35:"/_next/static/chunks/0_6qdgbl18thf.js" 36:"/_next/static/chunks/3316o_1ddm9ab.js" 37:I[96601,["$1","$22","$23","$35","$36"],"default"] 38:I[58709,["$1","$22","$23","$35","$36"],"default"] 39:I[80423,["$1","$22","$23","$35","$36"],"default"] 3a:T6fd,import numpy as np import matplotlib.pyplot as plt rng = np.random.default_rng(66) n = 400 runs_at_10 = rng.normal(85, 12, n).clip(45, 130) wickets_at_10 = rng.integers(0, 7, n).astype(float) venue_par = rng.choice([160, 170, 180, 190], n).astype(float) final = 60 + 1.55 * runs_at_10 - 6 * wickets_at_10 + 0.35 * (venue_par - 175) + rng.normal(0, 14, n) X = np.column_stack([runs_at_10, wickets_at_10, venue_par, np.ones(n)]) idx = rng.permutation(n) tr, te = idx[:300], idx[300:] coef, *_ = np.linalg.lstsq(X[tr], final[tr], rcond=None) fitted = X[tr] @ coef resid = final[tr] - fitted local = np.random.default_rng(7) preds = [] for _ in range(1000): y_star = fitted + local.choice(resid, size=len(tr), replace=True) c, *_ = np.linalg.lstsq(X[tr], y_star, rcond=None) noise = local.choice(resid, size=len(te), replace=True) preds.append(X[te] @ c + noise) preds = np.vstack(preds) lo = np.percentile(preds, 5, axis=0) hi = np.percentile(preds, 95, axis=0) point = X[te] @ coef covered = float(np.mean((final[te] >= lo) & (final[te] <= hi))) mae = float(np.mean(np.abs(final[te] - point))) worst = int(np.argmax(np.abs(final[te] - point))) print("coverage", round(covered, 3), "mae", round(mae, 2)) print("coverage in 0.85 to 0.95", 0.85 <= covered <= 0.95) print("worst miss, predicted", round(float(point[worst]), 1), "actual", round(float(final[te][worst]), 1)) show = np.arange(len(te)) yerr = np.vstack([np.maximum(point[show] - lo[show], 0), np.maximum(hi[show] - point[show], 0)]) plt.errorbar(show, point[show], yerr=yerr, fmt="o", color="#2563eb", ecolor="#93c5fd", label="90 percent interval") plt.scatter(show, final[te][show], color="#b45309", s=18, label="actual", zorder=3) plt.xlabel("held-out innings") plt.ylabel("final score") plt.legend() plt.show() 3b:T5a8,import numpy as np rng = np.random.default_rng(66) n = 1000 km = rng.uniform(0.5, 10, n) prep = rng.normal(14, 5, n).clip(3, 40) hour = rng.integers(11, 24, n) rain = (rng.random(n) < 0.2).astype(float) peak = (((hour >= 12) & (hour <= 14)) | ((hour >= 19) & (hour <= 22))).astype(float) minutes = 6 + 2.4 * km + 0.8 * prep + 6 * rain + 5 * peak + (rng.gamma(2, 2, n) - 4) X = np.column_stack([km, prep, rain, peak, np.ones(n)]) idx = rng.permutation(n) tr, te = idx[:800], idx[800:] mu, sigma = X[tr, :4].mean(0), X[tr, :4].std(0) sigma = np.where(sigma == 0, 1, sigma) def pack(rows): Z = rows.copy() Z[:, :4] = (Z[:, :4] - mu) / sigma return Z coef, *_ = np.linalg.lstsq(pack(X[tr]), minutes[tr], rcond=None) pred_tr = pack(X[tr]) @ coef pred_te = pack(X[te]) @ coef resid = minutes[tr] - pred_tr promise = pred_te + np.percentile(resid, 90) mae = float(np.mean(np.abs(minutes[te] - pred_te))) kept = float(np.mean(minutes[te] <= promise)) print("mae", round(mae, 2), "arrived by the promise", round(kept, 3)) print("mae under 4", mae < 4, "at least 88 percent kept", kept >= 0.88) print("promise rule: prediction plus the 90th percentile of training residuals") abs_err = np.abs(minutes[te] - pred_te) boot_idx = rng.integers(0, len(te), size=(2000, len(te))) mae_boot = abs_err[boot_idx].mean(axis=1) mae_lo, mae_hi = np.percentile(mae_boot, [2.5, 97.5]) print("mae interval", round(float(mae_lo), 2), round(float(mae_hi), 2)) 3c:T31e6,import numpy as np text = "arjun plays a cover drive\nthe cover drive races to the boundary\nfour runs for arjun\narjun plays a pull shot\nthe pull shot races to the boundary\nfour runs for arjun\narjun plays a straight drive\nthe straight drive races to the boundary\nfour runs for arjun\narjun plays a late cut\nthe late cut races to the boundary\nfour runs for arjun\narjun faces a yorker\nthe bowler bowls a yorker\narjun faces a bouncer\nthe bowler bowls a bouncer\narjun faces a full toss\nthe bowler bowls a full toss\narjun faces a slower ball\nthe bowler bowls a slower ball\narjun takes a single\narjun calls for a second run\nwicket falls and arjun walks back\nrohan plays a cover drive\nthe