In addition, this function plots replicated datasets from model at each MCMC sample generated by HMC.
Using the Hamiltonian Monte Carlo Sampling: HMC.
we can draw the MCMC
samples of size \(n\), say $$\theta_1, \theta_2, \theta_3,...,\theta_n $$,
namely,
$$\theta_1 \sim \pi(.|D), $$
$$\theta_2 \sim \pi(.|D), $$
$$\theta_3 \sim \pi(.|D),$$
$$...,$$
$$\theta_n \sim \pi(.|D).$$
where \(\pi(\theta|D)\) is the posterior for given data \(D\).
Then, the function plots the following datasets \(y_1^1,y_2^1,...,y_I^J\).
$$y_1^1,y_1^2,y_1^3,...,y_1^j,....,y_1^J \sim L ( . |\theta_1),$$
$$y_2^1,y_2^2,y_2^3,...,y_2^j,....,y_2^J \sim L ( . |\theta_2),$$
$$y_3^1,y_3^2,y_3^3,...,y_3^j,....,y_3^J \sim L ( .|\theta_3), $$
$$...,$$
$$y_i^1,y_i^2,y_i^3,...,y_i^j,....,y_i^J \sim L ( . |\theta_i),$$
$$...,$$
$$y_I^1,y_I^2,y_I^3,...,y_I^j,....,y_I^J \sim L ( . |\theta_I),$$
where \(L ( . |\theta_i)\) is a likelihood at parameter \(\theta_i\).
Let \( \chi(y|\theta) \) be a chi square goodness of fit statistics of our hierarchical Bayesian Model
$$\chi(y|\theta) := \sum_{r=1}^R \sum_{m=1}^M \sum_{c=1}^C ( \frac { ( H_{c,m,r}-N_L\times p_{c,m,r})^2}{N_L\times p_{c,m,r}} + \frac{(F_{c,m,r}-(\lambda_{c} -\lambda_{c+1} )\times N_{L})^2}{(\lambda_{c} -\lambda_{c+1} )\times N_{L} }).$$
and a chi square goodness of fit statistics of our non-hierarchical Bayesian Model
$$\chi(y|\theta) := \sum_{c=1}^C \biggr( \frac{( H_{c}-N_L\times p_{c})^2}{N_L\times p_{c}} + \frac{(F_{c}-(\lambda_{c} -\lambda_{c+1} ) )\times N_{L}]^2}{(\lambda_{c} -\lambda_{c+1} )\times N_{L} }\biggr).$$
where a dataset \(y\) denotes \( (F_{c,m,r}, H_{c,m,r}) \) in MRMC case and \( (F_{c}, H_{c}) \) in a single reader and a single modality case,
and model parameter \(\theta\).
Then we can calculate the posterior predictive p value for a given dataset \(y_0\).
$$ \int \int I( \chi(y|\theta) > \chi(y_0|\theta) ) f(y|\theta) \pi(\theta|y_0) d \theta d y $$
$$ \approx \int \sum_i I( \chi(y|\theta_i) > \chi(y_0|\theta_i) ) f(y|\theta_i) d y $$
$$ \approx \sum_{j=1}^J \sum_{i=1}^I I( \chi(y_i^j|\theta_i) > \chi(y_0|\theta_i) ) $$
When we plot these replicated data \(y_i^j\), we use the jitter() which adds a small amount of noise to avoid overlapping points.
For example, jitter(c(1,1,1,1)) returns values: 1.0161940 1.0175678 0.9862400 0.9986126, which is changed
from 1,1,1,1 to be not exactly 1 by adding tiny errors to avoid overlapping. I love you. 2019 August 19
Nowadays, I cannot remove my self from some notion, such as honesty, or pain, or,.. maybe these thing is no longer with myself.
This programm is made to fix previous release calculation. Now, this programm calculates correct p value.
So... I calculate the ppp for MCMC and Graphical User Interface based on Shiny for MRMC, which should be variable such as
number of readers, modalities, to generate such ID vectors automatically. Ha,... tired! Boaring, I want to die...t, diet!!
Tinko, tinko unko unko. Manko manko. ha.
Leberiya, he will be die, ha... he cannot overcome, very old, old guy.
I will get back to meet him. Or I cannot meet him? Liberiya,...very wisdom guy,
Ary you already die? I will get back with presents for you. Ball, I have to throgh ball, and he will catch it.
The reason why the author made the plot of data drawn from Posterior Predictive likelihoods with each MCMC parameters is
to understand our programm is correct, that is, each drawing is very mixed. Ha,.... when wright this,... I always think who read it.
I love you, Ruikobach. Ruikobach is tiny and tiny, but,... cute. Ruikosan...Ruiko...
But he has time only several years. He will die, he lives sufficiently so long, ha.
Using this function, user would get reliable posterior predictive p values, Cheers! Pretty Crowd!
We note that the calculation of posterior perdictive p value (PPP) relies on the law of large number.
Thus, in order to obtain the relicable PPP, we need to enough large MCMC samples to approximate
the double integral of PPP.
For example, the MCMC samples is small, then R hat is far from 1 but, the low MCMC samples leads
us to incorrect p value which sometimes said that the model is correct even if the R hat criteria
reject the MCMC results.