chi_square_at_replicated_data_and_MCMC_samples_MRMC: chi square at replicated data drawn (only one time) from model with each MCMC samples.

Description

In order to pass this result to posterior predictive p value calculator.

Usage

chi_square_at_replicated_data_and_MCMC_samples_MRMC(StanS4class,
  summary = TRUE, seed = NA, serial.number = NA)

Arguments

StanS4class

An S4 object of class stanfitExtended which is an inherited class from the S4 class stanfit. This R object can be passed to the DrawCurves(), ppp() and ... etc

summary

Logical: TRUE of FALSE. Whether to print the verbose summary, i.e., logical; If TRUE then verbose summary is printed in the R console. If FALSE, the output is minimal. I regret, this variable name should be verbose.

seed

This is used only in programming phase. If seed is passed, then, in procedure indicator the seed is printed. This parameter is only for package development.

serial.number

An positive integer or Character. This is for programming perspective. The author use this to print the serial numbre of validation. This will be used in the validation function.

Value

From any given posterior MCMC samples $\theta_1,\theta_2,...,\theta_i,....,\theta_n$ (provided by stanfitExtended object), it calculates a return value as a vector of the form $\chi(y_i|\theta_i),i=1,2,....$, where each dataset $y_i$ is drawn from a likelihood $likelihood(.|\theta_i)$, namely,

$$y_i ~ likelihood(.| \theta_i).$$

The return value also retains $y_i$.

Revised 2019 Dec. 2

Details

For a given dataset $D_0$, let $\pi(|D_0)$ be a posterior of the given data $D_0$, then we can draw poterior samples.

$$\theta_1 ~ \pi(.| D_0),$$ $$\theta_2 ~ \pi(.| D_0),$$ $$\theta_3 ~ \pi(.| D_0),$$ $$....,$$ $$\theta_n ~ \pi(.| D_0).$$

We let $f(|\theta)$ be a likelihood function. Then we can draw samples in only one time from the collection of likelihoods $likelihood(\\theta_1),likelihood(\\theta_2),...,likelihood(\\theta_n)$.

$$y_1 ~ likelihood(.| \theta_1),$$ $$y_2 ~ likelihood(.| \theta_2),$$ $$y_3 ~ likelihood(.| \theta_3),$$ $$....,$$ $$y_n ~ likelihood(.| \theta_n).$$

Altogether, using these pair of samples $(y_i, \theta_i), i= 1,2,...,n$ we calculates the return value of this function. That is,

$$\chi(y_1|\theta_1),$$ $$\chi(y_2|\theta_2),$$ $$\chi(y_3|\theta_3),$$ $$....,$$ $$\chi(y_n|\theta_n).$$

This is contained in a return value,

so the return value is a vector of length is the number of MCMC iterations except the burn-in period.

Application of this return value: calculate the so-called Posterior Predictive P value.

In other functions, the author use this function with many seeds, namely, chaning seed, we can obtain

$$y_1^1,y_1^2,y_1^3,...,y_1^j,....,y_1^J ~ likelihood ( . |\theta_1), $$ $$y_2^1,y_2^2,y_2^3,...,y_2^j,....,y_2^J ~ likelihood ( . |\theta_2),$$ $$y_3^1,y_3^2,y_3^3,...,y_3^j,....,y_3^J ~ likelihood ( .|\theta_3),$$ $$...,$$ $$y_i^1,y_i^2,y_i^3,...,y_i^j,....,y_i^J ~ likelihood ( . |\theta_i),$$ $$...,$$ $$y_I^1,y_I^2,y_I^3,...,y_I^j,....,y_I^J ~ likelihood ( . |\theta_I),$$

And thus, we will obatin

whih are used when we calculate the so-called Posterior Predictive P value.

Revised 2019 Sept. 8 Revised 2019 Dec. 2

Examples

Run this code

# NOT RUN {

# }
# NOT RUN {
  fit <- fit_Bayesian_FROC( ite  = 1111,  dataList = ddd )
 a <- chi_square_at_replicated_data_and_MCMC_samples_MRMC(fit)

# }
# NOT RUN {

# }

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