making baseline hierarchical bayes models for CIVA

[hyunjimoon]

1. Model Structure:

   Level 1 (Individual Startup):
   log(Valuation_it) = β_0i + β_1i * Stage_it + β_2i * Revenue_it + β_3i * Employees_it + ε_it

   Where:
   i = individual startup
   t = time period
   ε_it ~ N(0, σ^2)

   Level 2 (Startup-specific parameters):
   β_0i = γ_00 + γ_01 * Industry_i + γ_02 * FounderExperience_i + u_0i
   β_1i = γ_10 + γ_11 * Industry_i + u_1i
   β_2i = γ_20 + u_2i
   β_3i = γ_30 + u_3i

   Where:
   u_0i, u_1i, u_2i, u_3i ~ MVN(0, Σ)

2. Prior Distributions:
   γ_00, γ_01, γ_02, γ_10, γ_11, γ_20, γ_30 ~ N(0, 10)
   σ^2 ~ InvGamma(0.01, 0.01)
   Σ ~ InvWishart(I, 5)

This model structure allows for:

1. Firm-specific effects: Each startup has its own intercept and slopes, capturing heterogeneity.

2. Industry effects: The impact of industry on both the baseline valuation and the effect of funding stage.

3. Founder experience: Incorporated at the startup level to affect baseline valuation.

4. Time-varying covariates: Stage, Revenue, and Employees can change over time for each startup.

5. Correlated random effects: The multivariate normal distribution for the u terms allows for correlation between the random effects.

6. Captures heterogeneity across startups and industries.

7. Allows for both startup-specific and population-level inferences.

8. Can handle unbalanced panel data (startups observed at different time points).

9. Incorporates prior knowledge through the prior distributions.

10. Can be easily extended to include more covariates or levels as needed.

using Gen

@gen function startup_valuation_model(num_startups, num_timepoints, industry, founder_experience, stage, revenue, employees)
    # Hyperparameters
    γ_00 ~ normal(0, 10)
    γ_01 ~ normal(0, 10)
    γ_02 ~ normal(0, 10)
    γ_10 ~ normal(0, 10)
    γ_11 ~ normal(0, 10)
    γ_20 ~ normal(0, 10)
    γ_30 ~ normal(0, 10)

    σ² ~ inv_gamma(0.01, 0.01)
    Σ ~ inv_wishart(4, eye(4))

    # Startup-level parameters
    for i in 1:num_startups
        β_0i ~ mvnormal([
            γ_00 + γ_01 * industry[i] + γ_02 * founder_experience[i],
            γ_10 + γ_11 * industry[i],
            γ_20,
            γ_30
        ], Σ)
    end

    # Observations
    for i in 1:num_startups
        for t in 1:num_timepoints
            μ = β_0i[1] + β_0i[2] * stage[i, t] + β_0i[3] * revenue[i, t] + β_0i[4] * employees[i, t]
            log_valuation = {:log_valuation => (i, t)} ~ normal(μ, sqrt(σ²))
        end
    end
end

# Data preparation (you'd replace this with your actual data)
num_startups = 100
num_timepoints = 5
industry = rand(1:5, num_startups)
founder_experience = rand(0:20, num_startups)
stage = rand(1:4, num_startups, num_timepoints)
revenue = rand(LogNormal(10, 2), num_startups, num_timepoints)
employees = rand(LogNormal(3, 1), num_startups, num_timepoints)

# Observed log valuations (you'd replace this with your actual data)
observed_log_valuations = rand(Normal(15, 2), num_startups, num_timepoints)

# Create observations dictionary
observations = Dict()
for i in 1:num_startups
    for t in 1:num_timepoints
        observations[(:log_valuation, i, t)] = observed_log_valuations[i, t]
    end
end

# Perform inference
traces, weights = importance_resampling(startup_valuation_model, (num_startups, num_timepoints, industry, founder_experience, stage, revenue, employees), observations, 1000)

# Extract and analyze results
# (You would add code here to extract parameter estimates, calculate credible intervals, etc.)

