Questions tagged [dirichlet-process]
A family of stochastic processes whose realizations are probability distributions
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Parallel Tempering and Bayesian Non-Parametrics
In parallel tempering, we run multiple MCMC chains in an ascending temperature ladder, where the posterior density of the $i$th chain is exponentiated to the reciprocal of the temperature of the $i$th ...
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Bayesian Hierarchical Clustering prior update
I am working through Heller and Ghahramani's "Bayesian Hierarchical Clustering" paper (https://www2.stat.duke.edu/~kheller/bhc.pdf) and things aren't quite working out the way I expect with ...
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Inference on latent variable with observation of its convolution with itself
Problem
I have an inference problem where the data observed are univariate random numbers whose distribution is obtained as follows. A latent random variable X is first sampled from a parametric ...
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Dirichlet distribution with correlated components?
I am working with models that use Dirichlet distributions. However, I want to account for correlations between components. If this question is a duplicate, I'd also appreciate any pointers to the ...
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Stick-breaking construction of Dirichlet distribution vs Dirichlet process
Let $F_0$ be some probability measure and $\alpha > 0$ be the concentration parameter. I can draw a random distribution from $F\sim \mathrm{DP}(\alpha, F_0)$ using the stick-breaking construction:
\...
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Expected value (and variance) of a Dirichlet Process
Suppose I have a measure $G$ that follows a Dirichlet Process,
$$G \sim DP(H_0,\alpha)$$
where $H_0$ is some base measure. Is there a closed form solution for the expected value of $G$?
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If a scalar is added to a Dirichlet process with a normal base distribution, is the resulting process also Dirichlet?
Let $\theta_i | G \sim G$ and $G|\alpha \sim DP(\alpha, G_0)$ with $G_0 \sim N(0,\sigma^2)$. For a scalar k, does $\theta_i + k | G' \sim G'$ with $G'|\alpha \sim DP(\alpha, G'_0)$ and $G'_0 \sim N(k,\...
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Changing scan order of Gibbs Sampler on each iteration
I'm implementing an algorithm that requires the use of Gibbs Sampling and, due to the nature of the way I store the values, it would be efficient to change the order of the updates on each component ...
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Dirichlet Process posterior with partially observed data
Suppose I dipose of a set of independant observed couples $(x_1,y_1),...,(x_N, y_N)$ from a joint distribution $P(x,y)$. Furthermore, I suppose that the random distribution $P$ as a Dirichlet prior
$P\...
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Stick-breaking: break sticks of decreasing lengths
The stick-breaking construction used for Dirichlet Processes can create an infinite sequence of probabilities $ \boldsymbol{\pi} $ (stick lengths) that sum to 1 via the following formulae:
$\nu_i \sim ...
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Understanding the blocked gibbs sampler for Dirichlet process
I've always implemented DP MM's using chinese restaurant process, which necessitates sequential sampling of cluster assignments (as cluster weights depend on current number of observations in cluster, ...
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Let $H$ be the base distribution of a Dirichlet process. How is this process well-defined in case $H(B_1) = 0$?
I have read that the parameters of Dirichlet distribution must be strictly positive.
The Dirichlet distribution of order $K \geq 2$ with parameters $\alpha_{1}, \ldots, \alpha_{K} \color{blue}{> 0}...
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Estimating the likelihood of a Dirichlet process
I am not sure if what I'm trying to achieve makes sense or is even possible, but I'd like to do MLE on a Dirichlet process mixture model. My reasoning is the following:
If we can write out the ...
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Can we use Dirichlet process to simultaneously estimate the number of mixtures and component distribution of a Bernoulli mixture?
Suppose I have a random sample on a Bernoulli random variable $\{X_i\}_{i=1}^N$ generated from model $p=\sum_{k=1}^K\pi_kp_k$,where $p\equiv Pr(X=1)$ and $p_k\equiv Pr(X=1|k)$, and $\pi_k$ are the ...
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Mixtures of Dirichlet multivariates or Dirichlet processes
I am exploring the properties of Dirichlet distributions and their parameters. When mixing two Dirichlet distributed random bivariates
$$\mathbf{X}\equiv(X_1,X_2)\sim\text{Dir}(\alpha_1,\alpha_2)$$
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