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In the original paper Latent Dirichlet Allocation, the authors said that the function

$$p(\mathbf{w} \mid \alpha, \beta)=\frac{\Gamma\left(\sum_{i} \alpha_{i}\right)}{\prod_{i} \Gamma\left(\alpha_{i}\right)} \int\left(\prod_{i=1}^{k} \theta_{i}^{\alpha_{i}-1}\right)\left(\prod_{n=1}^{N} \sum_{i=1}^{k} \prod_{j=1}^{V}\left(\theta_{i} \beta_{i j}\right)^{w_{n}^{j}}\right) d \theta$$

is intractable due to the coupling between $\theta$ and $\beta$ in the summation over latent topics. Then they went on with "the posterior distribution is intractable for exact inference". IMHO, the exact estimation of $\alpha,\beta$ is not possible because we do not observe $\theta$.

Can you explain what "intractable" means in this case?

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It means that it's not possible to calculate in a reasonable amount of time with a computer. In this case, it's probably because it iterates over a very large number of elements.

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