: Bayesian Inference, Causal Identification, and Alternative Text Data

This chapter starts from frozen constructed results and retains the boundaries on template generalization and trust in summaries; no training or model call was performed.

Main thread: Prediction, probability updating, and intervention decisions in financial research have distinct mathematical foundations. Bayesian inference uses posterior probabilities to represent uncertainty with small samples and across groups; causal identification uses potential outcomes and DAGs to express confounding paths and identifies intervention effects under explicit assumptions; alternative data and large language models require feature extraction under strict contracts for point-in-time (PIT) eligibility, delayed ingestion, version revisions, and training-corpus cutoffs. Fitting a vocabulary across time, confusing retrospective smoothing with real-time prediction, or treating an LLM pretraining cutoff as historical PIT eligibility can all create fatal look-ahead bias. Prerequisites: Chapter 6: Probability and Chapter 7: Mathematical Statistics (conditional probability, Bayes’ theorem, expectation, and variance); Chapter 8: Linear Regression (least squares and regularization); Chapter 12: Fundamental Quantitative Analysis and PIT (financial-statement releases and point-in-time availability); Chapter 22: Research Design (causal inference and confounding bias); and Chapter 31: Label Purging and Embargo and Chapter 38: Machine-Learning Pipelines (strict time-series validation and leakage-resistant features). By the end, you should be able to: derive the Beta-Binomial conjugate posterior and the joint predictive covariance of two future periods; calculate by hand partial-pooling shrinkage for a normal hierarchical mean with known hyperparameters and its posterior predictive variance; build a causal DAG and identify the ATE by stratified standardization under identification assumptions, calculate the reversal mechanism in Simpson’s paradox, and diagnose violations of positivity; construct a rigorous after-hours earnings-call availability ledger and filter eligible versions under time-zone conversion and delayed ingestion; fit a TF-IDF vocabulary independently within the training window and compare text Ridge regression with a historical-mean baseline; calculate cosine similarity for dense word embeddings and illustrate sentiment reversal under negation; and design an offline LLM research ledger that includes prompt-version hashes and a review of training-corpus cutoffs.

