Week 12 — Ethics, Adversarial ML & the Semester Retrospective第12週 — 倫理・敵対的ML・学期の総括

Friday, July 17, 2026 · the last session2026年7月17日（金）· 最終回

Prof. Joseph Austerweilオウステウェイル　ジョセフ教授

Housekeeping事務連絡

• Final paper due next Friday, Jul 24, 8 PM (~6 pages).
• Today, in order: a short ethics lecture → the paper presentation → your final-project presentations → the semester retrospective.
• Presenters: submit your slides via the course site before or right after class.

• 最終レポート締切：来週金曜 7月24日 20:00（約6ページ）。
• 本日の流れ： 短い倫理講義 → 論文発表 → 最終プロジェクト発表 → 学期の総括。
• 発表者：スライドを提出（授業前後にコースサイトから）。

The last session, and it’s packed: keep housekeeping to under a minute. Order of the day: ~15 min ethics lecture, then the paper presentation (~20 min), then the three final-project presentations (~12 min each), then the ~29 min retrospective. Final paper is due next Friday. Remind presenters to submit slides.

Today本日の予定

• Ethics & adversarial ML — the machinery has a shadow倫理と敵対的ML — 仕組みには影がある 0:05
• Paper presentation論文発表 0:20
• Final-project presentations ×3最終プロジェクト発表 ×3 0:40
• Semester retrospective — the map of everything学期の総括 — 全体の地図 1:16

The shape of the day. The lecture content is deliberately tiny — one tight ethics segment — because the presentations and the retrospective own the session. The retrospective is the real climax.

Ethics & adversarial ML倫理と敵対的ML

Block 1, ~15 min. The spine: every tool this semester inherits or amplifies what it learns. Don’t lecture broadly — anchor each beat on a prior week so ethics falls OUT of the course rather than being bolted on. The deep treatment is in the Part IX textbook chapters; this is the trailer.

The machinery has a shadow仕組みには影がある

Everything we built learns from data — and inherits what the data carries. Three shadows, each a direct callback to a tool you already own:

• Bias — the embedding geometry from Week 11 encodes society’s stereotypes.
• Fairness — “fair” turns out to be several conditional probabilities that conflict.
• Brittleness — a tiny perturbation flips a confident classifier.

私たちが作ったものはすべてデータから学び — データが運ぶものを受け継ぐ。三つの影、それぞれすでに手にした道具への直接の呼び戻し：

• バイアス — 第11週の埋め込み幾何は社会のステレオタイプを符号化する。
• 公平性 — 「公平」は互いに衝突する複数の条件付き確率だと分かる。
• 脆さ — 微小な摂動が自信満々の分類器を反転させる。

Name the three beats and their callbacks up front so the segment feels like a payoff, not a new topic. Bias -> embeddings (Week 11 vectors). Fairness -> conditional probability (Part I). Brittleness -> the learned boundary (Part VIII). Then take each in one slide.

Embeddings carry society’s biases埋め込みは社会のバイアスを運ぶ

Week 11’s word vectors put occupation words on a gender axis nobody designed (Caliskan et al., 2017) — bias inherited from the corpus, in the geometry doing the useful work.

第11週の単語ベクトルが、誰も設計していない性別軸に職業語を並べる（Caliskan et al., 2017） — バイアスはコーパスから受け継がれ、有用な働きをする幾何に潜む。

Caliskan is the required reading — anchor here. The WEAT finding: word2vec/GloVe carry the same implicit-association biases as people. The callback that makes it land: this is the SAME geometry as the embedding widget from Week 11 — bias isn’t a separable module, it’s distributed in the vectors. Full treatment: the bias-in-data chapter (Part IX).

Fairness is a conditional-probability statement公平性は条件付き確率の主張

Write \(\hat{Y}\) = decision, \(Y\) = truth, \(G\) = group, \(\hat{S}\) = score.\(\hat{Y}\) = 判定、\(Y\) = 真実、\(G\) = 集団、\(\hat{S}\) = スコア。

• Demographic parity: \(P(\hat{Y}{=}1 \mid G)\) equal across groups
• Equalized odds: \(P(\hat{Y}{=}1 \mid Y, G)\) equal — matched error rates
• Calibration: \(P(Y{=}1 \mid \hat{S}, G)\) equal — a score means the same thing

• 人口統計的平等： \(P(\hat{Y}{=}1 \mid G)\) が集団間で等しい
• 均等化オッズ： \(P(\hat{Y}{=}1 \mid Y, G)\) が等しい — 誤り率が一致
• 較正： \(P(Y{=}1 \mid \hat{S}, G)\) が等しい — スコアの意味が同一

• Each is a conditional-probability equality — the objects from Part I.
• Fairness isn’t vague: you must choose which.

