisson() has been simplified to genpoisson0 by only ...
- 61
4
votes
How do I specify a Bayesian Beta binomial model, with predictor variables, for R2jags?
It is not really a "how to code it in JAGS" problem, but it is about defining the appropriate model for your data. If you want to include predictor variables for your data, this means you need a ...
- 141k
4
votes
How do I carry out a significance test with Tarone's Z-statistic?
If you would like another explanation of the procedure, you can read the original paper by Tarone (1979), but the blog actually gives a longer and clearer explanation than the original paper. In any ...
- 133k
4
votes
Does order of events matter in Bayesian update?
In Bayesian inference, terms like "observation" and "event" are just conveniences; there is no fundamental importance to them, so don't get hung up on them.
In particular, there is no physical ...
- 675
4
votes
Accepted
Beta-Binomial regression or Poisson-Gamma model to account for uncertainty in (empricial Bayesian) prior? Explained in simple terms?
OK, so this isn't the greatest implementation, but I think it serves our purpose.
The beta binomial regression makes the assumption that the data are generated from a binomial distribution
$$ y_i\...
- 39.6k
4
votes
Accepted
Beta vs beta-binomial why beta has higher AIC
If you are modeling the proportion directly (i.e. you are measuring $\frac{\text{count of mature females}}{\text{count of all females}})$ then the beta distribution will have a much better AIC than ...
- 301
4
votes
Accepted
Discrepancy between binomial and beta in R?
Why would you expect to see similar results? Those are different distributions, used for modelling completely different things. First is a discrete distribution, second is a continuous distribution. ...
- 141k
4
votes
Significant dispersion test
I'm the developer of DHARMa. This is a pretty common question, and there is a section in the vignette giving guidance on that, but I just realised now that it's not very well placed, so I will change ...
- 8,699
4
votes
Accepted
Beta-Binomial mixture vs Beta-Binomial multilevel model?
They are both pieces of the beta-binomial model. In beta-binomial model, the predicted variable $y$ follows the binomial distribution, where the number of samples $n$ is known and we want to learn the ...
- 141k
4
votes
What is the correct implementation of MCMC
You've specified a class of MCMC algorithms referred to as Metropolis-Hastings.
With these algorithms, choosing the proposal distribution is the key question when specializing the algorithm. Under ...
- 21.6k
3
votes
Accepted
Comparing two groups with binomially distributed data
If you're willing to assign a prior Beta distribution on $p_1$ and $p_2$, you could do a Bayesian analysis. The model would be
\begin{align}
p_1&\sim \operatorname{Beta}(\alpha_1,\beta_1) \\
p_2&...
- 31.2k
3
votes
What is the intuition behind beta distribution?
In the cited example the parameters are alpha = 81 and beta = 219 from the prior year [81 hits in 300 at bats or (81 and 300 - 81 = 219)]
I don't know what they call the prior assumption of 81 hits ...
- 31
3
votes
How to implement generalized hypergeometric function to use in beta-binomial cdf, sf, ppf?
This does not answer your question directly, but if you are thinking of estimating the cumulative distribution function of beta-binomial more efficiently, then you can use a recursive algorithm that ...
- 141k
3
votes
Beta prior on (a,b)
In general the support space of the prior-if a subset of the likelihood parameter space-will be the support space of the posterior. For this case write the likelihood
$$L(x| p)= {n\choose{x} }p^{x}(...
- 4,379
3
votes
Accepted
Beta-Binomial Derivation
There are just two things you need to know for this.
First one: definition of the Beta function.
$$
B(x,y) := \int_0^1 t^{x-1}(1-t)^{y-1}\,\text dt.
$$
Second thing: the identity that
$$
B(x,y) = \...
- 20.8k
3
votes
Accepted
how to analyze overdispersed binary data
If you analyse the data using an ordinary generalised linear model this is what you get:
...
- 11.7k
3
votes
Accepted
Correct usage/understanding of Bayes Factor when comparing two proportions
The Bayes factor for testing $H_0:\ p\le q$ when $X\sim\mathcal{N}(n,p)$ and $Y\sim\mathcal{N}(m,p)$ is$$\mathfrak{B}_{01}(x,y)=\dfrac{\int_{p\le q} f_X(x|p)f_Y(y|q)\pi(p,q)\text{d}p\text{d}q}{\int_{p\...
- 108k
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