🙉2uc e/a ratio (ear), 🎲3sampling strategy (k*), 🇮🇸4step REIK alg.

[hyunjimoon]

smart sampling for startups with exchangeability

Section/Subsection	🔐Research Question	🧱Literature Brick	🔑Key Message	🖼️Figure / 🗄️table
2. Two Types of Uncertainty 2.1 Aleatoric vs. Epistemic 2.2 Unmeasured vs. Unmeasurable 2.3 Why It Matters	What? Distinguish irreducible randomness (aleatoric) from learnable gaps (epistemic). How? By identifying which features can be uncovered with more data versus those inherently unpredictable. Why? To avoid wasting resources on noise or missing discoverable signals.	- Gelman (2004) on iterative Bayesian phases ("learning" vs. "target distribution") - Lundberg (2021) on "origins of unpredictability" and unmeasured features in life‐trajectory tasks - Buildup of "unmeasured vs. unmeasurable" concept to highlight the portion that can be learned	- Pinpoint epistemic (reducible) uncertainty vs. aleatoric (irreducible) noise - If uncertainty is mostly epistemic, more sampling pays off; if it's aleatoric, extra data provide diminishing returns - Ties directly to Tesla's battery choice: Are consumers' preferences stable but unknown (epistemic), or truly random day-to-day (aleatoric)?	(No direct figure from #🖼️Figure for Section 2). See final "Figure 1 (🎲aek(k∗k^*))" in Section 3 and "Figure 2 (🇮🇸REIK)" in Section 4 for applied visuals.
3. Three Sampling Strategies 3.1 Fixed k 3.2 Single "Optimal" k 3.3 Dynamic k	What? How many tests (surveys) do we run before a yes/no decision? How? By comparing a simple fixed‐size approach, a one‐time "optimal k" based on priors, and an adaptive "dynamic k." Why? Different levels of prior knowledge and cost/time tradeoffs require tailored strategies.	- Gelman (Bayesian sampling): showing iterative data collection strategies - Monte Carlo / MCMC approaches inform the "dynamic sampling" logic - Extension of Lundberg's unmeasured features: each additional survey reduces the unknown portion if it's epistemic	- Fixed k: Easiest but may over/under-sample - Single "Optimal" k: Good if we have a narrower prior (e.g. p≈[0.7,0.9]) - Dynamic k: Most efficient—invest more sampling only when signals are borderline or high stakes	Figure 1 (🎲aek(k∗k^*)): Shows how the optimal sample size k∗ shifts with prior range (A2E vs. E2K) and cost ratio. 🗄️table "Fixed vs. Single vs. Dynamic" also appears here, clarifying trade‐offs.
4. Four‐Step R-E-I-K Pilot 4.1 Random 4.2 Exchangeability 4.3 Ignorance 4.4 Knowledge	What? A structured procedure (Random → Exchangeability → Ignorance → Knowledge) to decide if you've found a real pattern vs. a coin‐flip. How? By applying de Finetti's theorem to pool partial observations into a latent parameter P. Why? Avoid over‐testing random noise; confirm signals that matter.	- de Finetti's Exchangeability: fundamental to Bayesian updating - Epistemic vs. Aleatoric synergy: each pilot cycle checks if new data reduce ignorance or confirm irreducible randomness - Connects to Lundberg's unmeasured features: repeated "R-E-I-K" cycles can uncover deeper structure	- Stepwise pilot approach ensures resources aren't sunk into pure randomness - "R-E-I-K" clarifies why adaptive sampling works (Section 3): if new data show a consistent tilt, keep going; if not, it's likely a coin‐flip	Figure 2 (🇮🇸REIK): Depicts how each pilot step (Random → Exchangeability → Ignorance → Knowledge) updates from raw data x̄₁,x̄₂ to a refined posterior P

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Abstract (One Paragraph)
Entrepreneurs and managers often face two fundamental types of uncertainty—some that can be discovered through more data (epistemic) and some that remains inherently random (aleatoric). This paper shows how to tell these two apart, then lays out three sampling strategies—ranging from a simple “fixed (k)” approach to a “dynamic (k)” method—that guide how many tests to run before deciding. Finally, we propose a four‐step R-E-I-K (Random, Exchangeability, Ignorance, Knowledge) pilot algorithm explaining why bigger samples help if uncertainty is learnable (epistemic) but add little if it’s irreducibly random (aleatoric). Using Tesla’s question of whether to adopt a US vs. Japan battery, we illustrate how clarifying which portion of uncertainty is truly unmeasured (and thus learnable) can transform decision-making from guesswork into a systematic, data‐driven process.

