I think for the users of this package the following definitions are no need to understand nor remember. To tell the truth, the author sometimes forget these definitions when I develop this packages or running this code. So, do not bother yourself even if you can not understand the following things. Especially, I think for the medical researcher or radiologists it is difficult to understand. I studied mathematics (Differential Geometry) in some university, so it is very easy for me, but I do not want to force the users of this package to understand these precise definitions, and it is the reason why I put these in Appendix.
\(\Phi()\) denotes the cumulative distribution function of the Gaussian distribution with mean 0 and variance 1. \(\Phi^{-1}()\) is the inverse mapping of \(\Phi()\).
Let \((x,y)\) be a Cartesian coordinate system on a plane. Then, for any non-negative real number \(\lambda\), the FROC curve is defined as \[ (x(\lambda),y(\lambda)) := \bigl(\lambda , 1-\Phi(b\Phi ^{-1}(\exp(-\lambda)) -a ) \bigr) \] Or
\[ (x(\lambda),y(\lambda)) := \bigl(\lambda , \text{Prob} \bigr\{ X_i > \Phi ^{-1}(\exp(-\lambda)) \bigl\} \bigr). \] It is easy to see that these two definition is same.
Note that AUC of FROC is infinity. So, in this package AUC means the next AFROC curve’s AUC.
Similarly, let \((\xi, \eta)\) be a Cartesian coordinate system for the square \(0\leq \xi \leq 1, 0\leq \eta \leq 1\). Then, for any non-negative real number \(\lambda\), we define an AFROC curve as \[(\xi(\lambda), \eta(\lambda)) := \bigl(1-\exp(-\lambda) , 1-\Phi(b\Phi ^{-1}(\exp(-\lambda)) -a ) \bigr). \]
we define \(C\) points as follows: \[ (x_{c},y_{c}):=( \sum_{c\leq c' \leq C } F_{c^\prime}/N_{I}, \sum_{c\leq c' \leq C } H_{c^\prime}/N_{L}), \] where \(c\) is the \(c^{\text{th}}\) confidence level, and we call these points the cumulative false positive-cumulative true positive (CFP-CTP) points over \((N_I, N_L)\).
In the case of single reader and single modality, this is used if ModifiedPoisson = FALSE in BayesianFROC::fit_Bayesian_FROC(). IF ModifiedPoisson = FALSE is true then the following points are depicted instead:
\[ (x_{c},y_{c}):=( \sum_{c\leq c' \leq C } F_{c^\prime}/N_{L}, \sum_{c\leq c' \leq C } H_{c^\prime}/N_{L}), \]
The following model are implemented in the function BayesianFROC::fit_MRMC_versionTWO().
\[H_{c,m,r} \sim \text{Binomial}( p_{c,m,r}, N_L ), \] \[F_{c,m,r} \sim \text{Poisson} ( ( \lambda _{c} - \lambda _{c+1})N_L ), \] \[\lambda _{c} = - \log \Phi (z_{c }), \] \[p_{c,m,r}
:= \Phi (\frac{z_{c +1}-\mu_{m,r}}{\sigma_{m,r}})-\Phi (\frac{z_{c}-\mu_{m,r}}{\sigma_{m,r}}), \]
\[A_{m,r} := \Phi (\frac{\mu_{m,r}/\sigma_{m,r}}{\sqrt{(1/\sigma_{m,r})^2+1}}), \]
\[A_{m,r} \sim \text{Normal} (A_{m},\sigma_{r}^2), \] \[\mu_{m,r} \sim \text{Normal} (\mu_{m},\sigma _{\mu,m}^2), \] \[\sigma_{m,r} \sim \text{Normal} (\sigma_{m},\sigma _{\sigma,m}^2), \] \[ a_{m} :=\mu_{m} / \sigma_{m} \] \[ b_{m} := 1/ \sigma_{m} \] \[dz_c := z_{c+1}-z_{c}, \] \[dz_c, \sigma_{m,r} \sim \text{Uniform}(0,\infty), \] \[z_{c} \sim \text{Uniform}( -\infty,100000), \] \[A_{m} \sim \text{Uniform}(0,1). \]
We introduce a new parameter: \[ dz_c:=z_{c+1}-z_{c}, \] where \(c=1,2,\cdots,C\). In practical modeling, we do not use the previous section’s parameters \(z_2,z_3,\cdots, z_C\), using \(z_{c}, dz_1,dz_2,\cdots, dz_{C}\) instead. To include an order constraint for Bayesian models, we assume a non-regular uniform prior for \(dz_c\): \[ dz_c \sim \text{Uniform}(0, \infty), \] where \(\text{Uniform}(0,\infty)\) indicates improper priors whose integrated values are not one. These priors are equivalent in that we assume monotonicity in the thresholds, as follows: \[ z_{1}\leq z_{2}\leq z_{3} \leq \dots \leq z_{C+1}.\] Once we apply the above priors for monotonicity to the breaking of symmetry, then the Poisson rates \(\lambda_{c}\) automatically inherit this order constraint, as follows: \[ \lambda_{1}\geq \lambda_{2}\geq \lambda_{3} \geq \dots \geq \lambda_{C+1}.\] Furthermore, we must restrict the highest threshold \(z_{C+1}\), which theoretically is infinity. However, computing the estimate requires assuming its upper bounds. \[ z_{C+1} \sim \text{Uniform}( -\infty,100000). \] where \(\text{Uniform}(-\infty,100000)\) means improper priors as above. If we do not place an upper bound o \(z_{C+1}\), then \(z_{C+1}\) is not convergent in Markov Chain Monte Carlo (MCMC) sampling. Our computation uses the upper bound of 100,000. Its value does not affect the other parameters significantly except for \(z_{C+1}\). Note that when we programmed in C++ for the Stan language, we used the array format for thresholds \(z_c\), leading to some unnecessarily strong constraints, as follows: \[ z_{c} \sim \text{Uniform}( -\infty,100000), \] for all \(c\) (not only \(C+1\)). We omit the graphical model of this modified version because of its complexity. Consequently, we have the Bayesian model \[ H_{c } \sim \text{Binomial} ( p_{c}, N_{L} ),\] \[F_{c } \sim \text{Poisson} ( (\lambda _{c} -\lambda _{c+1} )\times N_{I} ),\] \[\lambda _{c} = - \log \Phi ( z_{c } ),\] \[p_{c} = \Phi (\frac{z_{c +1}-\mu}{\sigma})-\Phi (\frac{z_{c}-\mu}{\sigma}). \] \[dz_c := z_{c+1}-z_{c},\] \[dz_c \sim \text{Uniform}(0,\infty),\] \[z_{c} \sim \text{Uniform}( -\infty,100000).\]
Our new model has parameters \(z_{1}, dz_1,dz_2,\cdots, dz_{C}\), \(\mu\), and \(\sigma\). Note that \(z_2, z_3, \cdots, z_C\) are no longer the parameters of this symmetry-breaking model.
If user has any questions, please tell me.
tsunoda.issei1111
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