1. Learning Objectives

1. Master conjugate Bayesian updating and predictive distributions: Starting from the prior and likelihood, derive the Beta-Binomial posterior parameters $(\alpha_{\mathrm{post}}, \beta_{\mathrm{post}})$ in full; derive the one-period future predictive mean $E[\tilde{y} \mid y]$ and the joint predictive probability for two future periods, $E[\tilde{y}_1 \tilde{y}_2 \mid y] = E[p^2 \mid y]$; prove that two future Bernoulli trials are conditionally independent given $p$ but correlated under the marginal posterior predictive distribution, with covariance exactly equal to the posterior variance $\operatorname{Var}(p \mid y)$.
2. Understand shrinkage through hierarchical models and partial pooling: Using Section 5.4 of Gelman et al. (BDA3) as a reference, derive the precision-weighted shrinkage weight for each continuous-return asset group’s posterior mean when hyperparameters $(\mu, \tau^2, \sigma^2)$ are known: $w_j = \frac{n_j / \sigma^2}{1/\tau^2 + n_j / \sigma^2}$. Calculate by hand how a small-sample group ($n=4$) and a large-sample group ($n=36$) shrink toward the whole-market prior mean $\mu=0.0$, and derive the posterior predictive variance $\operatorname{Var}_{\mathrm{pred}, j} = V_j + \sigma^2$, which includes uncertainty about the group and noise in a new observation. This chapter is limited to teaching with known hyperparameters; it does not claim to estimate unknown hyperparameters as empirical Bayes.
3. Build causal graphs (DAGs) and a potential-outcomes framework: Distinguish statistical association $P(Y \mid T=t)$ from causal intervention $P(Y \mid \operatorname{do}(T=t))$; list the four assumptions for identifying the average treatment effect (ATE): consistency, conditional exchangeability, positivity/overlap, and the stable unit treatment value assumption (SUTVA, including no interference between units and no hidden treatment versions); calculate the Simpson’s paradox example by hand, showing how an unadjusted difference in binary success rates of $+0.38$ (38 percentage points) becomes a negative observed difference after stratified standardization and, under the identification assumptions, an ATE of $-0.10$ (10 percentage points); stress that an adjusted association cannot be equated with unconditional true causality without the identification assumptions; and write an executable counterexample of collider bias and derive the negative covariance $-1/9$.
4. Specify a point-in-time (PIT) contract for unstructured alternative data: Build a multi-time-zone ledger containing event_time, available_at, ingested_at, revision_time, signal_time, trade_time, and label_available_at; use a U.S. after-hours earnings call at 16:05 to apply a 20-minute vendor delay and a 5-minute internal-ingestion delay, and work through daylight-saving-time (EST/EDT) and UTC conversions; implement exclusion of amended versions revised after the signal time and of records ingested too late; and distinguish public accessibility from authorization for commercial scraping, training, and redistribution.
5. Build a leakage-resistant text-prediction pipeline aligned with its baseline: Use 120 constructed earnings-call events generated from four repeated templates (with the CONSTRUCTED_CALL_ prefix and an explicit teaching-only label); enforce time-series splits of 59 training documents, 29 validation documents, and 30 test documents (training and validation labels must mature strictly before the first signal time in the following block, <); fit the vocabulary and IDF independently within the training window and freeze them before transforming validation and test documents; keep training features unchanged when unseen or out-of-vocabulary (OOV) terms appear in the test window; compare a global-mean baseline with TF-IDF + Ridge regression on the same test set; retain the negative low-SNR result (test MSE $2.3603 \times 10^{-4}$ versus baseline $2.3274 \times 10^{-4}$, relative $R^2 = -0.0141$; time-series summary Spearman rank correlation $+0.0763$; cross-sectional IC is undefined because there is only one asset per day, and rank correlation is undefined for the constant-mean baseline); this chapter is prediction-only and specifies an interface to downstream portfolio accounting and fee calculations without inventing turnover from asset rotation.
6. Understand the geometry beneath word embeddings and negation-based sentiment analysis: Give the formal TF-IDF definition and a complete, step-by-step hand calculation for two teaching documents; use a hand-calculated $6 \times 3$ dense embedding matrix to explain angles between word vectors and mean pooling; and use a bag-of-words model (clarifying that bag-of-words representations do not assume statistical independence among words) to show errors under the negation terms “not” and “no,” exposing the limits of unigram-frequency features and the need for n-grams or rule-based enhancements.
7. Maintain a rigorous offline research ledger for large language models (LLMs): Create an offline prompt template, version control, and a SHA-256 check of the complete actual input text; clearly mark the model as not called (not_run / unknown) and use the pretraining cutoff only as a hypothetical counterexample; analyze why an LLM pretraining-corpus cutoff cannot establish verifiable historical PIT eligibility; explain why zero temperature and a seed do not eliminate hardware or underlying nondeterministic drift; and assess hallucinations, reproducibility, and fact-checking in quantitative research.

All run directories must be new paths that do not already exist. Use a different trial number when rerunning an example.

Completion criteria: exercises in Sections 15–17 can be derived or run independently; code/ch43_bayesian_text.py runs successfully in an independent output directory and generates all figures and the complete summary.

2. Why Quant Needs This

The companies and earnings-call scenarios below are constructed teaching examples, not established market facts. Conclusions may still fail if the prior is wrong, samples are dependent, or causal assumptions do not hold.

Quantitative researchers often face three common cognitive traps and modeling failures:

The first trap is the illusion of point estimates with small samples. In low-frequency macro-hedging or event-driven strategies (such as defaults, CEO departures, or major acquisitions), the historical sample may be tiny, perhaps only a dozen observations or fewer. Frequentist point estimates such as the sample mean can fluctuate sharply. If only four companies in an industry report positive earnings surprises and all four rise, the sample-frequency point estimate is 100%; this does not mean that frequentist inference asserts a 100% population win rate. A small-sample interval still reflects uncertainty. Bayesian methods provide an axiomatic framework for updating uncertainty: a reasonable prior can shrink an extreme small-sample estimate toward the whole-market prior mean and yield a full posterior distribution and posterior predictive variance. Shrinkage can reduce sensitivity to small samples, but it does not guarantee stable optimizer settings; a misspecified prior can introduce bias.