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Section 1. Introduction (One Paragraph)

Imagine you’re tossing a coin and not sure if it’s a fair coin (exactly 50–50) or one that lands heads more often (say 70–30). You test it a few times: if you see about half heads and half tails every time, it might really be 50–50, so getting more data won’t help. But if you see it land heads more than half the time, you’ve learned something—that the coin isn’t fair. In the same way, Tesla does a few quick tests: if results always split evenly, there’s no hidden “better” option (true randomness). If results lean one way, you’ve discovered a real difference, and gathering more data can sharpen that insight.

Many high‐stakes decisions—like Tesla’s choice of battery supplier—fail precisely because leaders can’t distinguish two crucial types of unknowns: random fluctuations nobody can fix vs. knowledge gaps that more sampling can fill. Here, we use a 2–3–4 framework: two sorts of uncertainty, three ways to pick a sample size, and a four‐step pilot process. This framework ensures managers only invest big sampling resources when real insight remains discoverable, rather than wasting time on what’s irreducible noise.

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Section 2. Two Types of Uncertainty (One Paragraph)

We define aleatoric uncertainty as the intrinsically random part of reality (unmeasurable features or post‐event disruptions) and epistemic uncertainty as the portion we can reduce by measuring unmeasured features. Inspired by Gelman’s notion of two iterative phases—mixing toward the target distribution (learning) vs. moving around within it—our approach hinges on identifying whether additional data truly uncovers hidden signals or merely reconfirms random fluctuations. We focus on unmeasured features (those we can capture with bigger or better sampling) rather than unmeasurable ones, which remain unknown given current precision of measurement.

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Section 3. Three Sampling Strategies (One Paragraph)

Once we know how much of our unknowns are learnable vs. irreducibly random, we pick a sampling plan from three options. (1) Fixed (k): always survey a set number (e.g., 200). (2) Single informed (k): do a one‐time optimization for a known prior range of (p), such as Texans who typically favor the US battery at 70–90%. (3) Dynamic (k): adapt sample size on the fly, using small pilots that expand only if results are borderline. Each strategy balances ease vs. accuracy and fits different levels of uncertainty about consumer preference.

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Section 4. Four‐Step Pilot Process (One Paragraph)

Finally, the four-step R-E-I-K method—\textbf{Choose Random}, \textbf{Verify Exchangeability}, \textbf{Represent Ignorance}, \textbf{Make Knowledge}—explains \emph{why} sampling strategies in Sec.3 work with E/A ratio in Sec.2 to better distinguish two types of uncertainty.

1. Choose Random (blue): You begin by drawing raw data ((\bar{x}_1, \bar{x}_2)) from the environment, which may contain uncontrollable factors (“nature’s randomness”).
2. Represent Ignorance (green): You acknowledge there may be a latent parameter (P) that you don’t fully know. In other words, some portion of uncertainty is “unmeasured but learnable.”
3. Verify Exchangeability (red test statistics): You then apply test quantities (T(x)) to check whether your samples share the same probability structure—if so, they come from one \emph{underlying} (P). If data aren’t exchangeable, you may need a different model or a larger pilot.
4. Make Knowledge (red): Combining what you’ve learned (test results + repeated sampling) lets you unify observations ((x_1, x_2)) into a stable (\hat{P})—turning ephemeral “ignorance” into actionable knowledge. If that knowledge remains incomplete (e.g., borderline results), you return to step 1 (Choose Random) and run another pilot.

By cycling through these steps, you systematically reduce the “unmeasured” portion that can be discovered with enough data, while recognizing that \emph{true} irreducible randomness persists no matter what. This prevents wasting resources in purely aleatoric situations yet captures real patterns when they exist.

🙉2uc e/a ratio (ear)

• EAR in Tesla: agents who don't know underlying p, measures it E/A ratio of p via ## 🇮🇸4step REIK alg.
• HEAD vs HOAD interpret EAR differently 🙉2uc e/a ratio (ear), 🎲3sampling strategy (k*), 🇮🇸4step REIK alg. #271 (reply in thread)

🎲3sampling strategy (k*)

🇮🇸4step REIK alg.

is why and how of pilot test (exploration)

🇮🇸REIK algorithm to distinguish E/A, k* sampling strategy,

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p is success probability from .5 to 1 and below shows aleatoric uncertainty (Variance = p(1 - p)) by p.

  ^
  | 0.25  ● (p=0.5)
  |       .
  |        .
0.2|         .
  |          .  
0.15|           .
  |            .
0.1 |             .
  |               .
0.05|                .
  |                  ● (p=1, variance=0)
  +-------------------------------------> p
    0.5                1.0

[hyunjimoon]

🙉2uc e/a ratio (ear)

Below is a quick walk‐through of how Tesla could simulate different “true” preferences (for Japan vs.\ US batteries) across California (CA) and Texas (TX), then measure how well they learn each preference after a certain number of trials (k). This is how you’d compute the E/A ratio in practice:

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1. Set a Range for p

• Scenario: Tesla believes each region's "true probability that customers think Japan battery is better than US battery" (p) is somewhere between 0.5 and 1.0 (i.e., at least half of the customers might favor it, up to everyone favoring it).
• What it means: You're bounding the aleatoric uncertainty by saying, "We don't know exactly how strong the preference is, but it's in this plausible interval."

