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I want to understand this paper on brain tumour segmentation.

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How is this equation read?

I'm guessing $q_i(t_i)$ represents the likelihood of tumour on voxel i.Is q usually used to represent likelihoods?

What does the equal sign with the triangle mean?

I understand what each term on the conditional probability represents, but I don't get why it's called proportional to the product of the summation. I'm looking for a reading reference here.

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The equal sign with triangle usually means "is defined as". – mark999 Sep 14 '12 at 7:00
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The equation might be erroneously including a summation sign. If the observation is $y$, then the likelihood of $p(y|t_i, k_i)$, the joint probability is the likelihood times the prior probability $p(t_i,k_i)$. Thus, $p(y) = \sum_j p(y|t_j, k_j)p(t_j,k_j)$ is the probability of the observation and is the sum above. The posterior probability is $$p(t_i|y) = \frac{p(y|t_i, k_i)p(t_i,k_i)}{p(y)}$$ and is thus proportional (with proportionality factor $[p(y)^{-1}$) to the joint probability $p(y|t_i,k_i)p(t_i,k_i)$. It is not proportional to, but rather inversely proportional to the sum $p(y)$. – Dilip Sarwate Sep 14 '12 at 13:21

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I think the use of q for the function is the authors choice and not anything standard. I agree with mark999 about the triangle over the equal sign. From reviewing the article I see that the function p on the left side of the proportionality sign is a component of the posterior distribution for the t$_i$s and the proportionality piece is just the well-known relation dues to Bayes theorem, that the posterior distribution is proportion to the prior times the likelihood and in this case both sides are decomposible into products for the i$_t$$_h$ t.

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