# A question on Bayesian Search

I chanced on this article on wikipedia on Bayesian search. In the mathematics section, it states how the posteriors are estimated. While I understand how $p^{'}$ is calculated, I can't seem to figure out how the $r^{'}$ are updated. How does the posterior on other cells on the grid change given a no-find on a given grid cell? If someone could give any pointers it would be helpful. Thanks

I hope I can help you. First, remember the most simple statement of Bayes' theorem: $$P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$$ Let's assume that we have only two grid cells to search. Denote the prior probability that the wreck is in grid cell 1 as $P(W_{1})=p$, the prior probability that the wreck is in grid cell 2 as $P(W_{2})=r$, the probability of finding the wreck in a certain grid cell if it is really there as $P(F|W)=q$. Let's say that grid cell 1 has been searched but no wreck has been found. The probability that the wreck is in grid cell 1 although it hasn't been found there is given by: $$p'=P(W_{1}| \overline{ F}_{1})=\frac{P(\overline{ F}_{1}|W_{1})\cdot P(W_{1})}{P(\overline{F}_{1})}=\frac{P(\overline{ F}_{1}|W_{1})\cdot P(W_{1})}{P(\overline{F}_{1}|W_{1}) \cdot P(W_{1}) +P(\overline{F}_{1}|\overline{W}_{1}) \cdot P(\overline{W}_{1})} = \frac{(1-q)\cdot p}{(1-q)\cdot p + 1\cdot (1-p)}= \frac{(1-q)\cdot p}{1-pq}$$ For the other grid cell 2, we can proceed in the same manner: $$r'=P(W_{2}| \overline{ F}_{1})=\frac{P(\overline{ F}_{1}|W_{2})\cdot P(W_{2})}{P(\overline{F}_{1})}=\frac{1\cdot r}{(1-q)\cdot p + 1\cdot (1-p)}=\frac{1\cdot r}{1-pq}$$ Note that the denominator is the same for both situations. Also the probability of not finding the wreck in grid cell 1 when it is in fact in grid 2 (or in any other grid cell apart from cell 1) is assumed to be 1 (i.e. $P(\overline{ F}_{1}|W_{2})=1$).
• Not quite. Lets say there is another cell with its prior being set to $r$, then its posterior is $r\frac{1}{1 - pq}$. How is this arrived at? I understand why it would change and go up, but don't follow how it is calculated to that expression Jul 31 '13 at 0:41