2. Draw True p Values

• Procedure: For each simulation run, you randomly pick p_CA and p_TX within [0.5, 1.0]. For instance:
  • Run Main #1 might draw p_CA = 0.7 (70% prefer Japan) and p_TX = 0.5
  • Run #2 might draw p_CA = 0.8 and p_TX = 0.7, etc.
• Interpretation: Each pair (p_CA, p_TX) is a possible "true reality." By varying them across many runs, you capture the inherent randomness (aleatoric uncertainty) in how different states might actually feel about the two battery options.

3. Generate Observations and Estimate p̂

• Simulate Trials: Suppose you let Tesla survey or test with k customers in each region, collecting yes/no "Japan battery preference" data.
  • If p_CA = 0.7, you randomly generate 0/1 outcomes with a 70% chance of "1" ("prefer Japan"), repeated k times.
  • Then compute p̂_CA as the fraction of "1" outcomes (e.g. 68% if 68 of k=100 said yes).
• Epistemic Uncertainty: p̂_CA (the measured preference) might differ from the true p_CA because you only tested k times. The difference between "true" and "estimated" preference is your learnable uncertainty—if you tested more people, you'd presumably pin down p more accurately.

4. Repeat and Measure Variances

1. Var(p): Across many simulation runs, note how p_CA and p_TX vary. That's your "true" distribution's variance—the baseline aleatoric randomness.
2. Var(p̂|k): Within each run, see how your estimated p̂_CA or p̂_TX differs from the actual p. Then aggregate across runs to see the average variance of p̂ given k trials.

5. Compute the E/A Ratio

• Calculation:

  E/A = E[Var(p̂|k)] / Var(p)
  

  where Var(p̂|k) is the average variance of your estimates, and Var(p) is the variance of the true preferences.
• Interpretation:
  • A high E/A ratio means your estimates p̂ still bounce around a lot compared to the overall range of possible true p values—i.e., there's a lot more learnable uncertainty if you did more tests or refined your approach.
  • A low E/A ratio suggests your sampling is already capturing most of the relevant signal, so further data might not significantly improve certainty about who prefers Japan vs. US batteries.

Definition: E/A Ratio

The E/A ratio is the quotient of epistemic uncertainty (E) over aleatoric uncertainty (A). Informally, it answers "How much of our uncertainty is learnable versus irreducibly random?" A common operationalization in simulation is:

E/A = E[Var(p̂|k)] / Var(p)

where:

• p̂ is an estimated probability (e.g. success probability in a startup scenario)
• p is the true but unknown probability
• k is the number of samples (or test runs) chosen to estimate p

[hyunjimoon]

EAR interpretation differ by HEAD VS HOAD?

E/A Ratio	Homogeneity as Default	Heterogeneity as Default
High (lots of learnable structure vs. random noise)	- Similar results across multiple domains are not surprising.- You don’t question your assumptions just because things match. - Conflict in data is the main reason to doubt.	- Similar results across supposedly distinct contexts are very surprising.- You do question your assumptions because “they shouldn’t match unless something deeper is going on.” - Agreement triggers “methodic doubt.”
Low (data mostly noise)	- Similar or different results are mostly seen as random variation.- You rarely change assumptions unless extreme conflict appears.	- Even when you see some similarity, you assume it might be coincidence.- You only suspect a deeper link if you detect a strong pattern that overcomes the noise.

Concrete Example

• Sports Cars vs. Home Energy
  • Under Homogeneity as Default, seeing a similar battery performance in both uses doesn’t trigger much new thinking—“Of course it’s the same battery tech.”
  • Under Heterogeneity as Default, that same consistent performance is a red flag—“Wait, it’s surprising that cars and home storage are matching; maybe there’s a single underlying principle we haven’t accounted for.”

[hyunjimoon]

Section 2.3 (“Why Distinguishing Epistemic vs. Aleatoric Matters”),

each incorporating the suggestions you’ve provided—ranging from citing real‐world examples, referencing unmeasured features vs. unmeasurable ones, and acknowledging a new naming for your “one-time optimal” (k). You can choose which version resonates better, then revise accordingly.