Next: they can’t all hold.

• それぞれが条件付き確率の等式 — Part Iの対象。
• 公平性は曖昧ではない：どれかを選ぶ。

次：全部は満たせない。

The intellectual core — and the thing a probability course can give ethics that a normal ethics unit can’t. Each fairness notion is a conditional-probability equality (Part I). Define each in one clause. Don’t derive; state them and let the next slide deliver the twist: you can’t have all three.

You can’t have them all全部は満たせない

When base rates differ, no classifier satisfies all three (Kleinberg et al. 2016; Chouldechova 2017).基準率が異なるとき、三つすべてを満たす分類器は存在しない（Kleinberg et al. 2016; Chouldechova 2017）。

• The COMPAS debate: ProPublica and the vendor were both right — they invoked different criteria.
• Fairness is a modeling choice, stated in this course’s language: the math gives you the menu and the tradeoffs, not the answer.

• COMPAS論争： ProPublicaとベンダーはどちらも正しかった — 持ち出した基準が異なっていた。
• 公平性はモデリングの選択であり、この講義の言葉で述べられる：数学は選択肢とトレードオフを与えるが、答えは与えない。

THE RESULT (Kleinberg, Mullainathan & Raghavan 2016, “Inherent Trade-Offs in the Fair Determination of Risk Scores,” ITCS; Chouldechova 2017, “Fair Prediction with Disparate Impact,” Big Data): if two groups have different base rates \(p_G = P(Y{=}1 \mid G)\), no imperfect classifier or risk score can simultaneously be (a) calibrated within each group and (b) have equal false-positive AND false-negative rates across groups. The only escapes are degenerate: equal base rates, or a perfect predictor. Demographic parity conflicts with calibration under unequal base rates for the same arithmetic reason.

WALK THE FIGURE (~1 min), one clause per box. Calibration: among everyone the model scores 70%, about 70% actually reoffend — in every group (\(P(Y{=}1 \mid \hat{S}, G)\) equal). Equalized odds: people who would NOT reoffend get flagged at the same rate in both groups (\(P(\hat{Y}{=}1 \mid Y{=}0, G)\) equal — the false-positive rate), and likewise for misses. Demographic parity: the same fraction of each group gets flagged, full stop (\(P(\hat{Y}{=}1 \mid G)\) equal). Then the yellow badge: any two can coexist; all three cannot when base rates differ.

WHY — Chouldechova’s one-line algebra; say it slowly or board it. From the confusion matrix, within each group: FPR = p/(1−p) × (1−PPV)/PPV × TPR. Calibration forces PPV (the fraction of flagged people who actually reoffend) to be equal across groups. Hold PPV equal and TPR comparable, and the group with the higher base rate p MUST have the higher false-positive rate. The error rates are pushed apart by arithmetic, not by anyone’s intent.

COMPAS, WITH NUMBERS (Broward County FL; ProPublica/Angwin et al. 2016 vs Northpointe’s response — quote as “roughly”): two-year rearrest base rates ≈ 51% for Black defendants vs ≈ 39% for white. Northpointe’s defense: PPV ≈ 0.6 in BOTH groups — the score is (approximately) calibrated; that was their fairness claim. ProPublica’s finding: FPR ≈ 45% vs 23% — Black defendants who did NOT go on to reoffend were labeled high-risk at about twice the rate — with FNR running the other way (≈ 28% vs 48%). Both sides computed correctly from the SAME confusion matrices; they invoked different rows of the table, i.e. different fairness definitions. That is the theorem made flesh.

WHICH CRITERION TO WANT depends on where the harm lands: punitive settings (bail, sentencing) — false positives fall on individuals, so error-rate balance matters most; resource allocation (lending, screening) — the decision-maker needs calibrated scores; representation goals — demographic parity. The theorem’s lesson: this is an ethical choice; the math only exposes its price.