The second trap is treating a confounded association as an intervention effect. In the constructed supplemental-call example in this chapter, the treatment group’s overall target-attainment rate is 38 percentage points higher than the control group’s. Companies with high analyst attention are more common in the treatment group and also have higher baseline attainment rates. After stratifying by attention, the observed difference is −10 percentage points within each stratum. This reversal shows how group weights affect the overall association. Interpreting the standardized difference as an ATE still requires the identification assumptions below; the table does not establish the true mechanism of management behavior or actual trading profit and loss.

The third trap is look-ahead leakage in unstructured data and large-language-model applications. As alternative data (news, research reports, calls, and social media) and LLMs become more common, text has become a central resource in quantitative research. Yet its time structure is fragile:

1. Ingestion delay: A news item is published after the close at 16:05, takes 20 minutes to reach the vendor, and another 5 minutes to enter the company’s database. Before 16:30, the local quantitative research database cannot use this text. Generating a signal at 16:15 and assuming it could be executed after hours would amount to a time-traveling trade that is infeasible in live trading;
2. Retrospective revisions: Earnings-call transcripts are often corrected or supplemented by the issuer the following day. Backtesting a historical signal with the latest revised version leaks information about future corrections;
3. Pretraining-corpus contamination: LLM pretraining corpora are collected from the internet. Suppose a model’s training cutoff is the end of 2023; its corpus could include reports published after events in 2022. The cutoff statement alone cannot prove that an output for a 2022 historical signal contains no future knowledge. Whether contamination actually occurred requires a dedicated experiment.

This chapter connects Bayesian inference with causal identification mathematically and implements a rigorous point-in-time text-processing pipeline, giving quantitative research a firm boundary around time and causality.

3. Intuition

We will build a clear mental picture from three examples in finance and statistics:

3.1 Bayesian Conjugate Updating and Partial Pooling of Asset Returns

• Intuition for conjugate updating: Let the win probability of a quantitative strategy be $p \in [0, 1]$. Before live trading, set $p \sim \operatorname{Beta}(2, 2)$ based on prior knowledge of similar strategies across the market (prior expected win rate 0.5). If 8 of 10 constructed trades are profitable, the maximum-likelihood point estimate is 80%, but this point estimate does not express small-sample uncertainty. Under the prior specified here, Bayesian updating gives a posterior mean of approximately 71.4%, balancing prior knowledge with limited evidence.
• Partial pooling of continuous asset returns: Consider estimating continuous excess returns across industry or style groups. Let the center of the whole-market long-run unconditional excess-return prior be $\mu = 0.0$, between-group dispersion be $\tau = 0.02$, and the observation noise for an individual trade be $\sigma = 0.04$.
  • Group 1 is a newly listed technology stock with very little history ($n_1 = 4$) and sample mean $\bar{y}_1 = +0.03$ (+3%). Because the sample is small, the model assigns it a 50% shrinkage weight, pulling the posterior mean strongly back to +0.015;
  • Group 2 is a mature large-cap blue chip with a substantial history ($n_2 = 36$) and sample mean $\bar{y}_2 = +0.01$ (+1%). Given the stronger evidence, the model assigns the sample mean a 90% weight, yielding a posterior mean of +0.009. Partial pooling automatically balances “be cautious about groups with little evidence” against “respect groups supported by strong data.”

3.2 Simpson’s Paradox and Confounding by Analyst Attention

This constructed example contains 200 company records. The treatment is “a supplemental after-hours earnings call with updated guidance” ($T=1$), and the outcome is whether the next-day opening excess return meets its target ($Y \in \{0,1\}$). In the observed table, the target-attainment rates are 64% for the treatment group and 26% for the control group, an unadjusted difference of 38 percentage points. This is only an association. Stratify by analyst attention $Z$. In the high-attention stratum, 90 treated companies have 63 successes and 10 control companies have 8 successes, so the within-stratum difference is $0.70-0.80=-0.10$. In the low-attention stratum, 10 treated companies have 1 success and 90 control companies have 18 successes, so the difference is also $0.10-0.20=-0.10$. Ninety percent of the treatment group comes from the high-attention stratum, while 90% of the control group comes from the low-attention stratum. The overall association therefore has the opposite sign from the within-stratum differences. Using target-population weights of 50% for each stratum gives a standardized difference of $-0.10$. It can be interpreted as the target-population ATE only under consistency, conditional exchangeability, positivity, and the stated SUTVA assumptions. The four-cell table itself cannot verify those assumptions and is not evidence of a real market intervention.

