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Suppose we have a set of points $\mathbf{y} = \{y_1, y_2, \ldots, y_N \}$. Each point $y_i$ is generated using distribution $$ p(y_i| x) = \frac12 \mathcal{N}(x, 1) + \frac12 \mathcal{N}(0, 10). $$ To obtain posterior for $x$ we write $$ p(x| \mathbf{y}) \propto p(\mathbf{y}| x) p(x) = p(x) \prod_{i = 1}^N p(y_i | x). $$ According to Minka's paper on Expectation Propagation we need $2^N$ calculations to obtain posterior $p(x| \mathbf{y})$ and, so, problem becomes intractable for large sample sizes $N$. However, I can't figure out why do we need such amount of calculations in this case, because for single $y_i$ likelihood has the form $$ p(y_i| x) = \frac{1}{2 \sqrt{2 \pi}} \left( \exp \left\{-\frac12 (y_i - x)^2\right\} + \frac{1}{\sqrt{10}} \exp \left\{-\frac1{20} y_i^2\right\} \right). $$

Using this formula we obtain posterior by simple multiplication of $p(y_i| x)$, so we need only $N$ operations, and, so we can solve this problem for large sample sizes exactly.

I make numerical experiment to compare do I really obtain the same posterior in case I calculate each term separately and in case I use product of densities for each $y_i$. Posteriors are same. See enter image description here Where am I wrong? Can anyone make it clear to me why do we need $2^N$ operations to calculate posterior for given $x$ and sample $\mathbf{y}$?

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  • $\begingroup$ One operation per term and $N$ terms, so we need $O(N) $ operations. Also, I look through Minka's paper and Bishop's chapter on approximate inference again. Both suggest that we want estimate and obtain posterior for $x$. $\endgroup$ – Alexey Zaytsev Jul 14 '12 at 13:02
  • $\begingroup$ Am i understanding correctly that your $y_i$'s are univariate? If so, you can solve this in $O(n\log(n))$ which is considered tractable regardless of $n$ $\endgroup$ – user603 Jul 14 '12 at 13:28
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    $\begingroup$ @Alexey After re-reading this paragraph, I think the author does not mention $2^N$ operations. He just points out that "the belief state for $x$ is a mixture of $2^N$ Gaussians". $\endgroup$ – user10525 Jul 14 '12 at 13:46
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    $\begingroup$ @Procrastinator according to paper we want to use belief propagation, but can't use because we need to proceed mixture of $2^N$ gaussians. Then the question is why do we want to use BP? Another question arises in case we read chapter 10.7.1 in Bishop's PRML or watch videolecture by Minka. After that the answer isn't this clear. $\endgroup$ – Alexey Zaytsev Jul 14 '12 at 13:49
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    $\begingroup$ @Alexey I think the logic behind this is different. The author describes what happens if you use belief propagation, in order to emphasise some difficulties with it when $N$ is large, and then promoting his "expectation propagation". He mentions that belief propagation requires the use of a mixture of $2^N$ Gaussians for the belief state for $x$ which becomes complicated when $N$ is large. There is no mention to the number of operations required but to the complexity of the belief state for $x$. $\endgroup$ – user10525 Jul 14 '12 at 13:55
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You are right that the paper is saying the wrong thing. You certainly can evaluate the posterior distribution of $x$ at a known location using $O(n)$ operations. The problem is when you want to compute moments of the posterior. To compute the posterior mean of $x$ exactly, you would need $2^N$ operations. This is the problem that the paper is trying to solve.

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You missed the point that the distribution is a mixture of Gaussians: each sample $y_i$ is either distributed as per $p(y_i | x)$ with probability $1-w$ and as $p_c(y)$ (clutter distribution for $y$, independent of $x$) with probability $w$.

Let $c_i$ be the indicator variable indicating that sample $i$ was draw from the clutter distribution; thus, if it's $0$ it indicates that the sample was drawn from $p(y|x)$. Obviously, if the sample was drawn from the clutter distribution it's value is irrelevant for the estimation of $x$.

It's the presence of the $2^N$ possible joint states for these indicator variables that causes the problem.

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  • $\begingroup$ However, we can drop additional $c_i$ variables, because we need to obtain a maximum posterior solution of the problem. Posterior for $x$ has a clear form, so we are not forced to take into account all $2^N$ present states. So, the question is "Why do we need this amount of computations in case we want to find a maximum posterior solution?" $\endgroup$ – Alexey Zaytsev Dec 8 '12 at 8:44
  • $\begingroup$ The maximization needs to be taken over the states for the $c$ variables. $\endgroup$ – Dave Dec 8 '12 at 22:59
  • $\begingroup$ We don't know $c_i$, so we integrate out (sum up over) $c_i$. This can be done in a direct way, doesn't it? $\endgroup$ – Alexey Zaytsev Dec 9 '12 at 14:20
  • $\begingroup$ Direct yes, but the number of states (terms) grows like $2^N$, which can be computationally problematic. $\endgroup$ – Dave Dec 10 '12 at 13:41
  • $\begingroup$ We can do this for each observation in an independent way, so we have $O(n)$, not $O(2^n)$ complexity. $\endgroup$ – Alexey Zaytsev Dec 11 '12 at 8:06

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