BMEV with Marr3 cld reports inv_wishart function is not built into Gen, so we might need to use Stan.

data {
  int<lower=0> N; // Number of startups
  int<lower=0> T; // Number of time points
  int<lower=0> I; // Number of industries
  
  vector[N] industry; // Industry indicator for each startup
  vector[N] founder_experience; // Founder experience for each startup
  int<lower=1, upper=4> stage[N, T]; // Funding stage for each startup at each time point
  vector[N * T] revenue; // Revenue for each startup at each time point
  vector[N * T] employees; // Number of employees for each startup at each time point
  
  vector[N * T] log_valuation; // Observed log valuations
}

parameters {
  vector[7] gamma; // Population-level effects
  real<lower=0> sigma; // Residual SD
  
  matrix[4, N] z; // Matrix for non-centered parameterization
  cholesky_factor_corr[4] L_Omega; // Cholesky factor of correlation matrix
  vector<lower=0>[4] tau; // SDs of random effects
}

transformed parameters {
  matrix[4, N] beta; // Startup-specific effects
  
  // Non-centered parameterization of random effects
  beta = diag_pre_multiply(tau, L_Omega) * z;
  
  // Adding population-level effects
  beta[1] += gamma[1] + gamma[2] * industry + gamma[3] * founder_experience;
  beta[2] += gamma[4] + gamma[5] * industry;
  beta[3] += gamma[6];
  beta[4] += gamma[7];
}

model {
  vector[N * T] mu;
  
  // Priors
  gamma ~ normal(0, 10);
  sigma ~ inv_gamma(0.01, 0.01);
  tau ~ cauchy(0, 2.5);
  L_Omega ~ lkj_corr_cholesky(2);
  
  to_vector(z) ~ normal(0, 1);
  
  // Likelihood
  for (i in 1:N) {
    for (t in 1:T) {
      int idx = (i-1)*T + t;
      mu[idx] = beta[1, i] + beta[2, i] * stage[i, t] + beta[3, i] * revenue[idx] + beta[4, i] * employees[idx];
    }
  }
  
  log_valuation ~ normal(mu, sigma);
}

generated quantities {
  matrix[4, 4] Omega;
  Omega = multiply_lower_tri_self_transpose(L_Omega);
}

[hyunjimoon]

preparing meeting with jeff dotson

1. goal: illustrating essential heterogeneity to bayesian entrepreneurship scholars (#234)

hypothesis: due to lines of research on choice

• test two choose one Gans et al 19_found_es.pdf

• choosing technology Gans20_chooset.pdf,

• Enabling Entrepreneurial Choice

• textbook: ent.strat and choice
  it would be great if there exist some summary statistics of term sheet that captures essential heterogeneity from
  MackeyDotson24_Bayesian Stats in Mangement.pdf

Essential heterogeneity extends beyond simple heterogeneity by recognizing that variations in effects are not just random differences between units, but are systematically related to the units' choices. This correlation between choice and effect poses significant challenges for estimation, invalidating traditional methods like instrumental variables or simple regression approaches.

In the context of strategy research, such as studying diversification's impact on firm performance, essential heterogeneity acknowledges that firms choose to diversify based on their expected benefits. This necessitates a more complex modeling approach that jointly considers both the choice to adopt a strategy and its subsequent effects.

Bayesian methods, particularly hierarchical models, are well-suited for this task as they allow researchers to explicitly model the data-generating process, including the decision-making mechanism and its relationship to outcomes. This approach provides a more accurate representation of strategic decision-making and its consequences, overcoming the limitations of methods that assume randomization or independence between choices and their effects.

Aspect	Heterogeneity	Essential Heterogeneity
Definition	Varying effects across units	Varying effects correlated with choice
Cause	Natural unit differences	Strategic selection
Relation to choice	Independent	Correlated
Impact on estimation	Addressable with random effects	Invalidates traditional IV and regression approaches
Key implication	Effects differ by unit	Units choose based on expected effects
Modeling	Random effects sufficient	Joint choice- outcome modeling required

[hyunjimoon]

2. data to collect

based on Applying Essential Heterogeneity to Early-Stage Startup Valuations cld

three types of data:

• 🔢Quantitative startup metrics: Collect data on easily measurable factors such as revenue, number of employees, founder experience (in years), funding stage, and industry sector. This data would serve as the baseline for the valuation model.