cover drive races to the boundary\nfour runs for rohan\nrohan plays a pull shot\nthe pull shot races to the boundary\nfour runs for rohan\nrohan plays a straight drive\nthe straight drive races to the boundary\nfour runs for rohan\nrohan plays a late cut\nthe late cut races to the boundary\nfour runs for rohan\nrohan faces a yorker\nthe bowler bowls a yorker\nrohan faces a bouncer\nthe bowler bowls a bouncer\nrohan faces a full toss\nthe bowler bowls a full toss\nrohan faces a slower ball\nthe bowler bowls a slower ball\nrohan takes a single\nrohan calls for a second run\nwicket falls and rohan walks back\nmeera plays a cover drive\nthe cover drive races to the boundary\nfour runs for meera\nmeera plays a pull shot\nthe pull shot races to the boundary\nfour runs for meera\nmeera plays a straight drive\nthe straight drive races to the boundary\nfour runs for meera\nmeera plays a late cut\nthe late cut races to the boundary\nfour runs for meera\nmeera faces a yorker\nthe bowler bowls a yorker\nmeera faces a bouncer\nthe bowler bowls a bouncer\nmeera faces a full toss\nthe bowler bowls a full toss\nmeera faces a slower ball\nthe bowler bowls a slower ball\nmeera takes a single\nmeera calls for a second run\nwicket falls and meera walks back\nthe required rate is climbing\nthe field is set deep\nthe batter keeps strike\na no ball gives a free hit\nthe over ends with a dot ball\nthe crowd wants a six\nthe captain moves the field\ndrinks break and the chase continues\norder one masala dosa\nplease pack one masala dosa\norder two masala dosa\nplease pack two masala dosa\nmasala dosa will arrive in thirty minutes\nextra raita with the masala dosa\norder one paneer tikka\nplease pack one paneer tikka\norder two paneer tikka\nplease pack two paneer tikka\npaneer tikka will arrive in thirty minutes\nextra raita with the paneer tikka\norder one veg biryani\nplease pack one veg biryani\norder two veg biryani\nplease pack two veg biryani\nveg biryani will arrive in thirty minutes\nextra raita with the veg biryani\norder one butter chicken\nplease pack one butter chicken\norder two butter chicken\nplease pack two butter chicken\nbutter chicken will arrive in thirty minutes\nextra raita with the butter chicken\norder one idli sambar\nplease pack one idli sambar\norder two idli sambar\nplease pack two idli sambar\nidli sambar will arrive in thirty minutes\nextra raita with the idli sambar\norder one pav bhaji\nplease pack one pav bhaji\norder two pav bhaji\nplease pack two pav bhaji\npav bhaji will arrive in thirty minutes\nextra raita with the pav bhaji\norder one vada pav\nplease pack one vada pav\norder two vada pav\nplease pack two vada pav\nvada pav will arrive in thirty minutes\nextra raita with the vada pav\norder one filter coffee\nplease pack one filter coffee\norder two filter coffee\nplease pack two filter coffee\nfilter coffee will arrive in thirty minutes\nextra raita with the filter coffee\nthe rider is near the gate\nthe restaurant is preparing the order\nplease leave the order at the door\nthe soup is too spicy\nadd less chilli please\nthe bill is two hundred rupees\ncancel the extra roti\nthe kitchen is running late\nupi payment of rupees fifty received\npayment of rupees fifty failed please retry\nyou paid rupees fifty at the chai stall\ncashback of rupees twenty on rupees fifty\nupi payment of rupees one hundred received\npayment of rupees one hundred failed please retry\nyou paid rupees one hundred at the chai stall\ncashback of rupees twenty on rupees one hundred\nupi payment of rupees two hundred received\npayment of rupees two hundred failed please retry\nyou paid rupees two hundred at the chai stall\ncashback of rupees twenty on rupees two hundred\nupi payment of rupees five hundred received\npayment of rupees five hundred failed please retry\nyou paid rupees five hundred at the chai stall\ncashback of rupees twenty on rupees five hundred\nthe pin was incorrect\nthe bank declined the payment\na refund is on the way\ndo not share the otp\nthe train to pune is on platform one\nthe train to pune is on platform two\nthe train to pune is on platform three\nthe train to pune is on platform