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Draft A

2.3. Why Distinguishing Epistemic vs. Aleatoric Matters

When we talk about epistemic (learnable) vs. aleatoric (irreducible) uncertainty, we’re essentially separating “unmeasured features” from “unmeasurable features” (cf. Figure 4 in [cited paper]):

1. Unmeasurable: True randomness that no extra data can dispel (e.g., customers changing opinion at the last second for unknowable reasons).
2. Unmeasured: Important signals we could detect if we sample more.

Why it matters:

• If epistemic dominates (unmeasured features waiting to be uncovered), collecting more data—like adapting (k) or iterating the R-E-I-K steps—can drastically reduce our guesswork. Each additional survey reveals latent patterns we didn’t have before.
• If aleatoric dominates (unmeasurable features or pure randomness), bigger samples yield little new insight; you’ll merely keep confirming an irreducible noise factor.

Practical Consequence

• Wasted resources if we throw large samples at what’s truly irreducible.
• Missed opportunities if we undersample a situation that’s actually learnable.

Real-World Analogy

• Oil Drilling: If oil truly isn’t there, more wells don’t help. But if there is a reservoir, each new well can reveal a richer map of where the oil lies.
• Tesla Example: Suppose a new Japanese battery has a real advantage that 80% of folks will love if they see it. If you only sample 10 people, you might guess 50%. Testing 300 might converge to 80%—a big difference for a yes/no decision. On the other hand, if it’s actually a random 50–50 preference (genuinely aleatoric), going from 50 people to 500 changes almost nothing about your final conclusion.

Connection to Our “Three (k)” Approaches
In Section 3, we propose “fixed (k), single-informative-prior (k), and dynamic (k).” All revolve around deciding how many people to sample when you suspect a latent pattern (epistemic) vs. when you accept randomness (aleatoric). If you fail to tell them apart, you risk misusing resources or missing signals. As Gelman (2004) notes, the “first stage” (learning) is where big sampling moves you to the target distribution, while the “second stage” (already in the target) suggests you can’t refine it further. Our approach focuses on unmeasured features that better sampling can uncover—not unmeasurable or post‐event events.

(Optional Equation)
You may add an irreducible vs. reducible decomposition (like Eq. (1) below) if you want to highlight the mathematical synergy with the E/A ratio:
[
\text{Expected Squared Error}
;=;
\underbrace{\mathrm{E}\bigl[\mathrm{Var}[Y\mid X]\bigr]}{\text{Irreducible Error}}
;+;
\underbrace{\mathrm{E}!\bigl[\bigl(\hat{f}(X)-\mathrm{E}[Y\mid X]\bigr)^2\bigr]}{\text{Reducible (Learning) Error}}.
]
But if that complicates your main E/A ratio discussion, you can skip it.

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Draft B

2.3. Why Distinguishing Epistemic vs. Aleatoric Uncertainty Is Critical

Distinguishing epistemic (learnable) from aleatoric (inherent randomness) is the linchpin of our entire sampling logic. A scenario dominated by unmeasured (but potentially measurable) features calls for more data, whereas unmeasurable or random features yield diminishing returns, no matter how large the sample.

1. More Data Helps—But Only If There’s Something to Learn

   • Epistemic scenario:
     • Each new observation reveals deeper structure. If 70% of Californians truly prefer the US battery, a small pilot might mislead you at 58%, but bigger sampling converges near 70%.
   • Aleatoric scenario:
     • If half the population flips a coin each morning to decide which battery they like, 50 or 5,000 surveys produce the same noise. You can’t fix genuine unpredictability.

2. “Unmeasured” vs. “Unmeasurable”

   • Our method (Sections 3 and 4) primarily shrinks the set of unmeasured features—those we could measure if we either expand (k) or do more thorough R-E-I-K pilot tests.
   • Unmeasurable events (like last-minute mood swings or post-survey transformations) remain irreducible noise, no matter how sophisticated we get.

3. Gelman’s Two Phases of Iteration

   • In the “learning” stage, we invest in bigger experiments (if we suspect a discoverable signal).
   • In the “target distribution” stage, we accept random fluctuations as part of the environment, so additional sampling yields little benefit.

4. Implications for “Fixed vs. Single-Informed vs. Dynamic (k)”

   • If your domain is mostly unmeasurable randomness, “fixed (k)” might suffice (no reason for elaborate updates).
   • If you have an informative prior about (,p) (like “Texans typically 70–90% pro-US battery”), a one-time “optimal (k)” focuses resources effectively.
   • If the domain is quite unknown, a “dynamic (k)” approach invests heavily in those borderline cases—exactly because there’s a bigger epistemic payoff.

In short: If you misunderstand whether uncertainty is ephemeral “noise” vs. a real signal you haven’t measured, you might either (a) overspend on fruitless sampling or (b) skip essential tests that would have uncovered a crucial advantage. Our approach zeros in on the unmeasured features that truly matter—unlike unmeasurable features, which remain outside our sphere of influence regardless of sample size.