• 😎Qualitative startup information: Gather textual data from startup profiles, pitch decks, and company descriptions. This could include information about the startup's mission, market opportunity, competitive advantage, and team background. An LLM could be used to parse this text data and identify common themes or characteristics across startups.

• 💰Investor decision: Collect information on investor decisions, including which startups they chose to invest in, the terms of the investments (e.g., SAFE agreement details), and any public statements or rationales provided for their investment decisions. This data would help model the investor decision-making process, which is crucial for capturing the essential heterogeneity in startup valuations.

---

for 😎, 💰

table is sorted from the most efficient persuasion (high 💡/💰) to the least efficient (low 💡/💰), considering our goal of illustrating essential heterogeneity to Bayesian entrepreneurs. The top rows represent data categories that provide the most valuable information relative to the cost of collecting that data.

i built based on info on YC's SAFE document here

Data Category	Specific Data Points	Rationale	Cost	Info Amount	Persuasiveness per Unit Cost
Future Financing Provisions	- Equity Financing conversion terms - Liquidity Event payout terms - Dissolution Event payout terms - Most Favored Nation clause (if present)	These provisions indicate how the startup plans to handle future funding rounds and exit scenarios.	💰	💡💡💡	💡💡💡/💰
Quantitative Startup Metrics	- Purchase Amount (from SAFE) - Post-Money Valuation Cap (from SAFE) - Revenue (last 12 months) - Burn rate - Number of employees - Total addressable market (TAM) size - User/customer growth rate	These metrics provide objective, measurable data points that serve as a baseline for valuation models. The SAFE document directly provides some of these metrics.	💰	💡💡💡	💡💡💡/💰
Qualitative Startup Information	- Company stage (inferred from use of SAFE) - Funding strategy (use of SAFE vs. priced round) - Exit strategy options (from Liquidity Event definition) - Intellectual property status (from Company Representations) - Competitive advantage description (inferred from IP rights) - Governance structure (from amendment and voting provisions) - Risk factors (inferred from representations and warranties) - Corporate structure (e.g., state of incorporation)	This qualitative data, much of which can be extracted or inferred from the SAFE, captures nuanced aspects of a startup that influence investor decisions.	💰	💡💡💡	💡💡💡/💰
Capital Structure	- Existing Capital Stock composition - Converting Securities details - Liquidity Capitalization breakdown	Understanding the current and potential future capital structure provides insights into the company's financing strategy and potential dilution.	💰	💡💡	💡💡/💰
Legal and Regulatory Context	- Governing law jurisdiction (from miscellaneous section) - SEC compliance status (inferred from securities law references) - Tax treatment of SAFE (from section 5g) - Transferability restrictions (from section 5d)	The legal and regulatory context can impact the startup's operations and future funding options.	💰	💡💡	💡💡/💰
Founders and Team Information	- Founder names and titles (from signature page) - Board structure (inferred from voting rights and amendment provisions) - Option pool size (from Unissued Option Pool definition) - Promised Options (from definitions)	Information about the team and its structure provides insights into the startup's human capital and governance.	💰💰	💡💡💡	💡💡/💰💰
Investor Information	- Investor name and type (from signature page and representations) - Investor rights (e.g., information rights, pro-rata rights if mentioned) - Investor accreditation status (from Investor Representations) - Previous investors (if mentioned in capitalization table)	Understanding the investors involved can provide context about the startup's network and perceived potential.	💰💰	💡💡	💡/💰
Market and Industry Context	- Industry sector (if specified in company description) - Comparable company valuations - Recent exits in the sector - Regulatory environment specific to the company's industry	This data provides context for the valuations and helps account for sector-specific trends.	💰💰💰	💡💡	💡/💰💰
Post-Investment Performance Tracking	- Milestone achievements post-SAFE issuance - Follow-on funding success - Product/feature launches - Key hires made - Changes in metrics (revenue, users, etc.) since SAFE issuance	While not directly from the SAFE, tracking these points helps validate the model by comparing predicted outcomes with actual performance.	💰💰💰	💡💡	💡/💰💰

[hyunjimoon]

info on SAFE

current practice is post money safe with valuation cap 🧢 without discount according to here

I think SAFE which started from paul graham's spirit of high resolution funding, increasing the optimization dimension based on conversational inference is a good place to start.