LIKELY QUESTIONS. (1) “Just use different thresholds per group?” — that can equalize error rates but breaks calibration (the same score then means different things by group) and may constitute disparate treatment legally. (2) “So the model is the problem?” — no: ANY imperfect predictor faces this, including human judges; the tradeoff is in the problem, not the tool. (3) “Aren’t the base rates themselves suspect?” — yes, and that’s a PRIOR problem: Y here is re-ARREST, not re-offense, so measurement bias enters before any classifier — callback to the embeddings slide (bias in the data) two slides back.

~3 min. Full worked numbers + the runnable impossibility demo: the fairness-formalisms chapter (Part IX).

Poll — is equal accuracy fair?小テスト — 精度が同じなら公平？

A hiring model is equally accurate for every group. Is it fair?

ある採用モデルはどの集団に対しても同じ精度だ。これは公平？

• A. Yes — that’s what fairness means
• B. Not necessarily — it can hide unequal errors
• C. Only if base rates are equal across groups

• A. はい — それこそ公平の意味
• B. 必ずしも — 不均等な誤りを隠しうる
• C. 集団の基準率が等しい場合のみ

Commit before reveal. New for SP26 (no ethics quiz in the SP25 bank). Answer B (and C is a real subtlety). Equal overall accuracy is compatible with wildly different false-positive/false-negative composition — accuracy is not a fairness criterion. Let them sit with it.

Reveal答え

B — equal accuracy can hide very unequal errors.B — 等しい精度は非常に不均等な誤りを隠しうる。

• One shared accuracy number can hide who gets the false denials and who gets the false approvals.
• Accuracy is not a fairness criterion — name the conditional probability you care about.

• 一つの精度の数値は、誤った却下と誤った承認が誰に向かうかを隠しうる。
• 精度は公平性の基準ではない — 気にかける条件付き確率を名指しする。

Land it: accuracy hides error composition. This is exactly why you need the conditional-probability definitions — “fair” has to be made precise before it means anything. If students picked C — and some will, it echoes the impossibility theorem they just saw — do NOT litigate it here: the next slide takes C apart with two three-line counterexamples.

Why not C? Base rates are the wrong suspectなぜCではない？— 基準率は真犯人ではない

Equal base rates: still unfair

• Both groups: base rate 50%, accuracy 80%.
• A: 90% / 70% right on positives / negatives — B the reverse.
• A’s innocents flagged 3× as often.

基準率が等しくても不公平

• 両集団とも：基準率50%、精度80%。
• A：陽性/陰性で90% / 70%正解 — Bは逆。
• Aの無実の人は3倍誤検知。

Unequal base rates: still fair

• Both groups: 80% / 80% right on positives / negatives.
• So accuracy = 80% at any base rate.
• Base rates 30% vs 60%: equalized odds ✓.

基準率が違っても公平

• 両集団とも：陽性/陰性で80% / 80%正解。
• ゆえに精度はどの基準率でも80%。
• 基準率30%対60%でも：均等化オッズ ✓。

Unfairness hides in the error mix — not in the base rates.不公平は誤りの配合に隠れる — 基準率にではない。

WHY THIS SLIDE EXISTS: C is where the good students land — it transplants the base-rate condition from the impossibility theorem onto the accuracy question, and that transplant is wrong in BOTH directions. Deploy right after the reveal. If several students picked C, give it the full ~1.5 min; if nobody did, compress to 30 s (“C borrows the last slide’s villain — accuracy fails for its own reason”) and move on.

THE ALGEBRA TO BOARD (one line): within each group, accuracy = p·(share of positives caught) + (1−p)·(share of negatives caught) — ONE number blending TWO error rates, weighted by the base rate p. “Equal accuracy” equates the blends; fairness-by-error-rates (equalized odds, two slides back) equates the components. Equal blends with unequal components is trivial to build — the left card. And when the components are equal, the p-weights cancel — the right card.

LEFT CARD, spoken: both groups 50% base rate, both 80% accurate. Group A catches 90% of true positives and 70% of true negatives; group B is the mirror image. Accuracy is identical: 0.5·0.9 + 0.5·0.7 = 0.8 for both. But look only at the innocent people: A’s are falsely flagged 30% of the time, B’s 10% — a 3× disparity hiding behind one shared number, with base rates EQUAL. So equal base rates do NOT make equal accuracy sufficient for fairness.

