P-probmodel writes when the script runsDictionary of Applied Machine Learning · model
Numerical companion to the entry model: it recomputes what the entry states and prints one line per check
One block per paragraph of the entry (marked [P...]): each block verifies numerically what the corresponding statement asserts. Self-contained (numpy/matplotlib only), fixed seed.
Run it without installing anything:uv run https://dictionaryofml.org/terms/model.py
uv downloads this script and the pinned NumPy and Matplotlib it needs, then runs it; the script fetches any input file it uses. To keep the output files, download model.py into a folder and run uv run model.py there. With NumPy and Matplotlib already installed, python3 model.py, from any directory — it writes its output files into the current directory. Fixed seeds, so the printed numbers reproduce exactly. Download model.py · Notebook · Open in Colab
One cell per block of the script: the code, and what that code printed when it last ran here
"""
model.py — numerical companion to the glossary entry 'model'.
One block per paragraph of the entry (marked [P...]): each block verifies
numerically what the corresponding statement asserts. Self-contained
(numpy/matplotlib only), fixed seed.
Blocks
------
[P-hypospace] Sense 1 — the model as the hypothesis space: temperature
prediction commits to a set of candidate maps (three
linear maps, as in the entry's figure panel (a)); every
element is a map producing predictions.
[P-trained] Sense 2 — the trained model as the learned hypothesis:
the ML map A acts on a trainset and selects the single
element h-hat = A(D) of the hypothesis space
that best fits the trainset (panel (b)); software-library usage.
[P-probmodel] Sense 3 — the probabilistic model as a family of
probability distributions: each member is a data
generator, and datasets generated from the two members of
the family are statistically distinguishable (panel (c)).
Outputs
-------
model.png : preview figure (checking only).
Data generated by pythondemos/model.py.
"""
import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
from pathlib import Path
OUT_DIR = Path(__file__).parent
rng = np.random.default_rng(42)
report = []
def check(name, ok):
report.append((name, bool(ok)))
print(f" [{'ok' if ok else 'FAIL'}] {name}")
Sense 1 — the model as the hypothesis space: temperature prediction commits to a set of candidate maps (three linear maps, as in the entry's figure panel (a)); every element is a map producing predictions.
print("[P-hypospace] model as the hypothesis space (a set of maps)")
H = [lambda x: 0.8 * x + 5.0,
lambda x: 1.0 * x + 3.0,
lambda x: 1.2 * x + 1.0] # three candidate maps
x_morning = 6.0 # note: at x = 10 all three maps happen to agree
preds = [h(x_morning) for h in H]
check("the model is a set of candidate maps (three linear maps)",
len(H) == 3 and len(set(preds)) == 3)
check("every element is a map: it delivers a prediction",
all(np.isfinite(p) for p in preds))
[P-hypospace] model as the hypothesis space (a set of maps) [ok] the model is a set of candidate maps (three linear maps) [ok] every element is a map: it delivers a prediction
Sense 2 — the trained model as the learned hypothesis: the ML map A acts on a trainset and selects the single element h-hat = A(D) of the hypothesis space that best fits the trainset (panel (b)); software-library usage.
print("[P-trained] trained model = learned hypothesis A(D)")
m = 50
x_tr = rng.uniform(5, 20, m)
y_tr = 1.0 * x_tr + 3.0 + 0.3 * rng.normal(size=m) # truth = H[1]
emp_risk = [np.mean((y_tr - h(x_tr)) ** 2) for h in H]
h_hat = H[int(np.argmin(emp_risk))]
check("the ML map selects a single element of the hypothesis space",
h_hat in H)
check("A(D) picks the best-fitting map (the true map here)",
int(np.argmin(emp_risk)) == 1)
check("the trained model is itself a map (usable for prediction)",
np.isfinite(h_hat(12.0)))
[P-trained] trained model = learned hypothesis A(D) [ok] the ML map selects a single element of the hypothesis space [ok] A(D) picks the best-fitting map (the true map here) [ok] the trained model is itself a map (usable for prediction)
Sense 3 — the probabilistic model as a family of probability distributions: each member is a data generator, and datasets generated from the two members of the family are statistically distinguishable (panel (c)).
print("[P-probmodel] probabilistic model = family of distributions")
family = [{"mean": 0.0, "std": 1.0}, {"mean": 3.0, "std": 1.0}]
data0 = rng.normal(family[0]["mean"], family[0]["std"], 5000)
data1 = rng.normal(family[1]["mean"], family[1]["std"], 5000)
check("each member of the family generates iid data points",
data0.shape == (5000,) and data1.shape == (5000,))
check("datasets from different members are distinguishable",
abs(data0.mean() - data1.mean()) > 2.5)
check("the empirical means identify the generating member",
abs(data0.mean() - family[0]["mean"]) < 0.1
and abs(data1.mean() - family[1]["mean"]) < 0.1)
# ------------------------------------------------------------ preview
fig, ax = plt.subplots(1, 3, figsize=(10, 2.8))
xx = np.linspace(5, 20, 50)
for h, st in zip(H, ("--", "-", ":")):
ax[0].plot(xx, h(xx), st)
ax[0].set_xlabel("feature $x$"); ax[0].set_ylabel("prediction $h(x)$")
ax[0].set_title("(a) hypothesis space")
ax[1].plot(x_tr, y_tr, "o", ms=3, alpha=0.5)
ax[1].plot(xx, h_hat(xx), "r-")
ax[1].set_xlabel("feature $x$"); ax[1].set_ylabel("label $y$")
ax[1].set_title("(b) trained model $\\hat{h} = A(D)$")
ax[2].hist(data0, bins=40, alpha=0.6, density=True)
ax[2].hist(data1, bins=40, alpha=0.6, density=True, hatch="//")
ax[2].set_xlabel("observed value"); ax[2].set_ylabel("probability density")
ax[2].set_title("(c) probabilistic model")
fig.tight_layout()
fig.savefig(OUT_DIR / "model.png", dpi=110)
print(f"\n{sum(ok for _, ok in report)}/{len(report)} checks passed")
assert all(ok for _, ok in report)
[P-probmodel] probabilistic model = family of distributions [ok] each member of the family generates iid data points [ok] datasets from different members are distinguishable [ok] the empirical means identify the generating member 8/8 checks passed
P-probmodel writes when the script runs