I appreciate your response and thank you for taking the time. I am pleased to inform that I have improved the predictions by simplifying the model. Specifically, I have removed the latent predictor variable and instead incorporated a log-link function. Please find the revised model attached for your review.
with pm.Model() as model:
alpha_noise = pm.Beta("alpha_noise", 2, 1)
beta_noise = pm.Beta("beta_noise", 2, 5)
gamma_noise = pm.Beta("gamma_noise", 2, 5)
count_data_est = at.exp(alpha_noise + beta_noise + gamma_noise)
r = pm.Exponential("r", 0.5)
obs_data = pm.NegativeBinomial("obs_data", mu=count_data_est, alpha=1/r, observed=xExpirm)
trace = pm.sample(1000, tune=1000,target_accept = 0.9)