four\nthe train to pune is running late\npassengers for pune please board\nthe train to jaipur is on platform one\nthe train to jaipur is on platform two\nthe train to jaipur is on platform three\nthe train to jaipur is on platform four\nthe train to jaipur is running late\npassengers for jaipur please board\nthe train to lucknow is on platform one\nthe train to lucknow is on platform two\nthe train to lucknow is on platform three\nthe train to lucknow is on platform four\nthe train to lucknow is running late\npassengers for lucknow please board\nthe train to surat is on platform one\nthe train to surat is on platform two\nthe train to surat is on platform three\nthe train to surat is on platform four\nthe train to surat is running late\npassengers for surat please board\nthe train to indore is on platform one\nthe train to indore is on platform two\nthe train to indore is on platform three\nthe train to indore is on platform four\nthe train to indore is running late\npassengers for indore please board\nthe train to nagpur is on platform one\nthe train to nagpur is on platform two\nthe train to nagpur is on platform three\nthe train to nagpur is on platform four\nthe train to nagpur is running late\npassengers for nagpur please board\nthe next train arrives in ten minutes\nmind the gap on the platform\nthis coach is reserved for the journey\nthe announcement says the train is delayed\na clean shot from arjun\narjun finds the gap with a clean shot\na loud shot from arjun\narjun finds the gap with a loud shot\na late shot from arjun\narjun finds the gap with a late shot\na quick shot from arjun\narjun finds the gap with a quick shot\na clean shot from rohan\nrohan finds the gap with a clean shot\na loud shot from rohan\nrohan finds the gap with a loud shot\na late shot from rohan\nrohan finds the gap with a late shot\na quick shot from rohan\nrohan finds the gap with a quick shot\na clean shot from meera\nmeera finds the gap with a clean shot\na loud shot from meera\nmeera finds the gap with a loud shot\na late shot from meera\nmeera finds the gap with a late shot\na quick shot from meera\nmeera finds the gap with a quick shot\nmint chutney with the masala dosa\nsweet chutney with the masala dosa\nplain chutney with the masala dosa\nmint chutney with the paneer tikka\nsweet chutney with the paneer tikka\nplain chutney with the paneer tikka\nmint chutney with the veg biryani\nsweet chutney with the veg biryani\nplain chutney with the veg biryani\nmint chutney with the butter chicken\nsweet chutney with the butter chicken\nplain chutney with the butter chicken\nmint chutney with the idli sambar\nsweet chutney with the idli sambar\nplain chutney with the idli sambar\nmint chutney with the pav bhaji\nsweet chutney with the pav bhaji\nplain chutney with the pav bhaji\nmint chutney with the vada pav\nsweet chutney with the vada pav\nplain chutney with the vada pav\nmint chutney with the filter coffee\nsweet chutney with the filter coffee\nplain chutney with the filter coffee\nthe morning train to pune is expected soon\nthe evening train to pune is expected soon\nthe morning train to jaipur is expected soon\nthe evening train to jaipur is expected soon\nthe morning train to lucknow is expected soon\nthe evening train to lucknow is expected soon\nthe morning train to surat is expected soon\nthe evening train to surat is expected soon\nthe morning train to indore is expected soon\nthe evening train to indore is expected soon\nthe morning train to nagpur is expected soon\nthe evening train to nagpur is expected soon\nthe masala dosa is hot today\nthe masala dosa is mild today\nthe masala dosa is fresh today\nthe paneer tikka is hot today\nthe paneer tikka is mild today\nthe paneer tikka is fresh today\nthe veg biryani is hot today\nthe veg biryani is mild today\nthe veg biryani is fresh today\nthe butter chicken is hot today\nthe butter chicken is mild today\nthe butter chicken is fresh today\nthe idli sambar is hot today\nthe idli sambar is mild today\nthe idli sambar is fresh today\nthe pav bhaji is hot today\nthe pav bhaji is mild today\nthe pav bhaji is fresh today\nthe vada pav is hot today\nthe vada