Events
(a) Equity Financing. If there is an Equity Financing before the termination of this Safe, on the initial closing of such Equity Financing, this Safe will automatically convert into the greater of: (1) the number of shares of Standard Preferred Stock equal to the Purchase Amount divided by the lowest price per share of the Standard Preferred Stock; or (2) the number of shares of Safe Preferred Stock equal to the Purchase Amount divided by the Safe Price.
In connection with the automatic conversion of this Safe into shares of Standard Preferred Stock or Safe Preferred Stock, the Investor will execute and deliver to the Company all of the transaction documents related to the Equity Financing; provided, that such documents (i) are the same documents to be entered into with the purchasers of Standard Preferred Stock, with appropriate variations for the Safe Preferred Stock if applicable, and (ii) have customary exceptions to any drag-along applicable to the Investor, including (without limitation) limited representations, warranties, liability and indemnification obligations for the Investor.
(b) Liquidity Event. If there is a Liquidity Event before the termination of this Safe, the Investor will automatically be entitled (subject to the liquidation priority set forth in Section 1(d) below) to receive a portion of Proceeds, due and payable to the Investor immediately prior to, or concurrent with, the consummation of such Liquidity Event, equal to the greater of (i) the Purchase Amount (the “Cash-Out Amount”) or (ii) the amount payable on the number of shares of Common Stock equal to the Purchase Amount divided by the Liquidity Price (the “Conversion Amount”). If any of the Company’s securityholders are given a choice as to the form and amount of Proceeds to be received in a Liquidity Event, the Investor will be given the same choice, provided that the Investor may not choose to receive a form of consideration that the Investor would be ineligible to receive as a result of the Investor’s failure to satisfy any requirement or limitation generally applicable to the Company’s securityholders, or under any applicable laws.
Notwithstanding the foregoing, in connection with a Change of Control intended to qualify as a tax-free reorganization, the Company may reduce the cash portion of Proceeds payable to the Investor by the amount determined by its board of directors in good faith for such Change of Control to qualify as a tax-free reorganization for U.S. federal income tax purposes, provided that such reduction (A) does not reduce the total Proceeds payable to such Investor and (B) is applied in the same manner and on a pro rata basis to all securityholders who have equal priority to the Investor under Section 1(d).
(c) Dissolution Event. If there is a Dissolution Event before the termination of this Safe, the Investor will automatically be entitled (subject to the liquidation priority set forth in Section 1(d) below) to receive a portion of Proceeds equal to the Cash-Out Amount, due and payable to the Investor immediately prior to the consummation of the Dissolution Event.

recommended video:

• https://youtu.be/Dk6JNTDec9I?si=WCi9Z-e1rKokVh42
• https://youtu.be/KLz5CY3jOEo?si=T_262726ycc5lXD3

[hyunjimoon]

[hyunjimoon]

after getting jeff's input, will continue with getting vikash's input for CIVA baseline models cld to ask vikash for his input

[hyunjimoon]

using Startup Valuation Model with jeff cld

data {
  int<lower=0> N_startups;  // Number of startups
  int<lower=0> N_timepoints;  // Number of time points
  int<lower=0> N_stages;  // Number of funding stages (e.g., 5 for Seed, A, B, C, D)
  int<lower=0> N_criteria;  // Number of decision criteria

  // Startup characteristics
  vector[N_startups] founder_experience;
  int<lower=1, upper=5> industry[N_startups];