(Optional Equation)
A popular decomposition from regression tasks is:
[
\text{Error} = \text{Irreducible Error} + \text{Learning Error}.
]
We emphasize that our “big sampling” (or dynamic pilot tests) primarily attacks the “learning error” portion, bridging it with our E/A ratio concept. The “irreducible” remainder stands for truly aleatoric unknowns that persist even after “infinite” data.

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Additional Notes / Explanations

1. “One‐time optimal”

   • A possible rename could be: “Single-Informed (k)” or “Prior-Based (k)” to emphasize that you have an informative prior range, e.g., ([0.7,0.9]).

2. Equation for Irreducible vs. Reducible

   • If you already define E/A ratio, adding the classical MSE decomposition is purely optional. You can mention it briefly to connect your approach with standard references (Hastie et al., 2009; Gelman, 2004) but keep it short if you don’t want to overshadow your own ratio concept.

3. Keep “fixed vs. single optimal vs. dynamic”

   • If “single optimal” might be replaced by “single informed,” that’s up to you. But if you do rename, highlight that it’s based on an “informative prior range” for (,p).

---

Use whichever draft (A or B) feels more natural. Each integrates your comment about focusing on unmeasured (not unmeasurable) features and clarifies the link between big sampling and genuinely learnable signals. Feel free to mix & match elements—like keeping Draft A’s “oil drilling” metaphor and Draft B’s bullet points.

[hyunjimoon]

🎲3sampling strategy (k*)

Figure 1 (🎲aek((k^*))):
Caption:
“Comparing sampling decisions for different prior ranges—A2E ((p\in[0.5,0.75])) vs. E2K ((p\in[0.75,1.0]))—as a function of the ‘time‐cost ratio.’ When decisions are very costly to reverse, both curves rise sharply, implying a bigger optimal sample size (k^*).”

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Explanation (Section 3 context):
This plot shows how many surveys ((k^)) you want to run before choosing the US or Japan battery. On the horizontal axis is the “time‐cost ratio,” representing how expensive it is to commit to a final decision relative to the cost of sampling. The lines show that if you suspect (p) (the “true” preference for the Japan battery) is closer to 0.5–0.75 (more uncertainty, A2E in blue), then you typically choose bigger (k^) as the cost of a wrong commitment grows. If your prior range is 0.75–1.0 (E2K in red), you expect you’re already pretty sure about your battery’s appeal, so you pick fewer surveys. This allocation of sampling resources helps you avoid either undersampling in murky scenarios or oversampling when you’re nearly certain.

Paragraph 1: Fixed (k)
In this first approach, managers pick a single, unchanging sample size—for example, “We always survey 200 customers, no matter what.” As highlighted in the table, this “Fixed (k)” method is easy to implement and keeps everyone on the same page (200 is 200—done!). It works adequately when the decision isn’t too costly or if there’s a company rule demanding a certain minimum sample size. The downside is clear: if you face a near‐sure situation ((p\approx0.95)), 200 might be overkill, but if it’s murky ((p\approx0.5)), 200 might not be enough to uncover a solid conclusion. Nevertheless, for routine or regulated decisions, a fixed (k) can be perfectly acceptable.

Paragraph 2: Single “Optimal” (k) (p in ([0.7, 0.9]))
A middle‐ground method—shown in the table as “Single (k) for the whole (p)‐range”—involves doing one-time math to find the best average sample size for, say, (p\approx0.7) to (p\approx0.9). If managers believe most projects fall in that moderately uncertain zone, they might pick (k=350) (or any number) that maximizes expected decision quality across ([0.7,0.9]). This yields fewer overshoot/undershoot risks than a blind fixed policy while still keeping a single standard for simplicity. As the table indicates, the trade-off is that truly borderline situations ((p\approx0.5)) or near‐locks ((p\approx1.0)) might deviate from that sweet spot—but overall, it’s more efficient than a rigid fixed (k) across a typical range of scenarios.

Paragraph 3: Dynamic (k) (“Tailored per Scenario”)
Finally, the most flexible method is dynamic sampling, seen in the table as “Dynamic (k).” Here, you adapt your sample size each time. If initial polls reveal (p\approx0.99) (practically a slam dunk), you stop sampling early and commit. If it’s murky ((p\approx0.55)), you keep sampling until you’re more certain. This approach ensures maximum efficiency—bigger (k) when you really need to clarify an iffy market, smaller (k) when things are obvious. The payoff is higher accuracy and less resource waste, but the table reminds us that dynamic (k) is more complex and typically requires a data‐savvy team monitoring results in real time.