3.3 Time Travel in Earnings-Call Text

A listed company ends an after-hours call at 16:05. You find its text in a historical database and use it to calculate a signal for a purchase at the next day’s 09:30 open. The timeline appears to move forward, but the database text may be a version revised at 10:00 the following morning after review by the company’s legal and investor-relations teams (for example, to correct a misspoken figure). If you calculate a sentiment score from that text at 09:00, your algorithm has effectively traveled through time. Freeze the historical ingestion ledger at the signal time and reject future versions strictly.

4. Mathematical Foundations

This chapter follows the course notation standard in notation.md. The prerequisite symbols, dimensions, and time contract are:

5. Formal Definitions

5.1 Conjugate Bayesian Updating

The prior $p(\theta)$ is a probability distribution for parameter $\theta$ before observing the data. The likelihood $p(y\mid\theta)$ is a function of the parameter for fixed data $y$; it is usually not a normalized density over the parameter. Bayes’ theorem combines them to form the posterior:

The denominator is the marginal likelihood, which normalizes the posterior to integrate to 1; for a discrete parameter, replace the integral with a sum. This does not conflate the probability of the data with the probability of the parameter.

In Bayesian statistics, a prior distribution $p(\theta)$ is called a conjugate prior for likelihood $p(y \mid \theta)$ when the posterior $p(\theta \mid y)$ belongs to the same distribution family as the prior. For a binomial likelihood $y \sim \operatorname{Binomial}(n, p)$, the conjugate prior is the Beta distribution $\operatorname{Beta}(\alpha, \beta)$. The posterior is also Beta:

5.2 Posterior Predictive Distribution

The posterior predictive distribution is the marginal probability distribution of a new future observation $\tilde{y}$ after observing existing data $y$. It is obtained by integrating over posterior uncertainty about the model parameter $\theta$:

Key result: For two future Bernoulli trials $(\tilde{y}_1, \tilde{y}_2)$ that share the same unknown parameter $p$, they are conditionally independent given $p$: $P(\tilde{y}_1=1, \tilde{y}_2=1 \mid p) = p^2$. After integrating out $p$, they are correlated under the marginal posterior predictive distribution:

When the posterior concentration is finite and both Beta parameters are positive, the covariance is exactly $\operatorname{Var}(p \mid y)>0$. It arises from uncertainty about the shared parameter; it does not mean one future outcome physically causes the other.

5.3 Normal Hierarchical Model and Partial Pooling

The closed form assumes independent group priors conditional on the known hyperparameters, and conditionally independent, homoskedastic normal observations within each group given its $\theta_j$. New-observation noise is independent of $\theta_j$ and the old observations. With within-group time dependence or heteroskedasticity, $\sigma^2/n_j$ and these fixed precision weights cannot be used directly.

Consider $J$ asset or analyst groups. The parameter for group $j$ has a common prior $\theta_j \sim \mathcal{N}(\mu, \tau^2)$, and within-group observations satisfy $y_{j, i} \sim \mathcal{N}(\theta_j, \sigma^2)$. The sampling variance of the sample mean $\bar{y}_j$ is $\sigma_j^2 = \sigma^2 / n_j$.

When the hyperparameters $(\mu, \tau^2, \sigma^2)$ are known, the posterior mean of group mean $\theta_j$ has the following closed-form partial-pooling solution (Equation 5.17 in Gelman et al., BDA3):

The posterior precision is the sum of the prior and sample precisions: $1/V_j = 1/\tau^2 + n_j/\sigma^2$. The posterior predictive variance of a new observation $\tilde{y}_j$ from this group is $\operatorname{Var}_{\mathrm{pred}, j} = V_j + \sigma^2$.

