فحوصات التنبؤ البعدي

ما فحوصات التنبؤ اللاحق؟

استخراج العينات اللاحقة

# library(rstan)
# mu_samples    <- extract(fit, pars = 'mu')$mu
# sigma_samples <- extract(fit, pars = 'sigma')$sigma
#
# cat('Samples drawn:', length(mu_samples), '
')
# cat('Posterior mean of mu:', mean(mu_samples), '
')
# cat('90% CI:', quantile(mu_samples, c(0.05, 0.95)), '
')

توليد yrep من العينات اللاحقة

# y <- c(2.1, 1.8, 2.4, 1.9, 2.3, 2.0, 1.7, 2.2)
# n_obs <- length(y)
# S     <- length(mu_samples)  # 4000 posterior draws
#
# yrep <- matrix(NA, nrow = S, ncol = n_obs)
# for (s in seq_len(S)) {
#   yrep[s, ] <- rnorm(n_obs, mean = mu_samples[s], sd = sigma_samples[s])
# }
# dim(yrep)  # [4000, 8]

ppc_dens_overlay() — مقارنة الكثافات

# library(bayesplot)
#
# ppc_dens_overlay(y, yrep[1:50, ])
#
# Interpretation:
# - Dark line (y_obs) surrounded by light lines (yrep): good fit
# - Dark line systematically outside the cloud: model misfit
# - Light lines much wider than dark: overdispersed model
# - Light lines much narrower than dark: underdispersed model

ppc_stat() — فحص إحصاء الاختبار

# library(bayesplot)
#
# ppc_stat(y, yrep, stat = 'mean')   # does model capture the mean?
# ppc_stat(y, yrep, stat = 'sd')     # does model capture spread?
# ppc_stat(y, yrep, stat = 'max')    # does model capture extremes?
#
# If observed stat is in the tail of the histogram,
# the model fails to reproduce that statistic.

القيمة الاحتمالية البايزية

# Bayesian p-value for the mean:
# obs_mean <- mean(y)
# rep_means <- apply(yrep, 1, mean)
# pval <- mean(rep_means >= obs_mean)
# cat('Bayesian p-value (mean):', round(pval, 3), '
')
# # 0.5 is perfect; < 0.05 or > 0.95 suggests misfit

دوال PPC إضافية في bayesplot

# library(bayesplot)
#
# # Grid of 8 simulated histograms vs the observed
# ppc_hist(y, yrep[1:8, ])
#
# # Scatter: y_obs (x) vs mean of yrep (y) — should hug diagonal
# ppc_scatter_avg(y, yrep)
#
# # 50% and 90% posterior predictive intervals around each y_i
# ppc_intervals(y, yrep)

تفسير مخططات PPC — سوء ملاءمة النموذج

# Example: if data has a long right tail but yrep does not,
# consider switching:
# y ~ normal(mu, sigma)  =>  y ~ student_t(nu, mu, sigma)
#
# Or for count data:
# y ~ poisson(lambda)  =>  y ~ neg_binomial_2(mu, phi)  (overdispersion)
cat('PPCs guide model improvement by revealing specific failure modes
')

PPCs في كتلة نموذج Stan

# Stan model with generated quantities:
# '
# generated quantities {
#   array[N] real y_rep;
#   for (n in 1:N) {
#     y_rep[n] = normal_rng(mu, sigma);
#   }
# }
# '
# Then extract in R:
# yrep <- extract(fit, pars = 'y_rep')$y_rep  # [S, N] matrix

التحقق المتقاطع مع ترك مشاهدة واحدة

# library(loo)
# loo1 <- loo(fit1)  # normal model
# loo2 <- loo(fit2)  # student-t model
#
# comparison <- loo_compare(loo1, loo2)
# print(comparison)
#
# Model with elpd_diff > 0 is preferred
# se_diff > |elpd_diff| means difference is not reliable

أفضل ممارسات PPC

# Workflow:
# 1. Fit model -> extract() -> generate yrep matrix
# 2. ppc_dens_overlay(y, yrep[1:50,])  -- visual global check
# 3. ppc_stat(y, yrep, stat='mean')    -- check mean
# 4. ppc_stat(y, yrep, stat='sd')      -- check spread
# 5. ppc_stat(y, yrep, stat='max')     -- check tails
# 6. If misfit found -> revise model -> refit -> re-check

تحقق سريع: تفسير القيمة الاحتمالية البايزية

مراجعة فحوصات التنبؤ اللاحق