Three k* in tesla

Strategy	Basic Idea (In Manager Terms)	When You'd Use It	Upsides	Downsides
Fixed k (e.g. "always sample 200")	One-size-fits-all. You pick a single, unchanging sample size—like a standard "We always test with 200 customers" policy.	• If there's a company-wide standard or regulation for sampling (e.g., safety/legal compliance) • When decisions aren't very costly or you don't have time for deeper analysis	• Very easy to roll out • Everyone knows the same rule ("Take 200 samples, done")	• Could overshoot in obvious cases, wasting time/costs • Could undershoot in very uncertain situations, leading to risky guesses
Single k for the whole p-range ("optimal one-size-fits-all")	You do a one-time analysis of the likely range of outcomes (p)—for example, "p might be anywhere from 0.7 to 0.9." Then you pick one "best" k that balances average cost/benefit across that entire range.	• If you have a rough idea of where p will lie (say 0.5–0.9) • If you can do a simple cost-benefit calculation to find a single "optimal" k • Great middle ground when you can't tailor every single project individually but still want more rigor than a "fixed 200"	• Better than a blind fixed number • Ensures you're not drastically under- or over-sampling on average across typical scenarios	• Still a one-size approach: real projects that are near-certain or super-uncertain might not get exactly the right number of samples
Dynamic k ("tailored per scenario")	For each new project (or each major scenario), you adapt the sample size based on how uncertain/stakes are. If early feedback suggests "Yes, it's a slam dunk," you do fewer surveys. If it's murky, you do more.	• High-stakes decisions (multi-million/billion) • Large variability in "true preference" (i.e. p could be anything from 0.5 to almost 1.0) • When you have time/budget for careful "pilot → see results → keep sampling if needed"	• Most accurate final decision • Saves resources in slam-dunk scenarios; invests more effort in uncertain ones	• Complex to implement (needs real-time or iterative decisions) • Requires analytics/data science staff to keep adjusting sample sizes

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Example: Tesla Manager’s Quick Usage Scenarios

1. Fixed ( k ) Approach

   • Manager’s Choice: “We always run 200 pilot surveys.”
   • When:
     • If Tesla’s internal policy or a government regulation says: “Minimum 200 sample tests for safety compliance.”
     • Decision cost is not huge, or there’s a tight deadline.
   • Outcome: Quick to do but might be overkill if ( p \approx 0.9 ), or insufficient if ( p \approx 0.52 ).

2. Single “Optimal” ( k ) for the Whole Range

   • Manager’s Choice: “We expect ( p ) to be 0.5–0.9, so our analysis suggests 350 samples hits the best cost–benefit on average.”
   • When:
     • Tesla sees moderate uncertainty about preference for Japan vs. US battery but not extreme uncertainty.
     • They want a “golden rule” that most product teams can apply without rethinking every time.
   • Outcome: More fine-tuned than a blunt 200-sample rule. But if a future scenario is suddenly near 0.99 or 0.51, it might still not be perfect.

3. Dynamic ( k ) for Each Scenario

   • Manager’s Choice: “We do a small pilot (say 50 surveys). If results are borderline (e.g. 55% in favor?), we ramp up to 300 more to confirm. If it’s already 80% yes, we finalize quickly.”
   • When:
     • A big strategic decision—like selecting the battery supplier for a brand-new Tesla vehicle or deciding on a multi-year $1B contract.
     • Enough data/analytics resources to adapt sampling on the fly.
   • Outcome: Highest accuracy, largest flexibility. Uses more sophisticated methods (but also more overhead).

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Bottom Line for Tesla Managers:

• If your decision is routine (risk is low, or you have standard compliance rules): Fixed ( k ) is simplest.
• If you can do a bit more analysis (and you have a decent idea of your customer preference range): pick one “average-optimal” ( k ).
• If each new project is high stakes (or the preference is truly unknown and you can’t afford a bad call): consider dynamic ( k ) so you only invest deeply when it’s uncertain.

Use this table to decide how much sampling investment is justified for each new battery-supplier or feature decision.

[hyunjimoon]

🇮🇸4step REIK alg.

Figure 2 (🇮🇸REIK):
Caption:
“The R-E-I-K cycle (Random, Exchangeability, Ignorance, Knowledge) highlights why the sampling strategies in Section 3 work. Each step uses de Finetti’s exchangeability principle to unify partial observations ((\bar{x}_1,\bar{x}_2)) into a latent parameter (P), converting ephemeral randomness into actionable knowledge.”

Explanation (Section 4 context):
This diagram illustrates four steps to update beliefs using small pilots:

1. Choose Random (blue): gather fresh data from the environment, which might contain uncontrollable or truly random elements (aleatoric).
2. Represent Ignorance (green): formalize the learnable piece (epistemic) by positing a latent parameter (P) that underlies the new observations.
3. Verify Exchangeability (red): check if these data can be regarded as coming from one shared (P); if they do, we can pool them to refine our estimate of (P).
4. Make Knowledge (red): unify it all, updating ((\bar{x}_1,\bar{x}_2)\to X\mid P). If the result remains borderline, we repeat the cycle.