  // Time-varying covariates
  int<lower=1, upper=N_stages> stage[N_startups, N_timepoints];
  vector[N_startups * N_timepoints] revenue;
  vector[N_startups * N_timepoints] employees;
  vector[N_startups * N_timepoints] term_sheet_sentiment;
  vector[N_startups * N_timepoints] liquidity_event_protection;

  // Observed log valuations
  vector[N_startups * N_timepoints] log_valuation;

  // Decision criteria importance (from discrete choice experiments)
  matrix[N_criteria, N_stages] criteria_importance;
}

parameters {
  // Global parameters
  vector[N_criteria] beta_global;
  real<lower=0> sigma_global;

  // Stage-specific parameters
  matrix[N_criteria, N_stages] beta_stage;
  vector<lower=0>[N_stages] sigma_stage;

  // Startup-specific parameters
  matrix[N_criteria, N_startups] beta_startup;
  vector<lower=0>[N_startups] sigma_startup;

  // Industry effects
  vector[5] industry_effect;

  // Time trend
  real time_trend;

  // Uncertainty reduction parameter
  real<lower=0, upper=1> uncertainty_reduction;
}

transformed parameters {
  vector[N_startups * N_timepoints] mu;
  
  for (i in 1:N_startups) {
    for (t in 1:N_timepoints) {
      int idx = (i-1) * N_timepoints + t;
      int current_stage = stage[i, t];
      
      mu[idx] = 0;
      
      // Global effects
      mu[idx] += beta_global[1] * founder_experience[i];
      mu[idx] += beta_global[2] * revenue[idx];
      mu[idx] += beta_global[3] * employees[idx];
      mu[idx] += beta_global[4] * term_sheet_sentiment[idx];
      mu[idx] += beta_global[5] * liquidity_event_protection[idx];
      
      // Stage-specific effects
      for (c in 1:N_criteria) {
        mu[idx] += beta_stage[c, current_stage] * criteria_importance[c, current_stage];
      }
      
      // Startup-specific effects
      for (c in 1:N_criteria) {
        mu[idx] += beta_startup[c, i] * criteria_importance[c, current_stage];
      }
      
      // Industry effect
      mu[idx] += industry_effect[industry[i]];
      
      // Time trend
      mu[idx] += time_trend * (t - 1);
    }
  }
}

model {
  // Priors
  beta_global ~ normal(0, 5);
  sigma_global ~ cauchy(0, 2.5);
  
  for (s in 1:N_stages) {
    beta_stage[:, s] ~ normal(0, 1);
    sigma_stage[s] ~ cauchy(0, 2.5);
  }
  
  for (i in 1:N_startups) {
    beta_startup[:, i] ~ normal(0, 1);
    sigma_startup[i] ~ cauchy(0, 2.5);
  }
  
  industry_effect ~ normal(0, 1);
  time_trend ~ normal(0, 0.1);
  uncertainty_reduction ~ beta(2, 2);

  // Likelihood
  for (i in 1:N_startups) {
    for (t in 1:N_timepoints) {
      int idx = (i-1) * N_timepoints + t;
      real sigma_total = sigma_global + sigma_stage[stage[i, t]] + sigma_startup[i];
      sigma_total *= (1 - uncertainty_reduction * (stage[i, t] - 1) / (N_stages - 1));
      log_valuation[idx] ~ normal(mu[idx], sigma_total);
    }
  }
}

generated quantities {
  vector[N_startups * N_timepoints] log_valuation_pred;
  
  for (i in 1:N_startups) {
    for (t in 1:N_timepoints) {
      int idx = (i-1) * N_timepoints + t;
      real sigma_total = sigma_global + sigma_stage[stage[i, t]] + sigma_startup[i];
      sigma_total *= (1 - uncertainty_reduction * (stage[i, t] - 1) / (N_stages - 1));
      log_valuation_pred[idx] = normal_rng(mu[idx], sigma_total);
    }
  }
}

