Very Very Very Brief Description

Issei Tsunoda


Time is money, and I have time but no money. Why? Nowadays, many people work in eight days a week! But, I work in zero days a week! Thus, time is no money.

Anyway, the author thinks the conventional notation would helps such no time people.

Conventional Notation

Recall that the conventional likelihood notation;

\[ f(y|\theta), \]

where \(y\) denotes data and \(\theta\) is a model parameter.

Data \(y\)

In case of a number of confidence levels \(C=5\).

Confidence Level No. of Hits No. of False alarms
5 = definitely present \(H_{5}\) \(F_{5}\)
4 = probably present \(H_{4}\) \(F_{4}\)
3 = equivocal \(H_{3}\) \(F_{3}\)
2 = probably \(H_{2}\) \(F_{2}\)
1 = questionable \(H_{1}\) \(F_{1}\)

where, \(H_{c},F_c \in \mathbb{N}\). Moreover Number of images \(N_I\) and Number of lesions \(N_L\).

So, in conventional notation we may write

\[y = (H_1,H_2,H_3,H_4,H_5, F_1,F_2,F_3,F_4,F_5 ;N_L,N_I) \in \mathbb{N}^{2C+2}.\]

Likelihood \(f(y|\theta)\)

\[ H_c \sim \text{Binomial}(p_c(\theta),N_L),\\ F_c \sim \text{Poisson}(q_c(\theta)).\\ \]

\[ p_c(\theta) := \int_{\theta_c}^{\theta_{c+1}}\text{Gaussian}_{}(x|\mu,\sigma)dx,\\ q_c(\theta) := \int_{\theta_c}^{\theta_{c+1}}N_I \times \frac{d \log \Phi(z)}{dz}dz. \]

where model parameter is \(\theta = (\theta_1,\theta_2,\theta_3,...\theta_C;\mu,\sigma)\) which should be estimated and \(\Phi\) denotes the cumulative distribution functions of the canonical Gaussian. Note that \(\theta_{C+1} = \infty\)



R script

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