By iterating this loop, you systematically separate “random fluctuations” from “hidden structure,” ensuring the sampling strategies from Figure 1 genuinely capture the portion of uncertainty that’s learnable rather than purely random.

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value
Value (V) in our R-E-I-K framework quantifies the relative importance of epistemic vs aleatoric uncertainty in a decision, helping optimize sampling strategy by determining whether additional data will reduce learnable uncertainty or merely confirm inherent randomness. When two samples are exchangeable (i.e., E[V(X₁|P)] = E[V(X₂|P)]), it indicates they share the same underlying probability structure P, which guides decisions like Tesla's battery supplier selection by distinguishing between learnable quality variations (epistemic) and inherent manufacturing variability (aleatoric).

Below is a table showing how the 🇮🇸REIK (Random, Exchangeability, Ignorance, Knowledge) steps map onto a small pilot test—explaining plan, why, possible outcomes, and a Tesla example at each step:

🇮🇸REIK Step	Plan	Why	Case A (Aleatoric) vs. Case B (Epistemic)	Tesla Example
R: Random	- Gather a small number of pilot samples (10–20) from your data‐generating process.- Ensure the sampling is unbiased (randomly chosen conditions, participants, etc.).	- You want each observation to come from the same underlying probability pp.- Guarantees that your data reflect the true process rather than a systematic bias.	- Aleatoric: If it’s a true coin toss, even these random samples won’t reveal any pattern.- Epistemic: If there’s a real pattern, even a few unbiased draws might show a skew (e.g., mostly “heads”).	Tesla might survey 10 random drivers to see if they prefer Battery A over Battery B. If there’s any systematic preference, random draws should hint at it.
E: Exchangeability	- Check if data look consistent with a single fixed pp.- Watch for suspicious time‐based trends or changing test conditions.	- If samples are “exchangeable,” de Finetti’s theorem says there is some  p,p behind them. Otherwise, you can’t reliably estimate a single probability.	- Aleatoric: Exchangeable data but still hovers around 50–50 → truly random.- Epistemic: Exchangeable data that tilt in one direction → a stable pattern.	Tesla ensures all test‐drivers face the same conditions (e.g., same road type, same battery temperature). If each test is truly the same scenario, they can treat data as one Bernoulli process.
I: Ignorance	- Form a quick Bayesian or frequentist update on p^\hat{p}.- See if you remain near 0.5 or if you get a clear skew (e.g., 8 out of 10 prefer Battery A).	- This is about quantifying your uncertainty. Do you have a wide interval around 0.5, or does the new data push you strongly above/below 0.5?	- Aleatoric: No matter what, results keep splitting close to 50–50. More samples might stay ambiguous.- Epistemic: Posterior moves away from 0.5, revealing a real directional preference.	After 10 surveys, Tesla sees “5 prefer A, 5 prefer B.” That suggests no strong winner. Alternatively, “8 prefer A, 2 prefer B” points to a genuine signal that A might dominate.
K: Knowledge	- Decide whether to invest in more samples or stop.- If data hint at a “real pattern,” keep sampling (if cost permits). If it’s just 50–50, you might stop.	- If you confirm a pattern, more data can refine that knowledge.- If you confirm a coin toss, extra tests won’t add value.	- Aleatoric: Abandon further sampling; the process is likely a real coin flip. No improvement beyond 50–50. - Epistemic: Invest in more sampling if it’s cost‐effective—there’s a meaningful difference to uncover (e.g., maybe p=0.7p=0.7).	If Tesla’s pilot is stuck at 5–5, they might see no point in spending money on more tests (coin flip). If Tesla sees 8–2, they might do additional 30–50 samples to confirm Battery A is truly better, thus refining the estimate.

Key Takeaway: A small pilot—implemented using these 4 steps—quickly tells you whether you’re dealing with (A) “coin‐flip” (aleatoric, no hidden pattern) or (B) “real signal” (epistemic, more data helps).

[hyunjimoon]

value
Value (V) in our R-E-I-K framework quantifies the relative importance of epistemic vs aleatoric uncertainty in a decision, helping optimize sampling strategy by determining whether additional data will reduce learnable uncertainty or merely confirm inherent randomness. When two samples are exchangeable (i.e., E[V(X₁|P)] = E[V(X₂|P)]), it indicates they share the same underlying probability structure P, which guides decisions like Tesla's battery supplier selection by distinguishing between learnable quality variations (epistemic) and inherent manufacturing variability (aleatoric).

