update

Author: AlexandreAndr

notebooks/Notebook_1/Notebook_1b_Noise_and_Modeling_Data.ipynb

@@ -418,41 +418,22 @@ def plot_pdf_cdf_accumulation_gif(
 print("standard deviation:", np.std(v))
 
 # %% [markdown] id="nb1b-summaries-are-lossy"
-# #### What the Mean and Variance Miss
-#
-# The mean says *where* the data sits, the variance *how far it spreads*. Both are
-# useful, and neither pins down the distribution. Three samples make the point:
-#
-# - **narrow**: normal, $\mu = 0$, $\sigma = 1$
-# - **wide**: normal, $\mu = 0$, $\sigma = 2.5$ — same center, different spread, so the mean alone cannot tell it from *narrow*
-# - **bimodal**: half the samples near $-2.4$, half near $+2.4$, each with spread $0.7$
-#
-# The bimodal one is the counterexample. By symmetry its mean is $0$, and its
-# variance splits into the spread *of* the modes plus the spread *within* a mode:
-#
-# $$\sigma^2 = \underbrace{2.4^2}_{\text{between modes}} + \underbrace{0.7^2}_{\text{within a mode}} = 5.76 + 0.49 = 6.25 = 2.5^2$$
-#
-# So it matches *wide* on **both** statistics (up to sampling noise) while looking
-# nothing like it: for the normal, $0$ is the most common value; for the bimodal, $0$
-# is one of the *rarest*. The last column below makes that concrete.
+# These two numbers are a useful summary, but only a summary. Data with the same mean
+# can be tightly clustered or wildly spread out, and — less obviously — two samples
+# can agree on *both* the mean and the variance and still look nothing alike. All
+# three samples below are centered at $0$; the last two even share a spread, yet the
+# bimodal one almost never takes values near its own mean. Plot the distribution
+# before trusting its summary.
 
 # %% id="nb1b-summaries-are-lossy-data"
-narrow = rng.normal(loc=0.0, scale=1.0, size=5000)  # sigma = 1
-wide = rng.normal(loc=0.0, scale=2.5, size=5000)  # sigma = 2.5
-
-# Bimodal mixture built to match `wide` on mean AND variance.
-centers = np.repeat([-2.4, 2.4], 2500)  # exactly half the samples in each mode
-bimodal = centers + rng.normal(loc=0.0, scale=0.7, size=5000)
+narrow = rng.normal(loc=0.0, scale=1.0, size=5000)
+wide = rng.normal(loc=0.0, scale=2.5, size=5000)
+# Two lumps instead of one, tuned to match `wide`'s mean and spread.
+bimodal = np.repeat([-2.4, 2.4], 2500) + rng.normal(loc=0.0, scale=0.7, size=5000)
 
 samples = [("narrow", narrow), ("wide", wide), ("bimodal", bimodal)]
 for name, sample in samples:
-    mu, sd = np.mean(sample), np.std(sample)
-    # A shape probe the two statistics cannot see: how much mass sits near the mean.
-    near_mean = np.mean(np.abs(sample - mu) <= 0.5)
-    print(
-        f"{name:>7}:  mean = {mu:5.2f}   std = {sd:5.2f}"
-        f"   within 0.5 of mean = {100 * near_mean:4.1f}%"
-    )
+    print(f"{name:>7}:  mean = {np.mean(sample):5.2f}   std = {np.std(sample):5.2f}")
 
 # %%
 #| code-fold: true
@@ -463,17 +444,13 @@ def plot_pdf_cdf_accumulation_gif(
 
 for a, (_, sample), color, title in zip(ax, samples, colors, titles):
     sns.histplot(sample, bins=40, stat="density", kde=True, color=color, ax=a)
-    mu, sd = np.mean(sample), np.std(sample)
-    a.axvline(mu, color="red", linewidth=1.5, label="Mean")
-    for edge, lbl in [(mu - sd, "Mean $\\pm$ 1 std"), (mu + sd, None)]:
-        a.axvline(edge, color="gray", linestyle="--", linewidth=1, label=lbl)
-    a.set_title(f"{title}\n(mean {mu:.2f}, std {sd:.2f})")
+    a.axvline(np.mean(sample), color="red", linewidth=1.5, label="Mean")
+    a.set_title(f"{title}\n(mean {np.mean(sample):.2f}, std {np.std(sample):.2f})")
     a.set_xlabel("Value")
 
 ax[0].legend(fontsize=8)
 
-# Overlay the wide Gaussian on the bimodal panel: identical mean and variance,
-# completely different shape.
+# Overlay the wide normal on the bimodal panel: same mean and spread, different shape.
 sns.kdeplot(
     wide,
     color="darkorange",
@@ -484,25 +461,10 @@ def plot_pdf_cdf_accumulation_gif(
 )
 ax[2].legend(fontsize=8)
 
-plt.suptitle("Mean and Variance Do Not Determine the Distribution")
+plt.suptitle("Same Summary Statistics, Different Distributions")
 plt.tight_layout()
 plt.show()
 
-# %% [markdown] id="nb1b-summary-stats-lossy"
-# ::: {.callout-important title="Key concept: summary statistics are lossy"}
-# The mean and variance are a *compression* of the data, not a substitute for it.
-# Panels 1 and 2 share a mean; panels 2 and 3 share a mean *and* a variance. Plot the
-# distribution before trusting its summary — a two-lump histogram usually means the
-# data mixes two populations or regimes, and the "average" describes a value that
-# almost never occurs.
-# :::
-#
-# **Quick discussion.**
-#
-# 1. For the bimodal sample, is the mean a good prediction of a typical value? What would a better one-number summary be?
-# 2. If a model is trained with mean squared error on the bimodal data, what value will it be pulled toward, and why is that a problem?
-#
-
 # %% [markdown] id="Uri-wUVZniM9"
 # ### Why This Matters for Modeling (Now and Later)
 #