RIGHT CARD, spoken: now make the components equal — both groups get 80% of positives AND 80% of negatives right. Then accuracy = 0.8p + 0.8(1−p) = 80% no matter what p is. Let the base rates be 30% vs 60%: accuracy still equal, error rates still matched — equalized odds holds. Fair-by-error-rates with UNEQUAL base rates. So equal base rates are not necessary either. C fails both ways.

HANDOFF BACK TO THE THEOREM: base rates DO matter — one slide ago — but for a different pair of criteria: calibration ↔︎ equalized odds. Calibration ties scores to base rates (a calibrated “60%” must mean 60% in every group), which is exactly why unequal base rates force that trade-off. Accuracy never touches calibration, so base rates are not its problem. The moral, again: name WHICH conditional probability you mean — an aggregate one-number metric can’t settle fairness at any base rate.

Adversarial examples: brittle geometry敵対的事例：脆い幾何

• An invisible nudge flips the kiosk: tonkatsu → hamburger.
• The attack: the loss gradient w.r.t. the input — jax.grad at \(\mathbf{x}\).

• 見えないほどの摂動でキオスクが反転：トンカツ → ハンバーグ。
• 攻撃は入力に関する損失の勾配 — jax.grad を \(\mathbf{x}\) に。

The figure now stars Chibany — presenting their bento photo on the left, devastated by the HAMBURGER verdict on the right (canon: tonkatsu is their favorite; hamburger a distant second). Milk the reaction for a beat before the mechanics. The beautiful insight: the attack uses the model’s own gradient machinery, pointed at the input instead of the weights (FGSM, Goodfellow et al. 2015). Say the why-it-works aloud (it’s off the slide): in high dimensions the learned boundary is close to almost every point, so one tiny step in the right direction crosses it. Callback to the learned geometry from Part VIII. Honest: robustness is unsolved, adversarial training is an arms race, and human perception is robust here in ways we don’t fully understand (Lake’s critique again). Full runnable FGSM demo: the adversarial-examples chapter (Part IX).

Alignment: teaching values is inferenceアラインメント：価値を教えるのは推論

• Pretraining predicts text; helpful and safe is a second step — an inference problem you already know.
• RLHF (Week 9): fit a reward from human preferences (Bradley–Terry, \(P(A \succ B) = \sigma(r_A - r_B)\)), then optimize the policy.
• The reward is learned and imperfect, so the optimizer hacks it — Week 8’s reward-shaping trap.
• That’s Goodhart’s law: when a measure becomes a target, it stops being a good measure.

• 事前学習はテキスト予測；有用で安全は第2段階 — すでに知っている推論問題。
• RLHF（第9週）：人間の選好から報酬を当てはめ（Bradley–Terry, \(P(A \succ B) = \sigma(r_A - r_B)\)）、方策を最適化。
• 報酬は学習された不完全なものなので、最適化器はそれをハックする — 第8週の報酬整形の罠。
• それがグッドハートの法則：指標が目標になると、良い指標でなくなる。

Value inference is inverse planning (Part VI) with the stakes maxed out.価値の推論は、賭け金を最大にした逆計画（Part VI）。

Two verified callbacks: Week 9 taught RLHF/DPO with Bradley-Terry; Week 8 taught reward shaping + the positive-cycle/reward-hacking poll. The unifying frame: alignment is value inference — the inverse-planning problem from Part VI, target underspecified (ill-posed). Full treatment + the reward-hacking demo: the alignment-safety chapter (Part IX, the book’s last chapter).

GOODHART’S LAW, properly: Charles Goodhart (1975), a Bank of England economist, observed that once the Bank started TARGETING a money-supply measure, the previously reliable relationship between that measure and the economy broke down. The popular phrasing on the slide is Marilyn Strathern’s. The mechanism: a measure is a good proxy only while nothing is optimizing against it; put optimization pressure on the proxy and the optimizer finds exactly the places where proxy and true goal come apart. Classic examples to tell (pick one): the cobra bounty (colonial Delhi paid per dead cobra → people bred cobras); Soviet nail quotas by WEIGHT → factories made a few giant useless nails; teaching to the test. THE RLHF INSTANCE: the reward model is a learned proxy for human preference, and policy optimization applies enormous pressure to it — so the policy discovers responses the reward model loves but humans don’t: sycophancy, padded verbosity, confident-sounding nonsense. That IS reward hacking. Callback to Week 8: the agent circling the shaped checkpoint reward forever instead of reaching the goal is the same law in a gridworld. The alignment punchline: you can’t fully fix this by making the proxy better — any imperfect proxy fails under enough pressure — which is why the field cares about regularization toward the base policy (the KL term), oversight, and uncertainty-aware objectives.

