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I’ve recently read the article "Visual Tracking of Human Visitors under Variable-Lighting Conditions for a Responsive Audio Art Installation," A. Godbehere, A. Matsukawa, K. Goldberg, American Control Conference, Montreal, June 2012.

On page 4 it says:

Making use of Assumption I-C5, we let $p(f|F) = 1-p(f|B)$

[$f$ is the feature calculated and $F$ stands for foreground, $B$ for background]. Assumption I-C5 is given on page 3.

The quoted sentence is a non sense whatever assumption one makes.

$p(f|F)$ is then replaced in the following relation, from earlier on the same page (which is just Bayes’ rule):

$$p(B|f) = \frac{p(f|B)p(B)}{p(f|B)p(B) + p(f|F)p(F)}$$

and the final Boolean result comes from a threshold on the above formula, where $p(F)$ and $p(B)$ are adjustable parameters (now it's really $p(F) = 1- p(B)$).

I suspect that the authors noticed that this approach worked and accepted it.
Maybe the right way to proceed would have been to fix a constant p(f|F), as a uniform distribution models a situation where we have no way to know it, and so it will become a new adjustable parameter.

Like in the assumption made by the authors, we have an increasing function, $p(B|f)$, of variable $p(f|B)$ with a few constants.

It is equivalent to the above approach but with a different choice of parameters and a different threshold.

Any thoughts about it?

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  • $\begingroup$ It is difficult to evaluate the sense or nonsense of the quotation without knowing the context of the paper and the specifics of the assumptions they invoke to justify that relationship. If you could provide a link, that would help. In the meantime, $p(f|F)=1-p(f|B)$ does not look like outright nonsense, because it appears to express a symmetry akin to the binary symmetric channel (BSC) of communications theory. $\endgroup$ – whuber Sep 17 '13 at 18:00
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    $\begingroup$ @whuber, you are right, I have to provide more information. Feature f can assume a discrete number of values (in practise many of them), so if we consider the probability of a particular feature occurring in case of Background, p(f1|B), it should be a low number and all p(fi|B) should sum up to 1. The same happens for the foreground. p(f|F) = 1-p(f|B) is impossible. Assumption I-C5 states: “Pixels corresponding to visitors have a color distribution distinct from the background distribution” $\endgroup$ – Gino Strato Sep 17 '13 at 18:43
  • $\begingroup$ "p(f|F) = 1-p(f|B) is impossible" ... if this is so certainly the case, what do you need us for? Shouldn't you contact the authors? $\endgroup$ – Glen_b Sep 18 '13 at 3:51
  • $\begingroup$ It looks to me like if you interpret I-C5 in a very particular way, it might be quite possible to justify $p(f|F)=1-p(f|B)$, though such justification (i.e. explaining exactly how it follows from the assumption) appears to be absent. Why are you so certain it must be wrong? $\endgroup$ – Glen_b Sep 18 '13 at 3:57
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    $\begingroup$ It seems to me that it is apparent that the only case where that relation can be true is when fi can only assume 2 values. Suppose it has just three values, then $$p(f1|F)=1-p(f1|B) , p(f2|F)=1-p(f2|B), p(f3|F)=1-p(f3|B)\mapsto \sum p(fi|F) = \sum(1- p(fi|B))=3-1\neq1$$The second thing that seems apparent to me is that there is a natural inclination not to believe that in a paper a trivial error could occur. That's why I’m asking for a confirmation here and why people that have read all the information I’ve provided prefer thinking of miraculous conditions that can make the impossible be true. $\endgroup$ – Gino Strato Sep 18 '13 at 7:11

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