# Very Very Very Brief Description of MRMC

#### 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 )