The full treatment is in Part IX詳細はPart IXに

The trailer ends here — Part IX carries the full treatment:

• Adversarial examples — build the fast gradient sign method (FGSM)
• Fairness, formally — the impossibility result as runnable code
• Bias in data — the Word-Embedding Association Test (WEAT); the prior as power and liability
• Alignment & safety — fit a reward, watch it get hacked

予告編はここまで — Part IXが本格的に扱う：

• 敵対的事例 — 高速勾配符号法（FGSM）を自分で作る
• 公平性の形式化 — 不可能性定理を実行可能なコードで
• データのバイアス — 単語埋め込み連想テスト（WEAT）；力であり負債でもある事前分布
• アラインメントと安全性 — 報酬を当てはめ、ハックされるのを見る

Point them at Part IX for the depth this segment couldn’t give. Then transition: enough lecture — the rest of today is yours. Paper presentation next.

Paper presentation論文発表

Holder — the presenting student drives their own slides (~15-20 min + short discussion). Advance past this when they finish. If no presenter is confirmed this week, this time folds into the retrospective.

Final-project presentations最終プロジェクト発表

Three presentations · ~10 minutes each + brief Q&A · Background · Question · Method · Preliminary results

3件の発表 · 各約10分 + 短い質疑 · 背景 · 問い · 手法 · 予備的結果

Holder for the three final-project talks — each student drives their own slides. Rubric: 10 min + Q&A, scored on motivation/question, method, preliminary results, clarity, and keeping to time. Keep transitions tight; the retrospective still needs its ~29 minutes.

Project 1プロジェクト 1

Student 1 drives their own slides. ~10 min + Q&A.

Project 2プロジェクト 2

Student 2 drives their own slides. ~10 min + Q&A.

Project 3プロジェクト 3

Student 3 drives their own slides. ~10 min + Q&A.

Semester retrospective学期の総括

Block 2, ~29 min, the climax. Structure: a 3-min pulse, the 15-min map of the semester, a 7-min cog-sci -> modern-ML bridge, a 4-min close. Slow down here — this is what they walk out remembering.

Pulse — before we map itパルス — 地図にする前に

A quick go-around — one answer each, not a discussion:

• Which paper this semester surprised you most?
• Which one will you re-read in a year?

手短に一巡 — 各自ひとつ、議論ではなく：

• 今学期、最も驚いた論文は？
• 一年後に読み返すのはどれ？

3 min. A pulse, not a debate — one answer each. It primes them to think across the whole semester before the map appears, and it tells you what landed. Keep it moving.

The map of the semester学期の地図

Every paper you saw, on two axes: Marr’s level (what · how · neural) and the kind of computation (categories · structure · time · substrate). One science, many corners.

見てきたすべての論文を二軸で：Marrのレベル（何を・どのように・神経）と計算の種類（カテゴリ · 構造 · 時間 · 基盤）。一つの科学、多くの隅。

15 min — the centerpiece. Walk it one paper at a time, one sentence each: what it contributes and where it sits. Named anchors (corrected 2026-07-16): Tenenbaum/generalization = pure L1. Vul et al. 2009 ‘One and Done’ = the true L2 anchor — how many samples should a bounded mind take; the course’s process-level paper. Sanborn & Griffiths 2008 (MCMC with People) and Abbott et al. 2012 sit at the L1 EDGE, not deep L2 — both borrow algorithmic machinery (MCMC, random walks) to answer computational-level questions about what the mind represents. Daw et al. 2005 sits lowest — two neural controllers arbitrated by uncertainty, L2 shading into L3. Pereira 2018 = L3 (decoding meaning from fMRI). Mark Shohei Yoshida’s presented paper (Tenenbaum & Xu 2000). The point of the 2D layout: the same inference toolkit shows up at every level and every computation type.

