Improve description and add axes labels

app.py

@@ -32,8 +32,8 @@ def get_app_fn():
 
     def app_fn(n_samples: int, alpha_init: float, lambda_init: float):
         """Train a Bayesian Ridge regression model and plot the predicted points"""
-        
-        rng = np.random.RandomState(SEED) 
+
+        rng = np.random.RandomState(SEED)
         subset_idx = rng.randint(0, MAX_SAMPLES, n_samples)
 
         x_train, X_train, y_train = (
@@ -49,16 +49,18 @@ def get_app_fn():
         fig, ax = plt.subplots()
         ax.plot(x_test, y_test, color="blue", label="sin($2\\pi x$)")
         ax.scatter(x_train, y_train, s=50, alpha=0.5, label="observation")
-        ax.plot(x_test, ymean, color="red", label="predict mean")
+        ax.plot(x_test, ymean, color="red", label="predicted mean")
         ax.fill_between(
             x_test,
             ymean - ystd,
             ymean + ystd,
             color="pink",
             alpha=0.5,
-            label="predict std",
+            label="predicted std",
         )
         ax.set_ylim(-1.3, 1.3)
+        ax.set_xlabel("Cycles")
+        ax.set_ylabel("Amplitude")
         ax.legend()
         text = "$\\alpha={:.1f}$\n$\\lambda={:.3f}$\n$L={:.1f}$".format(
             reg.alpha_, reg.lambda_, reg.scores_[-1]
@@ -72,17 +74,26 @@ def get_app_fn():
 
 title = "Bayesian Ridge Regression"
 description = (
-    "This example shows the effect of different initial values for the regularisation paramters (alpha, lambda)."
-    "When starting from the default values (alpha_init = 1.90, lambda_init = 1.), the bias of the resulting curve is large, "
-    "and the variance is small. So, lambda_init should be relatively small (1.e-3) to reduce the bias."
+    "This demo is based on the [Bayesian Ridge Regression](https://scikit-learn.org/stable/auto_examples/linear_model/plot_bayesian_ridge_curvefit.html#curve-fitting-with-bayesian-ridge-regression) "
+    "example from scikit-learn.\n"
+    "The example shows the effect of different initial values for the regularisation parameters `alpha` and `lambda`. "
+    "When starting from the default values (`alpha_init = 1.90`, `lambda_init = 1.`), the bias of the resulting curve is large, "
+    "and the variance is small. So, `lambda_init` should be relatively small (e.g. `1.e-3`) to reduce the bias.\n"
+    "By evaluating log marginal likelihood (`L`) of these models, we can determine which one is better. A model with larger `L` is more likely."
 )
 with gr.Blocks(title=title) as demo:
     gr.Markdown(f"## {title}")
     gr.Markdown(description)
-    
-    n_samples_input = gr.Slider(minimum=5, maximum=100, value=25, step=1, label="#observations")
-    alpha_input = gr.Slider(minimum=0.001, maximum=5, value=1.9, step=0.01, label="alpha_init")
-    lambda_input = gr.Slider(minimum=0.001, maximum=5, value=1., step=0.01, label="lambda_init")
+
+    n_samples_input = gr.Slider(
+        minimum=5, maximum=100, value=25, step=1, label="#observations"
+    )
+    alpha_input = gr.Slider(
+        minimum=0.001, maximum=5, value=1.9, step=0.01, label="alpha_init"
+    )
+    lambda_input = gr.Slider(
+        minimum=0.001, maximum=5, value=1.0, step=0.01, label="lambda_init"
+    )
     outputs = gr.Plot(label="Output")
     inputs = [n_samples_input, alpha_input, lambda_input]
     app_fn = get_app_fn()