Thank you for reply.
For the following data,
| Confidence | number of Hits (TP) | number of False alarms (FP) |
|---|---|---|
| 3 = definitely present | H_{3} | F_{3} |
| 2 = probably present | H_{2} | F_{2} |
| 1 = questionable | H_{1} | F_{1} |
where, H_{c},F_c \in \mathbb{N}. Number of images N_I . Number of lesions N_L.
the likelihood function is defined to be
f(H_1,H_2,H_3,F_1,F_2,F_3;N_I,N_L|\theta) = \dbinom{N_L}{H_3}p_3(\theta)^{H_3}(1-p_3(\theta))^{N_L-H_3}
\times \dbinom{N_L-H_3}{H_2}p_3(\theta)^{H_2}(1-p_3(\theta))^{N_L-H_3-H_2}
\times \dbinom{N_L-H_3-H_2}{H_1}p_3(\theta)^{H_1}(1-p_3(\theta))^{N_L-H_3-H_2-H_1}
\times \frac{q_3(\theta)^{F_3} } {F_3!}\exp(-F_3)
\times \frac{q_2(\theta)^{F_2} } {F_3!}\exp(-F_2)
\times \frac{q_1(\theta)^{F_1} } {F_1!}\exp(-F_1).
In particular, if F_2=F_3=0, then
f(H_1,H_2,H_3,F_1,F_2,F_3;N_I,N_L|\theta) = \dbinom{N_L}{H_3}p_3(\theta)^{H_3}(1-p_3(\theta))^{N_L-H_3}
\times \dbinom{N_L-H_3}{H_2}p_3(\theta)^{H_2}(1-p_3(\theta))^{N_L-H_3-H_2}
\times \dbinom{N_L-H_3-H_2}{H_1}p_3(\theta)^{H_1}(1-p_3(\theta))^{N_L-H_3-H_2-H_1}
\times \frac{q_3(\theta)^{0} } {0!}\exp(-0)
\times \frac{q_2(\theta)^{0} } {0!}\exp(-0)
\times \frac{q_1(\theta)^{F_1} } {F_1!}\exp(-F_1).
and thus I guess the likelihood is well defined.
I never even thought of that actually, thank you.