pav is mild today\nthe vada pav is fresh today\nthe filter coffee is hot today\nthe filter coffee is mild today\nthe filter coffee is fresh today\nsplit the bill of rupees fifty\nsend rupees fifty to the hostel\nsplit the bill of rupees one hundred\nsend rupees one hundred to the hostel\nsplit the bill of rupees two hundred\nsend rupees two hundred to the hostel\nsplit the bill of rupees five hundred\nsend rupees five hundred to the hostel\narjun plays a cover drive and the ball races to the boundary\nrohan faces a yorker and takes a single\nmeera calls for a second run and keeps strike\norder one masala dosa and one filter coffee\nveg biryani will arrive in thirty minutes at the gate\nupi payment of rupees two hundred received from the chai stall\nthe train to pune is on platform four and running late\nplease pack two pav bhaji and extra raita" lines = [ln.strip() for ln in text.splitlines() if ln.strip()] rng = np.random.default_rng(66) order = rng.permutation(len(lines)) cut = max(1, int(0.8 * len(lines))) train_lines = [lines[i] for i in order[:cut]] test_lines = [lines[i] for i in order[cut:]] def tokenize(line): return ["~~"] + line.lower().split() + ["~~"] train = [tokenize(ln) for ln in train_lines] test = [tokenize(ln) for ln in test_lines] def counts(seqs, k): vocab = sorted({w for seq in seqs for w in seq}) stoi = {w: i for i, w in enumerate(vocab)} unk = len(vocab) size = unk + 1 uni = np.zeros(size) bi = np.zeros((size, size)) for seq in seqs: ids = [stoi.get(w, unk) for w in seq] for a, b in zip(ids, ids[1:]): uni[b] += 1 bi[a, b] += 1 uni_p = (uni + k) / (uni.sum() + k * size) bi_p = (bi + k) / (bi.sum(axis=1, keepdims=True) + k * size) return uni_p, bi_p, vocab, unk def index_of(word, vocab, unk): try: return vocab.index(word) except ValueError: return unk def ppl(seqs, uni_p, bi_p, vocab, unk, kind): logs = [] for seq in seqs: ids = [index_of(w, vocab, unk) for w in seq] for a, b in zip(ids, ids[1:]): prob = uni_p[b] if kind == "uni" else bi_p[a, b] logs.append(np.log(prob)) return float(np.exp(-np.mean(logs))) val_cut = max(1, int(0.8 * len(train))) val = train[val_cut:] train_fit = train[:val_cut] best_k, best_ppl = None, None for k in (1.0, 0.1, 0.01): uni_p, bi_p, vocab, unk = counts(train_fit, k) score = ppl(val, uni_p, bi_p, vocab, unk, "bi") print("k", k, "val perplexity", round(score, 2)) if best_ppl is None or score < best_ppl: best_k, best_ppl = k, score k = best_k train_fit = train print("chosen k", k) uni_p, bi_p, vocab, unk = counts(train_fit, k) print("lines", len(lines), "vocab", len(vocab)) print("uniform perplexity", len(vocab) + 1) print("unigram test perplexity", round(ppl(test, uni_p, bi_p, vocab, unk, "uni"), 2)) print("bigram test perplexity", round(ppl(test, uni_p, bi_p, vocab, unk, "bi"), 2)) print("smallest bigram probability", float(bi_p.min())) local = np.random.default_rng(67) def sample_sentence(temperature): word = "" out = [] for _ in range(12): row = bi_p[index_of(word, vocab, unk)] logits = np.log(row) / temperature logits = logits - logits.max() probs = np.exp(logits) probs = probs / probs.sum() nxt = int(local.choice(np.arange(len(row)), p=probs)) token = vocab[nxt] if nxt < len(vocab) else "" if token == "": break out.append(token) word = token return " ".join(out) for temperature in (0.7, 1.2): print("temperature", temperature) for _ in range(3): print(sample_sentence(temperature)) 3d:T6fd,import numpy as np import matplotlib.pyplot as plt rng = np.random.default_rng(66) n = 400 runs_at_10 = rng.normal(85, 12, n).clip(45, 130) wickets_at_10 = rng.integers(0, 7, n).astype(float) venue_par = rng.choice([160, 170, 180, 190], n).astype(float) final = 60 + 1.55 * runs_at_10 - 6 * wickets_at_10 + 0.35 * (venue_par - 175) + rng.normal(0, 14, n) X = np.column_stack([runs_at_10, wickets_at_10, venue_par, np.ones(n)]) idx = rng.permutation(n) tr, te = idx[:300], idx[300:] coef, *_ = np.linalg.lstsq(X[tr], final[tr], rcond=None) fitted = X[tr] @ coef resid = final[tr] - fitted local = np.random.default_rng(7) preds = [] for _ in range(1000): y_star = fitted + local.choice(resid, size=len(tr), replace=True) c, *_ = np.linalg.lstsq(X[tr], y_star, rcond=None) noise = local.choice(resid, size=len(te), replace=True) preds.append(X[te] @ c + noise) preds = np.vstack(preds) lo = np.percentile(preds, 5, axis=0) hi = np.percentile(preds, 95, axis=0) point = X[te] @ coef covered = float(np.mean((final[te] >= lo) & (final[te] <= hi))) mae = float(np.mean(np.abs(final[te] - point))) worst = int(np.argmax(np.abs(final[te] - point))) print("coverage", round(covered, 3), "mae", round(mae, 2)) print("coverage in 0.85 to 0.95", 0.85 <= covered <= 0.95) print("worst miss, predicted", round(float(point[worst]), 1), "actual", round(float(final[te][worst]), 1)) show = np.arange(len(te)) yerr = np.vstack([np.maximum(point[show] - lo[show], 0), np.maximum(hi[show] - point[show], 0)]) plt.errorbar(show, point[show], yerr=yerr, fmt="o", color="#2563eb", ecolor="#93c5fd", label="90 percent interval") plt.scatter(show, final[te][show], color="#b45309", s=18, label="actual", zorder=3) plt.xlabel("held-out innings") plt.ylabel("final score") plt.legend() plt.show() 29:["$","div",null,{"className":"min-h-screen bg-gray-50 dark:bg-gray-900 pt-20","children":[["$","$L37",null,{"topicSlug":"ml-math","topicTitle":"Math for Machine Learning","dayNumber":66,"dayTitle":"Capstone: IPL, delivery time, or a bigram model","sprintNumber":3,"estimatedTime":"60-90 minutes","totalDays":66}],["$","div",null,{"className":"w-full px-4 sm:px-6 lg:px-8 py-8","children":["$","$L38",null,{"dayNumber":66,"topicSlug":"ml-math","children":["$","$L39",null,{"day":{"dayNumber":66,"sprintNumber":3,"title":"Capstone: IPL, delivery time, or a bigram model","slug":"capstone","description":"Pick one: predict an IPL score and show an honest range, promise a delivery time you can keep, or build a bigram model and score it by perplexity. All three generate their data in the snippet, with a fixed seed, so nothing depends on the network. The IPL option is a linear model of the final score from the score at 10 overs, with a 90 percent prediction interval from a residual bootstrap. The delivery option is a linear model of minutes, plus a promise set at the 90th percentile of the training residuals. The language option is the corpus from days 57 and 59, all of it this time, with the smoothing chosen on a slice of the training lines.","duration":"60-90 minutes","lastUpdated":"2026-10-28","learningGoals":[{"id":"d66-g1","text":"Choose one capstone and run it to its done rule","measurable":"The printed numbers meet the table under the code."},{"id":"d66-g2","text":"Report a range or a baseline, not only a point","measurable":"Coverage, a promise rate, or a perplexity against the unigram."},{"id":"d66-g3","text":"Name one failure in a sentence","measurable":"The worst miss, a late order, or a nonsense sample."}],"whyThisMatters":{"productionStory":"Pick one: predict an IPL score and show an honest range, promise a delivery time you can keep, or build a bigram model and score it by perplexity.","realWorldContext":"All three generate their data in the snippet, with a fixed seed, so nothing depends on the network. The IPL option is a linear model of the final score from the score at 10 overs, with a 90 percent prediction interval from a residual bootstrap. The delivery option is a linear model of minutes, plus a promise set at the 90th percentile of the training residuals. The language option is the corpus from days 57 and 59, all of it this time, with the smoothing chosen on a slice of the training lines.","industryRelevance":"As of 2026, a model card wants the data, the metric, an interval or a baseline, and a failure. A single impressive number is not a result. The browser can run all three. There is no GPU and no download."},"coreContent":{"sections":[{"id":"why","type":"concept","title":"Why this matters","order":1,"content":{"explanation":"Pick one: predict an IPL score and show an honest range, promise a delivery time you can keep, or build a bigram model and score it by perplexity."