Parameter	Explanation
beta_stage	Captures how the effects of different decision criteria change across funding stages (Seed, Series A, B, C, D). Critical for understanding evolving valuation dynamics.
criteria_importance	Input matrix reflecting the importance of different factors at each stage. Directly interacts with beta_stage to determine stage-specific effects on valuation.
uncertainty_reduction	Quantifies how the variance in valuations decreases in later stages. Key for understanding risk profiles across different investment stages.
beta_startup	Captures startup-specific effects, accounting for unique characteristics of each company. Explains consistent over- or under-performance relative to peers.
industry_effect	Accounts for systematic differences in valuations across industries. Important for benchmarking and comparative analysis across sectors.

liquidity_event_protection?, founder_experience and revenue are input data, while beta_global and sigma_startup are estimated by the model

Parameter	Definition
beta_global	Global effects of various factors on valuation across all startups and stages
sigma_global	Global variance in valuations
beta_stage	Stage-specific effects of decision criteria on valuation
sigma_stage	Stage-specific variance in valuations
beta_startup	Startup-specific effects, capturing the unique characteristics of each startup
sigma_startup	Startup-specific variance in valuations
industry_effect	Effect of different industries on startup valuation
time_trend	Overall trend in valuations over time
uncertainty_reduction	Parameter capturing how uncertainty in valuations decreases in later stages
founder_experience	Measure of the founding team's prior experience
industry	Categorical variable indicating the industry of each startup
stage	Current funding stage of the startup (e.g., Seed, Series A, B, C, D)
revenue	Revenue of the startup at each time point
employees	Number of employees in the startup at each time point
term_sheet_sentiment	Measure of the favorability of the term sheet
liquidity_event_protection	Measure of protections against liquidity events in the funding agreement
criteria_importance	Matrix capturing the importance of different decision criteria at each funding stage
mu	Predicted log valuation for each startup at each time point
log_valuation	Observed log valuation for each startup at each time point
log_valuation_pred	Predicted log valuation for model checking and future predictions

[hyunjimoon]

shared below to jeff by explaining the concept of "thinking", "language understanding" in rational meaning construction (RMC) language, with the goal of writing abstract to introduce the idea of perceptual unpacking with RMC

[hyunjimoon]

using designing utility functions for f&i cld

Figure for January Conference:

1. Table 1: Descriptive Statistics (Jeff's lead)

   • Panel A: Summary of Term Sheet Features
   • Panel B: Distribution of Perceptual Attributes
   • Panel C: Founder-Investor Characteristics

2. Table 2: Phantom Attribute Inference Results (Joint)

   • Panel A: Observable Terms → Perceived Control
   • Panel B: Observable Terms → Perceived Fairness
   • Panel C: Observable Terms → Perceived Fit

3. Table 3: Stan Model Results (Angie's lead)

   • Panel A: Parameter Estimates for Meaning Construction
   • Panel B: Model Fit Statistics
   • Panel C: Predicted vs Actual Outcomes

4. Table 4: Validation Tests (Joint)

   • Panel A: Out-of-Sample Predictions
   • Panel B: Robustness Checks
   • Panel C: Alternative Specifications

Abstract:

[Empirical Motivation - Jeff lead]
"This paper examines how founders and investors construct meaning during term sheet negotiations, particularly in ventures where standard funding models create fundamental misalignments. Building on phantom attribute theory, we analyze how observable terms trigger inferences about control, fairness, and partnership fit."

🙋‍♀️Q1. add the following Yes / No ", specifically in contexts where business development requirements conflict with traditional venture metrics. (e.g. micro-mobility where investors misperceive founder's business model as revenue-centered software where in reality, it's asset heavy)"

[Computational Framework - Angie lead]
"We develop a probabilistic programming framework that formalizes meaning construction as Bayesian inference over latent attributes, explicitly modeling the divergence between investor expectations and venture requirements. Our model, implemented in Stan, maps observable terms to constructed meanings through a hierarchical inference process that captures both misaligned incentives and varying interpretations of milestone-based progress metrics."

