Questions tagged [multinomial-dirichlet-distribution]
11 questions
6
votes
2 answers
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 $n$ be the number of total trials. Then we have…
Greenteamaniac
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5
votes
1 answer
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 of -1 in "αi+xi−1" (I know that the…
Mosab Shaheen
- 252
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4
votes
0 answers
Using categorical data to build a Dirichlet distribution
I am building a graphical model. I have some categorical data $\boldsymbol{\mu}$ where they are generated by $p(\boldsymbol{\mu}|\boldsymbol{s},\mathbf{A})=\prod_k\prod_j\mathbf{A}_{ij}^{\mu_is_j}$. I'd like to use $\boldsymbol{\mu}$ to be mapped…
Dalek
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3
votes
1 answer
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
$Y_{i}=\frac{X_{i}}{\sum_{j=1}^{n}X_{j}}$
then…
bbecon
- 43
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3
votes
1 answer
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 species $(N_A,N_B,N_C)$. This is a Bayesian…
diagonalisable
- 31
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2
votes
0 answers
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 alternative (if exists) in the…
Stefano Vespucci
- 385
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1
vote
0 answers
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 time spent on different activities in that day.
Each…
Jeff
- 141
- 3
1
vote
1 answer
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:
$\pi_i = α_i+ x_i / \sum_{j} α_j+ x_j$
what are the other statistics and formulas…
Mosab Shaheen
- 252
- 2
- 9
0
votes
0 answers
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 likelihoods are Dirichlet-Multinomials, that…
Rylan Schaeffer
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0
votes
0 answers
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 error in the code or some other technical problem.
I…
Roger Vadim
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0
votes
1 answer
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 number of flowers at the end is $n.$ If $X=(X_1,…
hello
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