Knightian, Aleatoric, Epistemic Uncertainty (Knowing the know)

[hyunjimoon]

From the diagram (BD chp.1), latent layers are more hard to know, just as lower layer hierarchy parameters are difficult to identify as reported in #96.

Quotes like

You know, I know that you know (typical psychological game) 
You would know what we know as we know it (The Newsroom Season1.Ep7)

Distinction between brain and mind as hardware and software, sequential dependency in short term, but feedback in long term was new to me. Hard-soft seems to be another double-edged binary sword .

[tomfid]

https://youtu.be/E2y40U2LvKY

[hyunjimoon]

how to argue hierarchy, bayesian, dynamics are three sides of one dice🎲?

One may say "hierarchical structures and dynamics that can approximate each other, justifies agent's bayesian inference".

• "potentially hierarchical, context, Bayesian inference can distinguish between aleatory uncertainty (randomness) and epistemic uncertainty (ignorance)" from The Boxer, the Wrestler, and the Coin Flip: A Paradox of Robust Bayesian Inference and Belief Functions

• hierarchy or dynamics - I got verification from Andrew Gelman that hierarchy comes before Bayesian and Jim Duggan (I think) once mentioned Bayesian and dynamics are two sides of one coin. I'd be surprised if there doesn't exist a general framework where hierarchy, bayesian, dynamics are three sides of one dice.

• I wonder whether I could connect "I want you to be bayesian because that means sequential decision rule does not affect your posterior probability i.e. no additional information is given by the by the stopping rule" from andrew's talk here (title:it's about time) with the "test two choose one" stopping rule (which gives justification for using simulations in business) from
  Gans16_foundations_entrep_strategy.pdf - with claude's help, i fished some food for thoughts:

1. Stopping rule independence: The "no additional information is given by the stopping rule" principle means entrepreneurs can flexibly adjust their experimentation process without biasing their results. This allows for more adaptive and responsive strategies in testing business ideas.
2. Focus on content over process: Since the stopping rule doesn't provide additional information, entrepreneurs should concentrate on designing informative experiments rather than overthinking when to stop. The quality and relevance of the data gathered is more important than the decision of when to conclude testing.
3. Practical implementation of "Test Two, Choose One": This approach provides a concrete way to apply the stopping rule independence principle. Entrepreneurs can continue experimenting until they have two equally viable options, without worrying that the act of stopping at that point is itself informative. This balances the need for sufficient information with the practicality of making timely decisions.

Girsanov Theorem (the idea is, if you append stochastic process with stopping time, it becomes markovian (if my memory is correct)) which leads (?) to bayes rule. excerpt from ioannis' brownian motion and stochastic calculus textbook: girsanov_bayes_rule.pdf

will first ask prof.andrew (and escalate to nathaniel, ioannis, david) for verification on the relevance, once this thought gets more formalized🙏

[hyunjimoon]

i asked to andrew

could you point me to literature relevant to your comment "I want you to be bayesian because that means sequential decision rule does not affect your posterior probability i.e. no additional information is given by the by the stopping rule". I thought the girsanov_bayes_rule.pdf(excerpt from Ioannis' brownian motion and stochastic calculus textbook) might be a starting point. This line of literature can help me build a prior for an audience in the entrepreneurship field that can understand the stopping rule "Test Two, Choose One" from
Gans16_foundations of entrepreneurial strategy.pdf, in the context of Bayesian and simulation (-based calibration checking).

andrew's answer

I had no particular theorem in mind, just the general principle that all information in the design should be included in the analysis, hence if there is a sequential design the model should allow for dependence in time. We discuss the general principles in chapter 8 of BDA3.

led me to revisit chapter 8 of BDA3, below on Sequential designs

Consider a randomized experiment in which the probabilities of treatment assignment for unit i depend on the results of the randomization or on outcomes for previously treated units. Appropriate Bayesian analysis of sequential experiments is sometimes described as essentially impossible and sometimes described as trivial, in that the data can be analyzed as if the treatments were assigned completely at random. Neither of these claims is true. A randomized sequential design is ignorable conditional on all the variables used in determining the treatment allocations, including time of entry in the study and the outcomes of any previous units that are used in the design. See Exercise 8.15 for a simple example. A sequential design is not strongly ignorable.

with llm's help, i improved my understanding on the last comment to "Strong ignorability typically assumes that treatment assignment is independent of potential outcomes, given observed covariates. In a sequential design, this assumption doesn't hold because the treatment assignment probabilities change over time and depend on previous outcomes."

i sense there's something there i need to understand... and revisited every ioannis' lecture note searching for the trace of "bayes rule" and ran into a concept called "innovation conjecture" from filtering.

now my question became, what might be the relation between bayes and innovation? also what's the state of innovation conjecture and its intuitive relation with entropy?

Relevant excerpt from Karatzas88_StochAnal.pdf

For linear systems, we also have { F t N } = { F t Y } : the observations and the innovations carry the same information, because in that case there is a causal and causally invertible transformation that derives the innovations from the observations

It is not hard to see that the “innovations conjecture” { F_t^N} = { F_t^Y} holds for linear systems.

will ask ioannis (tbc)

[hyunjimoon]

• entropy and exchangeability are both measurement frameworks that don't require predetermined assumptions i.e.abstract msr
• 💡INNOV CONJECTURE : $\mathcal{Z}_t=\mathcal{U}t$ for $t\geq 0$ and $\hat{u}=\tilde{u}\circ Z$ with $\tilde{u}\in L_a^0(\mu,H)$ iff $H(Z(\nu_n)|\mu)=\frac{1}{2}E{\nu_n}[|\hat{u}_n|_H^2]$

Source	Key Quotes	Meaning in [[🧭🌏abE.moving with bayesian lens]]
exchangeability	🎲 Quotes from above thread "hierarchy or dynamics - I got verification from Andrew Gelman that hierarchy comes before Bayesian" "de Finetti's representation theorem as bypassing the issue of whether or not there is such a thing as chance." "potentially hierarchical, context, Bayesian inference can distinguish between aleatory uncertainty (randomness) and epistemic uncertainty (ignorance)" 🪁Eric's Need-Solution Search Method 1. "Today, technical advances in machine learning techniques for natural language understanding, such as semantic word space models and semantic network analytics, have made it practical to capture descriptions of early-stage, need-solution pairs" 2. Pattern Recognition Without Full Understanding: "The semantic 'word space model' filtering process...proves to be an effective method to automatically discover the semantic relatedness and semantic neighbours of specific user expressions" 3. Aggregation Without Mechanism: "Producers can now thus look for user innovations posted on the web that may involve either known or newly defined needs coupled to new solutions that are gaining traction"	🎲 Quotes from above thread 1. Suggests hierarchical structure underlies both Bayesian inference and functional transitions 2. Directly connects to how exchangeability helps reason about chance without assuming it exists 3. Shows how hierarchical thinking helps categorize different types of uncertainty, similar to how entropy measures different types of functional transitions 🪁Eric's Need-Solution Search Method 1. Measurement Without Complete Understanding: Just like Entropy H(Z(ν)|μ) measures information patterns without needing full mechanistic understanding in innovation Conjecture, semantic analysis identifies valuable patterns in user discussions without requiring understanding of how innovations emerge 2. Emergent Pattern Detection: just like entropy characterizes patterns that emerge naturally from system dynamics in Innovation Conjecture, Need-Solution Method discover semantic relationships that emerge naturally from user discussions in Word space models, rather than imposing predetermined categories
entropy	⚡️ Innovation Conjecture 1. i.e. entropy (H) quantifies the relationship between innovation process (Z) and its underlying dynamics (u) 2. "...entropy characterization of the almost sure invertibility of adapted perturbations of identity" 3. Thm6. Let $T_n=\inf(t: \int_0^t|E_P[\dot{u}_s|\mathcal{U}_s]|^2ds>n)$, define $\hat{u}_n=\hat{u}(t\wedge T_n)$, $U_n=Z+\hat{u}_n$. Then 💡INNOV CONJECTURE (above) for $n\geq 1$, where $d\nu_n=\rho(-\delta_Z\hat{u}_n)dP$ and $\rho(-\delta_Z\hat{u}_n)=\exp(-\int_0^{T_n}E_P[\dot{u}_s|\mathcal{U}_s]dZ_s-\frac{1}{2}\int_0^{T_n}|E_P[\dot{u}_s|\mathcal{U}_s]|^2ds)$ 🔍[[Karatzas88_StochAnal.pdf]] 1. For linear systems, we have { F t N } = { F t Y } : the observations and the innovations carry the same information, because in that case there is a causal and causally invertible transformation that derives the innovations from the observations. 📏 Exaptation Quantification 1. "To arrive at a formal understanding of exaptation in science and technology we start by briefly reviewing related perspectives on the origins of innovations." 2. "The forward normalised entropy can be thought of as a score for interdisciplinary impact." 3. "We focus on seminal publications that initially appear in a given domain and later receive acknowledgements and new functions from other domains."	⚡️ Innovation Conjecture 1.entropy can measure transformative processes i.e. how systems maintain identity during transformation - key to understanding functional shifts 2. shows necessary and sufficient condition using the entropic characterization of invertibility of adapted perturbations of identity Q1. intuition and implementation of stopping time from thm.5? $t_n=\inf \left(t: \int_0^t\left|\dot{v}_s\right|^2 d s>n\right)$ 🔍[[Karatzas88_StochAnal.pdf]] Q2. Design nonlinearity to increase innovation? 📏 Exaptation Quantification 1. Shows how entropy measures the spread of ideas across domains - similar to how exchangeability measures spread of probability across outcomes 2.Describes the exact process that entropy measures - transition from original to new functions 3.Frames exaptation as a measurable process in innovation

[hyunjimoon]

@jeff-dotson and angie will discuss on exchangeability: 10🪙chance_Finetti-UNIFICATION-2018.pdf

• 2021 angie's review on the book: "Starting from Cardano and Ramsey who thought chance and judgement could be measured it extends to how frequency is related to chance, and how chance, judgment, and frequency could be unified. Bayes laid the foundation of modern statistics, Hume explored the problem of induction, Kolmogorov laid general mathematical framework for probability. Application of computability to chance, why chance is essential to modern physics, and lastly we are psychologically predisposed to error when judging chance (Kahneman, Tversky) are introduced.")

• Chris' blog review

• Jeff mentioned ensemble learning

computer scientists are starting to realize models generate really great conditional expectation (deterministic, traditional expectations). But model is not good at defining uncertainty around expectations, when, in reality, everything is like stochastic. And so there's a movement now to try to figure out, okay, how do we quantify uncertainty within our models? And what they're doing is they're trying to find approximations to basically some some version of base. And one computationally feasible approach is attractive because it , ensembles on top of a deep learning model. e.g. if you were to fit classifier with a deep learning model train the model several times repeatedly (angie thinks: this procedural is not irrelevant to SBC and bootstrapping in that additional info exist assuming internal consistency). They argue those ensembled predictions are an approximation to a posterior distribution

Ensemble forecasting contributed to more accurate (majority voting) weather forecasting (book on the primacy of doubt)

[hyunjimoon]

from below three step process of test quantities update, ensemble extracts information from the second step by inspecting variance among conditional expectations i.e. q, p | T(x), X.

There are three types of opportunities depending on the level of epistemic to aleatoric uncertainty. The boundary of epistemic and aleatoric uncertainty keeps getting closer to p =.5, as humanity's measurement precision keeps increasing. e.g. weather after two weeks were aleatoric uncertainty in 18C, but now some of it is replaced by epistemic uncertainty.

1. high uncertainty opportunity (HO) is chosen by ⚖️IE who decides adding hierarchy or not (vision - judge and artist; given T1(x), \exist exchangeability?)

• medium uncertainty opportunity (MO) is chosen by 👩‍🔬👩‍🔬RA who updates prior to posterior (meaning - artist and scientist; given two sets of model class, what are possible result of competing model class)
  -- 👩‍🔬first RA chooses p-x to estimate p(p|x). (in: hierarchical model structure, out: posterior of p)
  -- 👩‍🔬second RA chooses q-p-x to estimate p(q, p|x). (in: hierarchical model structure, out: posterior of q and p)
  this comparison experiment is performed to judge whether deeper structure is economically viable.

• low uncertainty opportunity (LO) is chosen by 🏭who does posterior predictive check (skill - scientist and judge; given the posterior distribution (scientist) what model class is chosen and what are the next test quantities T2(x))

[hyunjimoon]

encouraged by @jeff-dotson, @chasfine, scott stern, diving into Exchangeability.

After re-reading my Bayesian bible, gelman(ebl) from BDA.pdf

• Exchangeability is about the ability to learn about underlying population parameters when individual observations share a common generative process
• It's not just about order invariance, but about the existence of a shared latent structure that can be learned from pooled observations
• Most importantly, it's about when and how we can pool information across seemingly different contexts

I made three seemingly different entrepreneurial contexts using the same "⛏️uncertainty mining with exchangeability" principle.

I didn't propose the word Platform, but gave up rejecting it (upon gpt's proposal). As it has some notion of "underlying structure (latent parameter)", let's give it a chance to expose it to charlie and jeff's evaluation.

⛏️mining evaluation metric with exchangeability

Process Step	💉 Therapeutic Platform	🏎️ Energy Storage Platform	💸 Digital Infrastructure Platform
1. Initial Observations X₁..Xₙ	- Cancer treatment responses- Vaccine antibody production- Rare disease protein levels	- EV battery performance- Home storage cycles- Grid backup responses	- Credit card processing- Bank transfer completions- Mobile payment outcomes
2. Initial T(X) Given Goal	Target: Therapeutic Success- Simple cell entry (yes/no)- Binary protein expression- Basic cell survival	Target: Vehicle Range- Battery life cycles- Peak power delivery- Temperature limits	Target: Payment Success- Basic success rate- Processing completion- Error frequency
3. Exchangeability Given T(X)	Discovery of similar success patterns by tracking protein uptake and expression across diseases	Discovery of similar performance patterns by measuring charge/discharge and thermal stability across uses	Discovery of similar reliability patterns by measuring transaction success and latency across methods
4. Latent Parameter θ	Universal delivery mechanism effectiveness independent of specific protein/disease	Core energy storage capability independent of application context	Fundamental transaction processing reliability independent of payment type
5. Distribution P(θ)	Variation in delivery platform effectiveness across therapeutic contexts and conditions	Distribution of storage capabilities across different environmental and load conditions	Range of processing reliability across different transaction types and volumes
6. Updated T(X)	Enhanced Metrics:- Pathway activation profiles- Cross-disease immune responses- Long-term expression stability	Enhanced Metrics:- System-level stability- Environmental resilience- Multi-use optimization	Enhanced Metrics:- Real-time verification- Cross-border reliability- System-wide security

For Bayesian statisticians: Uncertainty mining through exchangeability provides a systematic framework for discovering and leveraging shared latent structures across seemingly different contexts. This process begins with initial observations X₁..Xₙ and test quantities T(X) chosen for immediate goals, but its power emerges through identifying exchangeability patterns that reveal fundamental latent parameters θ. The key insight is not merely that probabilities remain invariant under permutation, but that there exists a generative process P(θ) that enables information pooling across contexts. This leads to a virtuous cycle where understanding the distribution of latent parameters helps refine test quantities, transforming simple success/failure metrics into sophisticated measures of system-level capabilities. This approach formalizes how initial domain-specific measurements evolve into platform-level insights through systematic verification of exchangeability.
For entrepreneurs and scholars: Consider how innovative companies transform initial context-specific experiments into universal platforms through systematic pattern recognition. A therapeutics startup might begin with simple protein uptake measurements across different diseases, only to discover that these test quantities reveal patterns pointing to a universal delivery mechanism. Similarly, an energy storage company starts with basic battery performance metrics but discovers that certain measurements of efficiency and stability remain meaningful across automotive, home, and grid applications. A payments infrastructure company follows the same path, finding that transaction reliability metrics reveal fundamental patterns across payment types. The key insight for entrepreneurs is that success lies not just in running experiments, but in choosing and evolving test quantities that can reveal shared structures across seemingly different contexts. By systematically verifying when and how information can be pooled, companies can transform aleatory uncertainty ("random" variation) into epistemic uncertainty (learnable patterns) that guides platform development.

[hyunjimoon]

goal conflict to integration was presented by stefano at TBV conference

angie commented

i'm very confused about how i should react to the word "pre-Bayesian" as i may be wrong but i think hierarchical bayesian modeling explains exactly how goals that seemed to be on the same level or conflicting can be integrated, how to discover the presence of latent parameter given exchangeability. for example biotech that is suprised by their technology working well in seemingly unrelated market: cancer and diabetes. this may lead to search of deeper reason and find mrna technology platform.

🌙angie's choice of committee member: jinhua vs bill (viab vs ops.fbl) -> mosche (tech.fbl + viab)

[hyunjimoon]

Mapping Markov Properties to Entrepreneurial Strategy and Calibration

Markov Property	Market Position Strategy	Customer Discovery	Internal Exchangeability	External Exchangeability
Irreducibility	All strategic positions remain mutually accessible (e.g., hardware → SaaS pivot possible)	Multiple paths for customer insight discovery	Technology choices preserve market options	All customer behavior patterns remain discoverable
Aperiodicity	Multiple timing patterns: - Quick experiments - Product development - Market positioning - Maintaining position	Variable feedback loops: - Daily metrics - Monthly cohorts - Market trends - Continuous monitoring	Variable timing in technology-market feedback	Multiple timescales for pattern emergence
Positive Recurrence	Return to proven positions via: - Core team retention - IP preservation - Capital efficiency	Return to previous segments via: - Early adopter relationships - Documented learnings - Flexible technology	Ability to leverage past learnings	Ability to validate discovered patterns
Ergodic Result	Stable yet adaptable market position with multiple paths to success	Robust customer understanding through varied exploration patterns	Consistent forward-backward reasoning about technology and market	Detection of hidden structures in seemingly random market responses

Key Applications for Simulation-Based Calibration

1. Coordinate System Design

   • Ensure full possibility space exploration
   • Multiple success paths
   • Reversible decisions
   • Balance exploration/exploitation

2. Uncertainty Navigation

   • Internal: Technology-market fit
   • External: Cross-venture patterns
   • Temporal: Multiple feedback timescales
   • Structural: Hidden market patterns

3. Success Metric Evolution

   • Continuous re-evaluation capability
   • Multiple valid perspectives
   • Pattern recognition across scales
   • Adaptive optimization

[hyunjimoon]

my intuition is, below three are all similar

1. synthetic population and iterative probability matching
2. state augmentation from Bersekas' dynamic programming which cites "life can live forward but can only be understood backward"

• State Augmentation and Other Reformulations
• some of the assumptions of the basic problem are violated. Generally, in such cases the problem can be reformulated into the basic problem format. This process is called state augmentation because it typically involves the enlargement of the state space. The general guideline in state augmentation is to include in the enlarged state at time k all the information that is known to the controller at time k and can be used with advantage in selecting u (🧲MARKOVIAN)
• the problem can be reduced to the basic problem format, by taking as augmented state
• "Forecasts: Finally, consider the case where at time & the controller has access to a forecast y that results in a reassessment of the probability distribution of wk and possibly of future disturbances." (🌙SIM-BASED CAL) - information relaxation.
• By combining the system equation with the forecast equation Yk+1 Eh, we obtain an augmented system given by (x_k+1, y_k+1) = (f_k(x_k, u_k, w_k), xi_k). now new state is $\tilde{x}_k = (x_k, y_k)$ and because the forecast y_k is known at time k, perfect state information prevails. new disturbance is $\tilde{w}_k = (w_k, xi_k)$ and its probability distribution is determined by distributions Q_i and probabilities p_i and depends explicitly on $\tilde{x}_k$(via y) but not on the prior disturbances. (🌙 connecting this with SBC

Dependent Variable	Intermediary	Independent Variable	Context
A	via C	B	General Case
θ''	via Y	θ'	SBC (Strict Feedback Case)
θ'	via y	xk\tilde{x}_kxk​	SAF (State Augmentation with Forecast)

DESIRE STATE = FORECAST

🌙 from charlie's forecast law (better accuracy with aggregate -> using hierarchical model extractable pattern and use it for individuals - example is learning investment trend and inertia)

• state augmentation, we can cast this problem into the frame-work of the linear-quadratic problem of Section 5.2
• Note also that if an upper bound, call it UPPER, is known for the optimal path length, it can be used to introduce the additional constraint and to make the test (2.14) more effective by augmenting it with the additional inequality ]h (24) + at (27 244) + d < UPPER.

4. clever lifting

[hyunjimoon]

using mapping uncertainty and variable across five papers cld ,

Comprehensive Mapping of Concepts Across Papers

Table 1: Variable Concept Mapping

moon25_testing_day2.pdf	hierarchy and uncertainty.pdf	knight uncertainty.pdf	bias variance decomposition.pdf
ϕ (idea's value/market viability)	Model Class (C)	F (true causal relationship)	F (truth function)
θ(·) (idea's implementation strategy)	Model (M)	p (probability of venture success)	f (statistical model)
MVT (Market Viability Test)	Parameter learning	Experiment (high λ₀, low λ₁)	-
GMT (Go-to-Market Test)	Model learning	Experiment (low λ₀, high λ₁)	-
Binomial demand process	Marginal likelihood	Probability distribution within F	-
Trial outcomes y	Data (D)	Signal s	Y (observed outcome)
α (prior strength parameter)	Prior strength	Size of F (set of probability distributions)	-
L-default (launch bias)	Model class selection	Inertia assumption	-
N-default (no-launch bias)	Parameter estimation	Status quo default	-
Sample size n	Sample size	Experiment type/size	Sample size
Entrepreneur's prior (Beta)	Prior distribution	Prior beliefs	Prior distribution

Table 2: Uncertainty Concept Mapping

moon25_testing_day2.pdf	hierarchy and uncertainty.pdf	knight uncertainty.pdf	bias variance decomposition.pdf
Irreducible uncertainty	Aleatoric uncertainty (inherent randomness)	Aleatory uncertainty	Knightian uncertainty (unmeasurable)
Reducible uncertainty	Epistemic uncertainty (knowledge gaps)	Epistemic uncertainty	Heterogeneous beliefs
Confidence measure	α (prior strength parameter)	E/A ratio	Size of F (smaller F = more confidence)
Uncertainty resolution	Bayesian updating via experiments	Hierarchical learning	Experiments to reduce uncertainty
Opportunity/venture uncertainty	Prior distribution on idea's value	Horizon uncertainty	Range of probability distributions in F
Market validation uncertainty	Validation test (are we making the right product?)	Need uncertainty	Uncertainty over value V
Implementation uncertainty	Verification test (are we making the product right?)	Solution uncertainty	Uncertainty over cost C
Uncertainty reduction goal - theory	Minimize ∆ (p̄ - p)	Reduce epistemic uncertainty	Narrow the set F
Uncertainty reduction goal - practice	Balance information gain vs. pilot cost	Optimize E/A ratio	Balance experiments vs. defaults
Decision approach under uncertainty	Default-based decisions with updating	Horizon-specific exploration patterns	Incomplete preferences with defaults

These tables map the key concepts across the four papers, showing how similar ideas about uncertainty and entrepreneurial decision-making appear under different terminologies and frameworks. The mapping reveals the conceptual bridges between statistical modeling approaches, entrepreneurial testing strategies, and uncertainty management.

[hyunjimoon]

find relationship between the following and my diamond model
from knight uncertainty.pdf

[hyunjimoon]

from bias variance decomposition.pdf above,

conceptual framework for statistical modeling and error decomposition. The first image defines truth as y = F(x), a statistical model as Y = g(X) + ε with an expected value E[Y|X] = f(X), and a sample estimate as ŷ = f̂(x). It distinguishes between explanatory modeling (matching f and F to gain theoretical insights) and predictive modeling (finding f̂ to make ŷ close to Y). The second image shows an error decomposition formula, breaking total error into three components: irreducible error (σ²ε), bias squared, and variance - highlighting that explanatory modeling focuses on minimizing bias² while predictive modeling aims to minimize the combination of bias² and variance.

[hyunjimoon]

imagine you're a researcher wishing to apply andrew's conjecture to answer "under which condition would entrepreneur prefer go to market testing (to market viability testing) using gpt

GPT and I agreed on the relevance of hierarchical Bayesian concepts with entrepreneurial testing strategies, abstracting an entrepreneurial idea as all potential implementations. Specifically, we identified:

• φ (phi): Represents the idea's overarching market value or viability.
• g(φ): Reflects the entrepreneur's initial (possibly subjective or imperfect) belief distribution about the idea’s value before extensive market validation—analogous to prototyping in entrepreneurship.
• θ (theta): Captures detailed implementation parameters tested through actual market experiments.

We systematically compared Tesla (exemplifying MVT with incremental calibration of detailed parameters) and Better Place (exemplifying GMT through rapid broad-market testing without incremental refinement). Drawing insights from "moon25_testing_day2.pdf," "hierarchy and uncertainty.pdf," and "zellweger-zenger-2023," we refined the entrepreneurial interpretation of Gelman's conjecture, emphasizing that detailed implementation parameters (θ) can still be reliably calibrated through extensive testing even under uncertain overarching assumptions (φ). This aligns closely with your core research question by clarifying conditions under which GMT (rapid calibration of implementation details) could still be robust despite broad initial uncertainty about idea value.

---

Applying Gelman’s Theorem as an Entrepreneurship Scientist (Research Proposal)

Research Goal:
To determine the conditions under which entrepreneurs rationally prefer the go-to-market test (GMT) over the market viability test (MVT), using Gelman’s hierarchical calibration conjecture from Bayesian statistics.

Theoretical Foundation:
Gelman’s conjecture suggests entrepreneurs do not necessarily need precise initial assumptions (φ) about the overarching value of an idea to reliably refine and validate detailed implementation parameters (θ) through iterative experimentation. When applied to entrepreneurial testing, this implies that detailed operational calibration can be robust even if broader market viability assumptions remain uncertain, provided the entrepreneur conducts sufficiently diverse and extensive tests.

Hypothesized Conditions Favoring GMT:
From Gelman’s conjecture, GMT should be preferred by entrepreneurs under three specific conditions:

1. High initial optimism (high prior mean) about the idea’s overarching viability reduces the perceived necessity of incremental validation (less need for careful refinement of φ through MVT).

2. Tighter initial belief distributions (low initial uncertainty) about idea viability imply confidence that direct market interactions will effectively calibrate implementation parameters, again diminishing the incremental value of MVT.

3. High cost ratio of MVT relative to GMT makes extensive hierarchical refinement less attractive economically, favoring rapid, real-world testing to empirically reduce variance and directly calibrate detailed operational parameters (θ).

Entrepreneurial Illustration:

• Tesla (MVT approach) incrementally tested and refined their broad assumption (EV viability) and detailed operational parameters (battery tech, vehicle features, pricing). This careful hierarchical refinement paid off, given high uncertainty and a lower cost ratio of incremental experimentation relative to broad market entry.
• Better Place (GMT approach), by contrast, rapidly entered the market with an optimistic initial belief in battery-swapping viability, but failed due to insufficient incremental calibration of detailed implementation parameters (θ), highlighting conditions under which GMT fails.

Conclusion (Intended Contribution):
This application of Gelman’s conjecture provides a rigorous Bayesian theoretical underpinning to entrepreneurial decision-making under uncertainty. It clarifies precisely the conditions (initial optimism, tighter priors, higher cost of incremental tests) under which GMT becomes a rationally preferred entrepreneurial strategy, contributing significantly to the Bayesian Entrepreneurship field by integrating advanced statistical theory into entrepreneurial practice.

[hyunjimoon]

Error Decomposition in Statistical Modeling

• Irreducible Error (σ²ε): The inherent randomness in the phenomenon that cannot be eliminated
• Bias²: The squared difference between expected prediction and true value, representing systematic error
• Variance: The variability of model prediction for a given point, representing sensitivity to sampling

Testing Strategies

• Market Viability Testing (MVT): Focuses on minimizing irreducible error + bias² through controlled experiments
• Go-to-Market Testing (GMT): Focuses on minimizing bias² + variance through real-world implementation

Sensor and Motion Models from Probabilistic Robotics

• Sensor Noise: Analogous to market feedback uncertainty (How reliable is the information we're receiving?)
• Motion Noise (p_noise, hd_noise): Analogous to implementation uncertainty (How accurately can we execute?)

Database Representation of Entrepreneurial Testing Framework

Creating this cohesive database table is a mandatory deliverable. The table must demonstrate both column-wise and row-wise cohesiveness, where relationships between cells maintain logical consistency.

Testing Approach	Primary Error Components	Founder Mindset	Under Low Uncertainty	Under High Uncertainty	Robotics Analogy
Market Viability Testing (MVT)	Irreducible Error + Bias²	Analytical, Risk-averse	Preferred by pessimistic founders	Shifts to being preferred by optimistic founders	High sensor noise, low motion noise - Need to improve perception before action
Go-to-Market Testing (GMT)	Bias² + Variance	Action-oriented, Experimental	Preferred by optimistic founders	Shifts to being preferred by pessimistic founders	High motion noise, low sensor noise - Can explore despite imperfect perception
Hybrid Approach	Dynamic balancing of all three	Adaptable, Learning-oriented	Used when noise sources are balanced	Becomes optimal as uncertainty increases in both dimensions	Simultaneous Localization and Mapping (SLAM) - Learning while acting
Mathematical Foundation	Error = σ²ε + bias² + variance	Quantitative decision framework	Error components can be estimated	Error becomes harder to decompose	Bayesian inference using probabilistic sensor and motion models