Update app.py

app.py

@@ -21,7 +21,7 @@ This demo computes a Bayesian Ridge Regression of Sinusoids.
 
 In general, when fitting a curve with a polynomial by Bayesian ridge regression, the selection of initial values of the regularization parameters (alpha, lambda) may be important. This is because the regularization parameters are determined by an iterative procedure that depends on initial values.
 
-In this demo, the sinusoid is approximated by a cubic polynomial using different pairs of initial values.
+In this demo, the sinusoid is approximated by a cubic polynomial using different pairs of initial values.  By evaluating log marginal likelihood (L) of these models, we can determine which one is better. It can be concluded that the model with larger L is more likely.
 
 The demo is based on the [scikit-learn docs](https://scikit-learn.org/stable/auto_examples/linear_model/plot_bayesian_ridge_curvefit.html#sphx-glr-auto-examples-linear-model-plot-bayesian-ridge-curvefit-py)
 """
@@ -40,37 +40,53 @@ def curve_fit(size, alpha, lam):
     X_test = np.vander(x_test, n_order + 1, increasing=True)
     reg = BayesianRidge(tol=1e-6, fit_intercept=False, compute_score=True)
 
-    fig, axes = plt.subplots(1, 2, figsize=(8, 4))
-    results = []
-    for i, ax in enumerate(axes):
+    fig1, ax1 = plt.subplots(1, 1, figsize=(8, 4))
+    fig2, ax2 = plt.subplots(1, 1, figsize=(8, 4))
+
     # Bayesian ridge regression with different initial value pairs
-        if i == 0:
-            init = [1 / np.var(y_train), 1.0]  # Default values
-        elif i == 1:
-            init = [alpha, lam]
-            reg.set_params(alpha_init=init[0], lambda_init=init[1])
-        reg.fit(X_train, y_train)
-        ymean, ystd = reg.predict(X_test, return_std=True)
+    init = [1 / np.var(y_train), 1.0]  # Default values
+    reg.fit(X_train, y_train)
+    ymean, ystd = reg.predict(X_test, return_std=True)
+    
+    ax1.plot(x_test, func(x_test), color="blue", label="sin($2\\pi x$)")
+    ax1.scatter(x_train, y_train, s=50, alpha=0.5, label="observation")
+    ax1.plot(x_test, ymean, color="red", label="predict mean")
+    ax1.fill_between(
+        x_test, ymean - ystd, ymean + ystd, color="pink", alpha=0.5, label="predict std"
+    )
+    ax1.set_ylim(-1.3, 1.3)
+    ax1.legend()
+    title = "$\\alpha$_init$={:.2f},\\ \\lambda$_init$={}$".format(init[0], init[1])
+    title += " (Default)"
+    ax1.set_title(title, fontsize=12)
+    text = "$\\alpha={:.1f}$\n$\\lambda={:.3f}$\n$L={:.1f}$".format(
+        reg.alpha_, reg.lambda_, reg.scores_[-1]
+    )
+    ax1.text(0.05, -1.0, text, fontsize=12)
+
+    init = [alpha, lam]
+    reg.set_params(alpha_init=init[0], lambda_init=init[1])
+    reg.fit(X_train, y_train)
+    ymean, ystd = reg.predict(X_test, return_std=True)
+
+    ax2.plot(x_test, func(x_test), color="blue", label="sin($2\\pi x$)")
+    ax2.scatter(x_train, y_train, s=50, alpha=0.5, label="observation")
+    ax2.plot(x_test, ymean, color="red", label="predict mean")
+    ax2.fill_between(
+        x_test, ymean - ystd, ymean + ystd, color="pink", alpha=0.5, label="predict std"
+    )
+    ax2.set_ylim(-1.3, 1.3)
+    ax2.legend()
+    title = "$\\alpha$_init$={:.2f},\\ \\lambda$_init$={}$".format(init[0], init[1])
     
-        ax.plot(x_test, func(x_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.fill_between(
-            x_test, ymean - ystd, ymean + ystd, color="pink", alpha=0.5, label="predict std"
-        )
-        ax.set_ylim(-1.3, 1.3)
-        ax.legend()
-        title = "$\\alpha$_init$={:.2f},\\ \\lambda$_init$={}$".format(init[0], init[1])
-        if i == 0:
-            title += " (Default)"
-        ax.set_title(title, fontsize=12)
-        text = "$\\alpha={:.1f}$\n$\\lambda={:.3f}$\n$L={:.1f}$".format(
-            reg.alpha_, reg.lambda_, reg.scores_[-1]
-        )
-        ax.text(0.05, -1.0, text, fontsize=12)
-        results.append(ax)
+    title += " (Chosen)"
+    ax2.set_title(title, fontsize=12)
+    text = "$\\alpha={:.1f}$\n$\\lambda={:.3f}$\n$L={:.1f}$".format(
+        reg.alpha_, reg.lambda_, reg.scores_[-1]
+    )
+    ax2.text(0.05, -1.0, text, fontsize=12)
     
-    return results[0], results[1]
+    return fig1, fig2
 
 
 with gr.Blocks(theme=theme) as demo: