Very Very Very Brief Description of MRMC

Issei Tsunoda

2019-08-02

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 \(y\)

2 readers, 2 modalities and 3 confidence levels.

Confidence Level Modality ID Reader ID Number of Hits Number of False alarms
3 = definitely present 1 1 \(H_{3,1,1}\) \(F_{3,1,1}\)
2 = equivocal 1 1 \(H_{2,1,1}\) \(F_{2,1,1}\)
1 = questionable 1 1 \(H_{1,1,1}\) \(F_{1,1,1}\)
3 = definitely present 1 2 \(H_{3,1,2}\) \(F_{3,1,2}\)
2 = equivocal 1 2 \(H_{2,1,2}\) \(F_{2,1,2}\)
1 = questionable 1 2 \(H_{1,1,2}\) \(F_{1,1,2}\)
3 = definitely present 2 1 \(H_{3,2,1}\) \(F_{3,2,1}\)
2 = equivocal 2 1 \(H_{2,2,1}\) \(F_{2,2,1}\)
1 = questionable 2 1 \(H_{1,2,1}\) \(F_{1,2,1}\)
3 = definitely present 2 2 \(H_{3,2,2}\) \(F_{3,2,2}\)
2 = equivocal 2 2 \(H_{2,2,2}\) \(F_{2,2,2}\)
1 = questionable 2 2 \(H_{1,2,2}\) \(F_{1,2,2}\)

where, each component \(H\) and \(F\) are non negative integers. By the multi-index notation, for example, \(H_{3,2,1}\) means the hit or the \(1\)-st reader over all images taken by \(2\)-nd modality with reader’s confidence level is \(3\).

So, in conventional notation we may write

\[y = (H_{c,m,r},F_{c,m,r} ;N_L,N_I).\]

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

\[ H_{c,m,r} \sim \text{Binomial}(p_{c,m,r}(\theta),N_L),\\ F_{c,m,r} \sim \text{Poisson}(q_c(\theta)).\\ \]

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

\[ A_{m,r} := \Phi (\frac{\mu_{m,r}/\sigma_{m,r}}{\sqrt{(1/\sigma_{m,r})^2+1}}), \\ A_{m,r} \sim \text{Normal} (A_{m},\sigma_{r}^2), \\ \]

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

Prior

\[ dz_c := z_{c+1}-z_{c},\\ dz_c, \sigma_{m,r} \sim \text{Uniform}(0,\infty),\\ z_{c} \sim \text{Uniform}( -\infty,100000),\\ A_{m} \sim \text{Uniform}(0,1).\\ \]

This is only example, and in this package I implement proper priors. The author thinks the above prior is intuitively most suitable non informative priors.

Of course I know it (uniform prior) is not suitable in some sence, Jeffrays prior is more .. but I won’t do such discussion. Only intuitive. Ha,.. I am only amateur. I am no responsiblity and no money, no position. Imagine there no position,…Oh,…I no need to imagine since I already realize it. Ha ha ha :-D but no possetion makes no money but it makes freedom, unfortunately, my body has pains from all my body by multiple chemical sensitivity. I want to d…

R script

d <- dataset_creator_new_version()



fit <- fit_Bayesian_FROC(
   ite  = 1111, 
   cha = 1,
   summary = TRUE,  
   Null.Hypothesis = F, 
   dataList = d )