[Integration & Contribution - Joint]
"By combining experimental data with computational modeling, we reveal how term sheet negotiations embody a form of rational meaning construction where venture development requirements often conflict with standardized investment expectations. Our model demonstrates how identical terms trigger different meanings between investors and founders, particularly around development timelines and operational milestones. This explains previously puzzling phenomena in entrepreneurial finance, such as why ventures accept seemingly suboptimal terms or alter their development trajectory under growth pressure. The results provide practical guidance for aligning investor expectations with venture development needs while advancing our theoretical understanding of meaning construction in high-stakes financial negotiations."

🙋‍♀️Q2. how important is understanding equilibrium for rational meaning function? do you see ourselves analyzing game theoretic situation?

🙋‍♀️Q3. building on Q2, does the following make sense? "...we show how SAFE agreements and term sheets represent equilibrium conditions settled between founders and investors through an iterative meaning construction process. Our model demonstrates how these equilibria emerge differently across founder-investor pairs despite identical terms, particularly around investor's understanding of industry business model which affects operational milestones."

---

Division of Labor:

• Jeff: Lead empirical design, data collection, descriptive analysis
• You: Lead Stan implementation, computational framework, model estimation
• Joint: Integrate findings, develop theoretical implications, prepare final presentation

Phase	Jeff's Tasks	Your Tasks	Joint Tasks	Timeline
Phase 1: Study Design	- Design conjoint for term sheet attributes - Define phantom attributes	- Implement Stan model above - Test with synthetic data	- Align on attributes to measure - Draft research questions	Nov 1-15
Phase 2: Pilot	- Design small-scale survey - Recruit 20-30 participants	- Debug Stan implementation - Create data processing pipeline	- Review pilot results - Refine approach	Nov 16-30
Phase 3: Main Study	- Launch full survey - Oversee data collection	- Finalize Stan models - Run inference on pilot data	- Monitor data quality - Initial analysis	Dec 1-15
Phase 4: Analysis & Writing	- Write empirical section - Draft implications	- Write computational section - Create visualizations	- Integrate findings - Prepare presentation	Dec 16-Jan 1
Phase 5: Finalization	- Review technical aspects - Polish presentation	- Finalize models - Document code	- Practice presentation - Submit materials	Jan 1-15

Key Deadlines:

• Nov 15: Complete study design
• Nov 30: Finish pilot and refinements
• Dec 15: Complete data collection
• Jan 1: Draft completion
• Jan 15: Final presentation ready

Weekly Checkpoints:

1. Every Monday: Brief sync on progress
2. Every Friday: Share weekly accomplishments and blockers

data {
  int<lower=0> N;  // Number of term sheets
  int<lower=0> K;  // Number of observable terms (valuation, investment amount, etc)
  int<lower=0> P;  // Number of perceptual dimensions (control, fairness, etc)
  matrix[N, K] X;  // Observable term sheet features
  vector[N] y;     // Observed choices/preferences
  
  // Prior parameters
  real mu_alpha;
  real<lower=0> sigma_alpha;
  real mu_beta;
  real<lower=0> sigma_beta;
}

parameters {
  // Latent perceptual variables
  matrix[N, P] phi;  // Perceived attributes
  
  // Weights for how terms map to perceptions
  matrix[K, P] W;    // Term-to-perception weights
  
  // Perception-to-utility weights
  vector[P] beta;    // Perception utility weights
  real alpha;        // Base utility
  
  // Variance parameters
  real<lower=0> sigma_phi;  // Perception noise
  real<lower=0> sigma_y;    // Choice noise
}

model {
  // Priors
  alpha ~ normal(mu_alpha, sigma_alpha);
  beta ~ normal(mu_beta, sigma_beta);
  
  // How observable terms generate perceptions
  for (n in 1:N) {
    phi[n] ~ normal(X[n] * W, sigma_phi);
  }
  
  // How perceptions influence utility/choice
  for (n in 1:N) {
    y[n] ~ normal(alpha + phi[n] * beta, sigma_y);
  }
}

generated quantities {
  // Predicted choices
  vector[N] y_pred;
  for (n in 1:N) {
    y_pred[n] = normal_rng(alpha + phi[n] * beta, sigma_y);
  }
}