Classical cog-sci → today’s AI古典的認知科学 → 今日のAI

This course taught…	…which is the seed of
Hierarchical Bayes	in-context learning in LLMs
Inverse RL	RLHF — learning reward from preferences
Monte Carlo / variational inference	diffusion models, VI in large models
Bayes nets & causality	alignment & reward modeling
Bayesian nonparametrics	open-ended concept learning

この講義が教えたこと…	…の種になっているもの
階層ベイズ	LLMの文脈内学習
逆強化学習	RLHF — 選好から報酬を学ぶ
モンテカルロ／変分推論	拡散モデル・大規模モデルのVI
ベイズネットと因果	アラインメント・報酬モデリング
ベイズ・ノンパラメトリクス	オープンエンドな概念学習

The left column isn’t history — still cog-sci’s working tools today.左の列は歴史ではない — 今も認知科学の現役の道具。

SAY THE DIM LINE OUT LOUD — it’s the point students most easily miss: the left column is not a museum. Bayesian models are still how computational cognitive science models human behavior, full stop, with or without the AI connection; the bridge COMPLETES the course’s other half (how modeling human learning connects to machine learning), it doesn’t replace the first half.

7 min. The bridge — each classical idea from this course is the seed of something in contemporary AI. Not a claim that they’re identical; a claim that the toolkit transfers. The hierarchical-Bayes -> in-context-learning row is the Week 11 flagship; IRL -> RLHF is Week 9. Read it as “you already have the concepts the 2026 papers are built on.”

The opportunityこの機会

AI has mostly stopped looking at Bayesian models of minds — an opening:

AIは心のベイズモデルをほぼ見なくなった — それは好機：

Cognition → AI

• AI lacks priors, structure, rational analysis.
• Use them to understand and improve today’s systems.

認知 → AI

• AIには事前分布・構造・合理分析が足りない。
• それで今日のシステムを理解し、改善する。

AI → cognition

• Classic Bayesian models need small, idealized data.
• With AI inside, they finally meet real-world data.

AI → 認知

• 古典的ベイズモデルは小さく理想化されたデータを要する。
• AIを組み込めば、ついに実世界データに届く。

Few people can speak both languages. As of this semester, you can.両方の言語を話せる人は少ない。今学期を終えたあなたは話せる。

This slide is the professor’s own stake in the ground — deliver it personally. The field’s attention moved to scale; the Bayes-and-cognition bridge that used to be crowded (the 2000s–2010s Tenenbaum-Griffiths era) is now under-populated. Two concrete directions, one per card.

COGNITION → AI: the toolkit this course taught is exactly what current systems lack an explicit account of — priors, structured hypotheses, rational analysis of the task. Concrete instances already seen: hierarchical Bayes as the lens on in-context learning (Week 11); RLHF as preference inference / inverse planning (Week 9); alignment as value inference under an ill-posed objective (this deck). Using the toolkit to probe, evaluate, and regularize modern systems is a wide-open research lane — and students entering it now face little competition.

AI → COGNITION: the classic embarrassment of Bayesian cognitive modeling is its data diet — conjugate toy stimuli, tiny artificial category structures, idealized psychophysics. Modern models offer representations and likelihoods over REAL data — text, images, naturalistic behavior — so the same rational-analysis discipline can finally be applied at ecological scale: cognitive models whose hypothesis spaces come from foundation models, LLM-derived priors/likelihoods for behavioral experiments, model-based analyses of naturalistic corpora.

The yellow line is the sell: this cohort is bilingual — probabilistic modeling of minds AND modern ML — and that intersection is scarce right now. ~1.5 min; then the closer.

What you leave with持ち帰るもの

Not a fixed set of answers — a toolkit. A generative model, a prior, an inference, the inversion of planning. You can now open a 2026 paper and see what problem it is trying to solve.

Chibany just wanted to know what was for lunch. That question — what’s hidden, given what I see? — turned out to be the whole science.

決まった答えの集合ではなく、道具箱。生成モデル、事前分布、推論、計画の反転。これで2026年の論文を開き、それがどんな問題を解こうとしているかが見える。

Chibanyはただ昼食が何かを知りたかっただけ。その問い — 見えるものから、隠れたものは何か？ — が、科学そのものだった。

Thank you. — J.A.ありがとうございました。— J.A.

4 min close. The toolkit, not answers. The Chibany callback ties the whole year together: the mascot’s lunch question IS the inference question the entire course formalized. Warm thank-you. End.