200 



FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



A'^,=the number of fish dying on the spawning 

 grounds during the i'" recovery period. 



Pi = 



The data available from a given experiment can 

 be laid out in a table as follows: 



Of course, 'E.T^^T and "^Ct^n. 



a i 



Now, the nuin])er of fish passing the tagging 

 point during a which die during period i might be 

 estimated by 



n „( = 



rria 



P.i 



(22) 



(I shall denote "estimate of" by the asterisk 

 herein) where Pai is the probability of a fish being 

 tagged during a and recovered during i. This 

 probability is unknown, and om' best available 

 estimate of it seems to be the joint probability 

 Piq^a, where these terms are as defined above. 

 This amounts to taking as the probability of 

 recovery the average probability of recovery of 

 all the fish passing the tagging point during a, 

 and as the probability of being tagged the average 

 probability of being tagged of all the fish dying 

 during period i. 



If the samples drawn for tagging and the samples 

 later drawn for tag ratios are representative of the 

 parts of the population from which they are 

 drawn, Pi and ga may be estimated from the data 

 as follows: 



'Z a = 



nia 



p-^^ m.j 



m.i 



The estimate of iiai is, then, given by 



«,' 



mat 



(24) 



which is equivalent to 



,(; la (yi 



it ai — '*'' a i • • • • 



m„. m.i 



The estimate of the total population is obtamod 

 bj' summing all these n*ai, thus 



TaCt 



N* = J2J2ma, 



(25) 



m„m.i 



A somewhat more rigorous derivation, based on 

 Bayes' theorem, has been suggested by Dr. 

 Crump: 



The problem is to estimate the Ua, and the qa- 

 if we can do this we can take as our estimate of A^, 



n * 

 a qc 



Let P(i/a) be the probability that a fish tagged 

 during the a'" period dies and is recovered during 

 the i'" recovery period. Now we have C, fish 

 taken diu"ing the i'" recovery period to be allo- 

 cated over the "a" tagging periods, and hence 

 we want the probability that a fish taken during 

 the i'" recovery period is one of those which passed 

 the tagging point during the a'" tagging period. 

 Denote by P{ali) the desired probability, and by 

 P{a) the true proportion of the ?i fish recovered 

 which passed the tagging point during the a'" 

 tagging period. Then by Bayes' theorem 



P{i/a)P(a) 



Piali)- 



'j:Piilcc)P{a) 



(26) 



we have the problem of estimating the P{i/a) 



and the P(a). 



Now, 



P(„)=-l-— 



and we may estimate P(a) by 



P*(a) = 



m. 



To estimate Pii/a) we may use 



rrici 



P*(ila) = 



rua 



Then our estimate of Pia/i) becomes 

 m..ma m.. 



P*{a/i) = 



nia 



Sr^ m„-ma 

 / vm..m„ 



m. 



(23) This gives us for an estimate of n„. 



■ri*ai=j;:c,p*iah)='z:r, 



m.i 



mai 



m.i 



(27) 



(28) 



(29) 



(30) 



