All Questions
Tagged with dirichlet-distribution multinomial-distribution
41
questions
2
votes
2
answers
344
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, $j$ ...
- 225
1
vote
1
answer
96
views
Concentration Bounds for categorial distribution with good Dirichlet prior
I would like to know if there are any standard methods for analyzing the concentration bounds (for example Hoeffding's bound) for a multinomial distribution modelled with a Dirichlet prior, with the ...
- 131
1
vote
0
answers
147
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 ...
- 150
7
votes
2
answers
4k
views
Dirichlet distribution vs Multinomial distribution?
Both Dirichlet and multinomial distributions are distributions over vectors, and both Dirichlet and multinomial distributions are constrained so that all of the elements of these vectors sum to a ...
- 2,955
1
vote
1
answer
502
views
Inference on Dirichlet hyper-parameter
I'm working on a Gibbs sampler for a (somewhat custom version of) Latent Dirichlet Allocation model.
In short, I have data that comes from a $K$-dimensional Dirichlet-Multinomial distribution, i.e.
$$...
- 153
1
vote
0
answers
104
views
Treating missing data in making Bayesian inference
Suppose we have two biased coins $X_1,X_2$ that are possibly correlated to each other.
In each round, when both the coins are tossed, there can be four possible outcomes: $(HH,HT,TH,TT).$ Let's ...
- 433
0
votes
1
answer
136
views
Bayesian inference using Dirichlet: muddled outcome case
In relation to my previous question (Bayesian inference for Beta distribution after an uncertain outcome),
Suppose that $$(x_1,x_2,x_3)\sim Dirichlet(a_1,a_2,a_3)$$
and an associated Mutinoulli ...
- 433
3
votes
1
answer
256
views
Multinomial-dirichlet with fractional counts
Suppose a lepidopterologist wants to estimate the relative proportions of three different species of butterfly. They go out into the field and count $N$ butterflies and record the number of each ...
1
vote
0
answers
347
views
Dirichlet Multinomial Posterior Predictive Distribution for Language Model
I have been trying to teach myself about Bayesian analysis, and whilst I have been through the theory several times, I am struggling to actually apply it. I have found some questions online to ...
- 571
1
vote
1
answer
510
views
What are the possible estimates of the parameters of the multinomial distribution?
The expected value of the parameters of the multinomial distribution (taking into account the Dirichlet prior $D(\alpha)$ and the posterior Dirichlet-Multinomial) is:
$\pi_i = α_i+ x_i / \sum_{j} α_j+...
- 272
2
votes
1
answer
4k
views
How to use the Dirichlet prior for estimating the multinomial parameters? [closed]
I know that the multinomial distribution gives the likelihood of some vector D of occurrences to happen given a probability vector (parameters) P' i.e. P(D|P'). Now with a Dirichlet prior we are ...
- 272
0
votes
0
answers
101
views
Recovering $\theta$ in Dirichlet-Multinomial (Polya) distribution
I'm working on Latent Dirichlet Allocation with Collapsed Gibbs Sampling.
LDA has two Dirichlet-Multinomial distribution and one of them is a document-topic distribution that determines the ...
- 245
1
vote
0
answers
685
views
Marginal Likelihood of Multinomial Dirichlet model
To find the marginal likelihood of the multinomial Dirichlet model, I tried the following:
$$\int_\theta p(N|\theta)p(\theta)d\theta=\frac{n!}{n_1!...n_K!}\frac{\Gamma(\sum_{k=1}^K\alpha_k)}{\Pi_{k=1}^...
- 153
1
vote
1
answer
2k
views
Maximum a posteriori on Multinomial distribution with a Dirichlet prior can result in negative probabilities?
I am doing a maximum a posteriori (MAP) estimation of a Multinomial distribution $M(c_1,\dots,c_n|p_1,\dots,p_n)$ with a Dirichlet prior $D(p_1,\dots,p_n|\alpha_1,\dots,\alpha_n)$. The experimental ...
- 113
0
votes
1
answer
124
views
multinomial model with some certain parameters
I might be asking a naive questions here, sorry.
Imaging I have 4 categories, each one has a probability of $\theta_i$ being selelcted, $i=1..4$ and sum of $\theta_i$ is 1. For this simple ...
- 69