}},{"id":"intuition","type":"concept","title":"Intuition","order":2,"content":{"explanation":"Read the table before you pick. Each option ends with a number and a reason to trust it. The IPL interval is honest when about 90 percent of held-out innings fall inside it. The delivery promise is honest when most test orders arrive by the time you quoted. The bigram is honest when it beats the unigram on lines it did not train on, and when an unseen pair does not score infinity. The first snippet is the IPL model, and it is what the editor opens. The other two are on the page."}},{"id":"formula","type":"concept","title":"The formula, in words","order":3,"content":{"explanation":"Read the symbols only after the sentence above makes sense.","formula":{"latex":"\\mathrm{report} = \\hat{y}\\ \\mathrm{with}\\ [q_{5\\%}, q_{95\\%}]\\ \\mathrm{or}\\ \\mathrm{PPL}_{\\mathrm{test}}","plainEnglish":"Every capstone ends with a number and its uncertainty, not a single boast.","symbols":[{"symbol":"y-hat","meaning":"the prediction"},{"symbol":"q","meaning":"a percentile of the bootstrap, or of the training residuals"},{"symbol":"PPL","meaning":"test perplexity"}]}}},{"id":"code","type":"hands-on","title":"Code it","order":4,"content":{"explanation":"Run one option. The editor opens the IPL model. The delivery model and the bigram are the next two snippets. Each finishes in a few seconds. Do not report a training-set score.","codeExamples":[{"title":"ipl.py","language":"python","code":"$3a","explanation":"400 synthetic innings, seed 66. A 90 percent interval is the 5th and 95th percentiles of 1,000 residual-bootstrap predictions, including a fresh residual so the interval covers the noise and not only the fitted line. The plot marks each held-out inning with its prediction interval and the actual score."},{"title":"delivery.py","language":"python","code":"$3b","explanation":"1,000 synthetic orders, seed 66. The promise is the prediction plus the 90th percentile of the training residuals. MAE should be under 4 minutes, and at least 88 percent of test orders should arrive by the promise. A percentile bootstrap of the test absolute errors prints a 95 percent interval for the MAE."},{"title":"bigram.py","language":"python","code":"$3c","explanation":"The full original corpus, seed 66. Smoothing is chosen on a slice of the training lines. Test perplexity must beat the unigram, and sampling must stay finite."}],"table":{"headers":["Option","Done when"],"rows":[["IPL score","90 percent interval coverage on 100 test innings is between 85 and 95 percent. Test MAE is printed. The plot shows predicted versus actual with interval bars. One sentence names the worst miss."],["Delivery time","Test MAE is under 4 minutes, and a bootstrap interval for that MAE is printed. At least 88 percent of test orders arrive by the promise. The promise rule is one sentence."],["Bigram model","Test perplexity beats the unigram. Unseen pairs stay finite. Three samples at temperature 0.7 and three at 1.2. One sentence names a failure."]]}}},{"id":"ml","type":"concept","title":"Where this shows up in ML","order":5,"content":{"explanation":"Whichever option you ran, the write-up is the same shape. What the data was. What you measured. The interval or the baseline. One failure you can point at. What you would measure next. That is a model card at the size of this course. As of 2026, that card is the difference between a demo and a result."}},{"id":"check","type":"practice","title":"Quick check","order":6,"content":{"quickCheck":[{"question":"What makes the IPL interval honest?","options":["About 90 percent of held-out innings fall inside the 90 percent interval","The training error is zero","The interval is as narrow as possible"],"answerIndex":0,"explanation":"Coverage on data the fit did not see is the test. A narrow interval that misses is not honest."},{"question":"Why can a bigram beat a unigram?","options":["It trains on the test lines","It uses the previous word as context","It has a larger vocabulary"],"answerIndex":1,"explanation":"The previous word narrows the menu. The unigram uses the same probability every time."},{"question":"A promise of prediction plus the 90th percentile of training residuals means","options":["Every order arrives exactly on the prediction","The MAE must be zero","About 90 percent of orders should arrive by the promise"],"answerIndex":2,"explanation":"Ninety percent of the past misses were smaller than that padding. The test set checks whether that still holds."