Questions tagged [multinomial-dirichlet-distribution]
The multinomial-dirichlet-distribution tag has no usage guidance.
13
questions
2
votes
0
answers
68
views
Posterior predictive distribution for Bernoulli (and categorical)
I'm trying to confirm something I've tried to figure out about the posterior predictive distribution for Bernoulli vs. Binomial (and categorical vs. multinomial) random variables after a Bayesian ...
- 28.2k
0
votes
0
answers
66
views
How to fit Multinomial-Dirichlet regression for correlated compositional count data in R?
I have compositional data where $K$ compositions (the abundance of $K$ cell populations) are measured on $n$ samples at baseline and repeated at 1 or more follow-up times. The $n$ samples are divided ...
- 93
2
votes
2
answers
241
views
Bayesian updates for Dirichlet-multinomial with Gamma prior
Let
$$
\begin{aligned}
X_i &\sim \text{Dir-multinom}(X\mid\lambda)\\
\lambda_{j} &\sim \text{Gamma}(\lambda_j\mid\alpha,\beta)\\
\end{aligned}
$$
where $i$ iterates over observations, $j$ ...
- 235
0
votes
0
answers
40
views
Expected value of log(gamma function(Dirichlet variable))
The following problem emerges from coordinate ascent variational inference in a mixture model with Dirichlet-Multinomial components. I want to compute the expectation of the log likelihood. Since my ...
- 771
1
vote
0
answers
202
views
Expectation maximization: does the likelihood always increase monotonically?
When working with (gaussian) mixture models, I always took it for a mathematical fact that the marginal likelihood increases with every iteration step. If it were not the case, it always meant an ...
- 3,390
5
votes
1
answer
237
views
Mean of Generalization of the Dirichlet Distribution
I know that if $X_{1},X_{2},...X_{n}$ are independent $\mathrm{Gamma}(\alpha_{i},\theta)$ - distributed variables (notice they all have the same scale parameter $\theta$) and
$Y_{i}=\frac{X_{i}}{\sum_{...
- 63
1
vote
0
answers
118
views
How to model proportions with a hierarchical structure?
I have thinking about how to model proportions for a problem with hierarchical structure.
In the problem, I have observations of users over multiple days, where each observation is a proportion of ...
- 151
0
votes
1
answer
76
views
How to find MLE for multinomial distribution and Expectation of X1 and X2
There are 3 types of flowers that can grow from planting a seed.
$$P(\text{Daisy}) = \theta_1$$
$$P(\text{Rose}) = (1-\theta_1)\theta_2$$
$$P(\text{Sunflower}) = (1-\theta_1)(1-\theta_2)$$
The total ...
- 1
6
votes
2
answers
1k
views
Multinomial distribution: probability that one outcome is greater than another
Consider a multinomial distribution with three outcomes. Let $x_i$ denote the number of occurences of the $i^{th}$ outcome, and the $i^{th}$ outcome occurs with probability $p_i$, $i=1,2,3$. Let $n$ ...
- 119
3
votes
1
answer
216
views
Multinomial-dirichlet with fractional counts
Suppose a lepidopterologist wants to estimate the relative proportions of three different species of butterfly. They go out into the field and count $N$ butterflies and record the number of each ...
5
votes
1
answer
1k
views
How is the mode in Dirichlet-Multinomial calculated?
The mode in Dirichlet-Multinomial is
$$
\mathrm{Mode}(\pi_i) = \frac{\alpha_i + x_i - 1}{\sum_{j=1}^k (\alpha_j + x_j -1)}
$$
Could you point out how is it calculated please?
What is the importance ...
- 262
1
vote
1
answer
460
views
What are the possible estimates of the parameters of the multinomial distribution?
The expected value of the parameters of the multinomial distribution (taking into account the Dirichlet prior $D(\alpha)$ and the posterior Dirichlet-Multinomial) is:
$\pi_i = α_i+ x_i / \sum_{j} α_j+...
- 262
2
votes
0
answers
224
views
Why there are not (long tail) alternatives to dirichlet-multinomial (while there are for posisson-gamma)
While there are a lot of long tail alternatives to poisson-gamma (negative binomial), for example
(Source)
I haven't found any work on replacing the dirichlet distribution with a more long tailed ...
- 397