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i was originally confused between value (summary statistics) with test quantities (anxiliary statistics for model checking). i thinks we should first focus on the former i.e. process of how summary statistics becomes sufficient as sigma of all quantity converge to be the filtration of required information. during this process the number of equivalence classes is added (with increasing number of summary statistics as long as V_n+1(X) is not fully explained by existing sigma (V_1...V_n)

Many high‐stakes decisions—like Tesla’s choice of battery supplier—fail precisely because leaders can’t distinguish two crucial types of unknowns: random fluctuations nobody can fix vs. knowledge gaps that more sampling can fill. Here, we use a 2–3–4 framework: two sorts of uncertainty, three ways to pick a sample size, and a four‐step pilot process. This framework ensures managers only invest big sampling resources when real insight remains discoverable, rather than wasting time on what’s irreducible noise.

[hyunjimoon]

Test quantities

We measure the discrepancy between model and data by defining test quantities, the aspects of the data we wish to check. A test quantity, or discrepancy measure, T(y, θ), is a scalar summary of parameters and data that is used as a standard when comparing data to predictive simulations. Test quantities play the role in Bayesian model checking that test statistics play in classical testing. We use the notation T(y) for a test statistic, which is a test quantity that depends only on data; in the Bayesian context, we can generalize test statistics to allow dependence on the model parameters under their posterior distribution. This can be useful in directly summarizing discrepancies between model and data. We discuss options for graphical test quantities in Section 6.4. The test quantities in this section are usually functions of y or replicated data y rep . In the end of this section we briefly discuss a different sort of test quantities used for calibration that are functions of both y i and y i rep (or ˜y i ). In Chapter 7 we discuss measures of discrepancy between model and data, that is, measures of predictive accuracy that are also functions of both y i and y i rep (or ˜y i ).

Checking the model

Defining test quantities to assess aspects of poor fit. The model was chosen to fit accurately the unequal means and variances in the two groups of people in the study, but there was still some question about the fit to individuals. In particular, the histograms of schizophrenics’ reaction times (Figure 22.1) indicate that there is substantial variation in the within-person response time variance. To investigate whether the model can explain this feature of the data, we compute s j , the standard deviation of the 30 log reaction times y ij , for each schizophrenic individual j = 12, . . . , 17. We then define two test quantities: T min and T max , the smallest and largest of the six values s j . To obtain the posterior predictive distribution of the two test quantities, we simulate predictive datasets from the normalmixture model for each of the 1000 simulation draws of the parameters from the posterior distribution. For each of those 1000 simulated datasets, y rep , we compute the two test quantities, T min (y rep ), T max (y rep ).

Graphical display of realized test quantities in comparison to their posterior predictive distribution. In general, we can look at test quantities individually by plotting histograms of their posterior predictive distributions, with the observed value marked. In this case, however, with exactly two test quantities, it is natural to look at a scatterplot of the joint distribution. Figure 22.2 displays a scatterplot of the 1000 simulated values of the test quantities, with the observed values indicated by an ×. With regard to these test quantities, the observed data y are atypical of the posterior predictive distribution—T min is too low and T max is too high—with estimated p-values of 0.000 and 1.000 (to three decimal places).

Example of a poor test quantity. In contrast, a test quantity such as the average value of s j is not useful for model checking since this is essentially the sufficient statistic for the model parameter σ y 2 , and thus the model will automatically fit it well.

Checking the new model

The expanded model is an improvement, but how well does it fit the data? We expect that the new model should show an improved fit with respect to the test quantities considered in Figure 22.2, since the new parameters describe an additional source of person-to-person variation. (The new parameters have substantive interpretations in psychology and are not merely ‘curve fitting.’) We check the fit of the expanded model using posterior predictive simulation of the same test quantities under the new posterior distribution. The results are displayed in Figure 22.3, based on posterior predictive simulations from the new model.

Once again, the × indicates the observed test quantity. Compared to Figure 22.2, the × is in the same place, but the posterior predictive distribution has moved closer to it. (The two figures are on different scales.) The fit of the new model, however, is by no means perfect: the × is still in the periphery, and the estimated p-values of the two test quantities are 0.97 and 0.05. Perhaps most important, the lack of fit has greatly diminished in magnitude, as can be seen by examining the scales of Figures 22.2 and 22.3. We are left with an improved model that still shows some lack of fit, suggesting possible directions for improved modeling and data collection.

[hyunjimoon]

will log everything on exchangeability here

exchangeability useful

• if the thought experiment doesn’t have a convincing link to reality in the specific situation of interest, we may not assign much meaning to it
• exchangeability is a great modelling tool if used temporarily with the license to drop it in the face of evidence that indicates against it (one could interpret this as saying that aleatory events disproves the model, or, from a fully epistemic point of view, that it shows that exchangeability wasn’t really what we believed in the first place)