}]}}]},"challenges":[{"id":"ml-d66-basecamp","difficulty":"basecamp","title":"Basecamp: run your option","description":"Run one of the three snippets and copy the numbers the done-table asks for.","instructions":["Pick IPL, delivery, or the bigram.","Keep the seed.","Print the coverage, the promise rate, or the two perplexities."],"starterCode":"print('Pick ipl.py, delivery.py, or bigram.py from the lesson')\n","hints":["The editor opens the IPL snippet. Paste another if you want it.","A training-set number does not count."]},{"id":"ml-d66-ridge","difficulty":"ridge","title":"Ridge: change one knob","description":"Change one feature, one promise percentile, or the smoothing, and show the effect with a second number.","instructions":["Change exactly one thing.","Rerun.","Print the before and after of the metric that should move."],"starterCode":"print('one change, two numbers')\n","hints":["For IPL, drop wickets and watch the MAE.","For the bigram, force k = 1 and watch the test perplexity."]},{"id":"ml-d66-summit","difficulty":"summit","title":"Summit: a short model card","description":"Print five lines: the data, the metric with its interval or baseline, one failure, and what you would measure next.","instructions":["Use the numbers you actually printed.","The failure is a sentence about a row or a sample, not a slogan.","Do not claim a cause."],"starterCode":"print('data:')\nprint('metric:')\nprint('interval or baseline:')\nprint('failure:')\nprint('next:')\n","hints":["Coverage, a kept-promise rate, or bigram versus unigram is the metric.","If you did not run the code, you do not have a card."]}],"commonPitfalls":[{"id":"d66-p1","title":"Reporting the training score","whyItHappens":"The model has seen those rows or those lines. The number is a souvenir, not a test.","howToAvoid":"Split first, with the fixed seed, and report the held-out metric."},{"id":"d66-p2","title":"Reading the formula before the picture","whyItHappens":"The symbols look like the important part because they are compact.","howToAvoid":"Say the idea in a sentence you could tell a friend, then match each symbol to a word."}],"connections":{"current":{"keyTakeaways":["A capstone is a prediction with a range, a promise you can keep, or a perplexity against a baseline.","The write-up names the data, the metric, a failure, and what you would measure next."]},"next":{"preview":"You finished Math for Machine Learning.","teaser":"You finished Math for Machine Learning. The AI Builder's Expedition roadmap links back here whenever a day needs math."}},"reflectionQuestions":[{"id":"d66-r1","question":"Which option did you pick, and which done-rule was the one you almost missed?","purpose":"Notice the gap while it is small."},{"id":"d66-r2","question":"What would you refuse to claim about your number?","purpose":"Tie the number to a consequence."}],"resources":[{"title":"Andrej Karpathy: building makemore","type":"video","url":"https://www.youtube.com/watch?v=PaCmpygFfXo","description":"A bigram language model, then a small network. English.","priority":"recommended"},{"title":"StatQuest: Bootstrapping Main Ideas","type":"video","url":"https://www.youtube.com/watch?v=Xz0x-8-cgaQ","description":"Resampling with replacement, in plain language.","priority":"essential"},{"title":"Mathematics for Machine Learning, chapters 6 and 8","type":"book","url":"https://mml-book.github.io/book/mml-book.pdf","description":"Deisenroth, Faisal, and Ong. Free PDF. Probability, then models meeting data.","priority":"recommended"},{"title":"Seeing Theory","type":"tool","url":"https://seeing-theory.brown.edu/","description":"Interactive pictures for probability, Bayes, intervals, and regression.","priority":"recommended"}],"metadata":{"estimatedTime":"60-90 minutes","difficulty":"summit","tags":["ml-math","sprint-3","capstone"]}},"topicSlug":"ml-math","navigation":{"previous":{"dayNumber":65,"title":"Reading ML notation in papers","sprintNumber":3},"next":null},"dayNumber":66,"totalDays":66,"mathEnabled":true,"starterCode":"$3d"}]}]}]]}]