# Very Very Very Brief Description

#### 2019-08-02

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 $$\theta_{C+1} = \infty$$

Omitted

## R script

##### Graphical User Interface: GUI
library(BayesianFROC)
BayesianFROC::fit_GUI() #   Enjoy fitting!

Or

library(BayesianFROC)
fit_GUI_dashboard() #   Enjoy fitting!

Or

library(BayesianFROC)
fit_GUI_simple() #   Enjoy fitting!
##### Characteristic User Interface: CUI

dat <- list(c=c(3,2,1),    #Confidence level. Note that c is ignored.
h=c(97,32,31), #Number of hits for each confidence level
f=c(1,14,74),  #Number of false alarms for each confidence level

NL=259,        #Number of lesions
NI=57,         #Number of images
C=3)           #Number of confidence level

#  where,
#      c denotes confidence level, i.e., rating of reader.
#                3 = Definitely deseased,
#                2 = subtle,.. deseased
#                1 = very subtle
#      h denotes number of hits (True Positives: TP) for each confidence level,
#      f denotes number of false alarms (False Positives: FP) for each confidence level,
#      NL denotes number of lesions,
#      NI denotes number of images,

fit <- BayesianFROC::fit_Bayesian_FROC(  dataList = d )
##### SBC
Simulation_Based_Calibration_single_reader_single_modality_via_rstan_sbc()