diff --git a/assets/stan_demos/ar_k.stan b/assets/stan_demos/ar_k.stan
index 3de67f4..360a9cf 100644
--- a/assets/stan_demos/ar_k.stan
+++ b/assets/stan_demos/ar_k.stan
@@ -1,6 +1,6 @@
 data {
-  int<lower=0> K;  // model order
-  int<lower=0> N;  // dataset size
+  int<lower=1> K;    // model order
+  int<lower=K+1> N;  // dataset size
   vector[N] y;
 }
 parameters {
diff --git a/assets/stan_demos/ar_k_fa.stan b/assets/stan_demos/ar_k_fa.stan
index f0b57c9..01722d7 100644
--- a/assets/stan_demos/ar_k_fa.stan
+++ b/assets/stan_demos/ar_k_fa.stan
@@ -1,5 +1,6 @@
 data {
-  int<lower=0> N, K;
+  int<lower=1> K;    // model order
+  int<lower=K+1> N;  // dataset size
   array[N] int<lower=0,upper=1> free;
   vector[N] y;
 }
diff --git a/assets/stan_demos/imdb.stan b/assets/stan_demos/imdb.stan
index a4304b4..9dc61c6 100644
--- a/assets/stan_demos/imdb.stan
+++ b/assets/stan_demos/imdb.stan
@@ -1,13 +1,13 @@
 data {
-  int N;  // num products
-  array[N, 5] int ratings;
+  int<lower=1> N;  // num products
+  array[N, 5] int<lower=0> ratings;
 }
 parameters {
   vector<lower=0>[5] alpha;
   array[N] simplex[5] d;
 }
 model {
-  d ~ dirichlet(alpha);
+  d ~ dirichlet(alpha);  // for all d[i]
   for (i in 1:N) {
     ratings[i] ~ multinomial(d[i]);
   }
diff --git a/assets/stan_demos/imdb_inference.stan b/assets/stan_demos/imdb_inference.stan
index 4b4d104..b64c3ba 100644
--- a/assets/stan_demos/imdb_inference.stan
+++ b/assets/stan_demos/imdb_inference.stan
@@ -1,6 +1,6 @@
 data {
   vector<lower=0>[5] alpha;
-  array[5] int rating;
+  array[5] int<lower=0> rating;
 }
 parameters {
   simplex[5] d;
diff --git a/assets/stan_demos/inhomogeneous_poisson_process.stan b/assets/stan_demos/inhomogeneous_poisson_process.stan
index eecfa1a..fba02d3 100644
--- a/assets/stan_demos/inhomogeneous_poisson_process.stan
+++ b/assets/stan_demos/inhomogeneous_poisson_process.stan
@@ -1,5 +1,5 @@
 data {
-  int<lower=0> N; // Number of observed events
+  int<lower=1> N; // Number of observed events
   real<lower=0> T; // End of observation interval
   vector<lower=0, upper=T>[N] t; // Times of events
 }
diff --git a/assets/stan_demos/lasso_multivariate.stan b/assets/stan_demos/lasso_multivariate.stan
index 0553809..6a3e316 100644
--- a/assets/stan_demos/lasso_multivariate.stan
+++ b/assets/stan_demos/lasso_multivariate.stan
@@ -1,6 +1,6 @@
 data {
+  int<lower=1> K;
   int<lower=0> N;
-  int<lower=0> K;
   matrix[N, K] x;
   vector[N] y;
   real<lower=0> lambda;
diff --git a/assets/stan_demos/mlp.stan b/assets/stan_demos/mlp.stan
index 796fca6..2b3f87b 100644
--- a/assets/stan_demos/mlp.stan
+++ b/assets/stan_demos/mlp.stan
@@ -1,5 +1,5 @@
 data {
-    int<lower=0> M, N, K;
+    int<lower=1> M, N, K;
     matrix[M, N] x;
     array[M] int<lower=0,upper=1> y;
 }
diff --git a/assets/stan_demos/ols_multivariate.stan b/assets/stan_demos/ols_multivariate.stan
index fe58934..2004e57 100644
--- a/assets/stan_demos/ols_multivariate.stan
+++ b/assets/stan_demos/ols_multivariate.stan
@@ -1,6 +1,6 @@
 data {
-  int<lower=0> N;
-  int<lower=0> K;
+  int<lower=1> K;
+  int<lower=K+1> N;
   matrix[N, K] x;
   vector[N] y;
 }
diff --git a/assets/stan_demos/ols_univariate.stan b/assets/stan_demos/ols_univariate.stan
index 0eac821..a5c5a21 100644
--- a/assets/stan_demos/ols_univariate.stan
+++ b/assets/stan_demos/ols_univariate.stan
@@ -1,5 +1,5 @@
 data {
-  int<lower=0> N;
+  int<lower=2> N;
   vector[N] x;
   vector[N] y;
 }
diff --git a/assets/stan_demos/poisson_glm.stan b/assets/stan_demos/poisson_glm.stan
index 1f79102..dc63de3 100644
--- a/assets/stan_demos/poisson_glm.stan
+++ b/assets/stan_demos/poisson_glm.stan
@@ -1,6 +1,6 @@
 data {
-  int<lower=0> N;
-  int<lower=0> K;
+  int<lower=1> N;
+  int<lower=1> K;
   array[N] int<lower=0> y;
   matrix[N, K] x;  // features
 }
diff --git a/assets/stan_demos/ridge_multivariate.stan b/assets/stan_demos/ridge_multivariate.stan
index 2849b74..30969c9 100644
--- a/assets/stan_demos/ridge_multivariate.stan
+++ b/assets/stan_demos/ridge_multivariate.stan
@@ -1,6 +1,6 @@
 data {
+  int<lower=1> K;
   int<lower=0> N;
-  int<lower=0> K;
   matrix[N, K] x;
   vector[N] y;
   real<lower=0> lambda;
diff --git a/assets/stan_demos/trivial_learning.stan b/assets/stan_demos/trivial_learning.stan
index 57a3d23..e98e819 100644
--- a/assets/stan_demos/trivial_learning.stan
+++ b/assets/stan_demos/trivial_learning.stan
@@ -1,9 +1,9 @@
 data {
-  int N;
-  array[N] int<lower=0> x;
+  int<lower=1> N;  // num samples
+  array[N] int<lower=0> x; // data
 }
 parameters {
-  real<lower=0> lambda;
+  real<lower=0> lambda;  // to learn
 }
 model {
   x ~ poisson(lambda);
diff --git a/assets/stan_demos/trivial_learning_custom.stan b/assets/stan_demos/trivial_learning_custom.stan
index 88e241f..0ddcd23 100644
--- a/assets/stan_demos/trivial_learning_custom.stan
+++ b/assets/stan_demos/trivial_learning_custom.stan
@@ -8,7 +8,7 @@ functions {
   }
 }
 data {
-  int N;
+  int<lower=1> N;
   array[N] int<lower=0> x;
 }
 parameters {