5.4 Potential Outcomes and the Average Treatment Effect (ATE)

Let the binary treatment variable be $T \in \{0, 1\}$. For each unit, define a pair of potential outcomes:

• $Y(1)$: the potential outcome for the unit if it receives the treatment ($T=1$);
• $Y(0)$: the potential outcome for the unit if it does not receive the treatment ($T=0$). The fundamental difficulty in causal inference is that only one of the two potential outcomes can be observed for any unit in practice: $Y = T \cdot Y(1) + (1-T) \cdot Y(0)$.

The average treatment effect (ATE) for the target population is defined as:

5.5 Four Causal Identification Assumptions

Following Hernán & Robins (2020), statistical contrasts in observed data can be interpreted as a causal ATE only if the following four foundational assumptions hold:

1. Consistency: If a unit actually receives treatment $T=t$, its observed outcome $Y$ equals its potential outcome $Y(t)$; that is, $Y = Y(T)$.

2. Conditional exchangeability / no unmeasured confounding: Conditional on adequately controlling for confounder covariates $Z$, the potential outcomes are conditionally independent of treatment assignment: $(Y(0), Y(1)) \perp T \mid Z$. In a DAG, this means that controlling for $Z$ blocks all backdoor paths connecting $T$ and $Y$.

3. Positivity / overlap: Within every confounder stratum $z$ in the target population (that is, where $P(Z=z) > 0$), the probabilities of receiving treatment and control must both be strictly positive:

   $$0 < P(T=1 \mid Z=z) < 1$$

   Positivity means every stratum has a positive probability of receiving either treatment; it does not mean that variables follow a normal distribution. If everyone in a stratum receives treatment or everyone receives control, the counterfactual outcome for the missing arm cannot be identified from the data. The algorithm must report an explicit error and reject the calculation; it must never fill the missing value silently.

4. Stable unit treatment value assumption (SUTVA): This includes two closely related dimensions:

   • No interference: One unit’s treatment does not affect another unit’s potential outcome (for example, one stock having an earnings call and being bought by a strategy does not directly change the potential excess return of a stock without a call in the parallel world);
   • No treatment variation / no hidden versions of treatment: For a stated treatment level $T=t$, there are no unobserved versions of treatment that lead to different potential outcomes (for example, whether a CEO or CFO leads the call, or how quickly they speak, does not make the treatment $T=1$ causally different under the declared treatment definition).

5.6 Stratified Standardization

When conditional exchangeability, positivity, consistency, and SUTVA all hold, the target-population causal ATE can be calculated by weighting stratum-specific conditional effects by the marginal distribution $P(Z)$ in the target population:

Emphasize that the numerical contrast from stratified standardization can be interpreted as the target-population causal ATE only when all four identification assumptions above hold. With unmeasured confounding, a SUTVA violation, or sample-selection bias, the adjusted statistical association remains a conditional association under a particular adjustment; it cannot be unconditionally claimed as a true causal effect.

5.7 Point-in-Time (PIT) Availability and Delay Contract

Inputs must have nonmissing, time-zone-aware values for event, revision, availability, ingestion, signal, trade, and label times. Under valid inputs and version semantics, the teaching selector requires one decision cutoff per source_id and positive, increasing integer version identifiers. Duplicate versions, contradictory times, and missing text cause errors; versions ingested too late are filtered out. License fields store source notes only; the program does not determine legal authorization. These checks also run at the direct training entry point.

Alternative data must record the complete lifecycle timestamps for every version:

• Eligibility rule (iff, under the valid-input and version-semantics assumptions above): a version is eligible at the decision time if and only if $\max(\operatorname{available\_at}, \operatorname{ingested\_at}) \le \operatorname{signal\_time}$.
• Version replacement: If an event has several revised versions, use only the latest compliant version ingested by the signal cutoff; never overwrite history retrospectively.

5.8 Bag-of-Words (BoW) and TF-IDF Feature Representation

• A bag-of-words model treats a text as an unordered multiset of term occurrences, ignoring word order, grammar, and syntactic dependencies.
• Important clarification: Bag-of-words feature representation itself does not assume that words are statistically independent. Treating feature conditional distributions as independent is an assumption of particular models such as the Naive Bayes classifier, not of the BoW vector representation. The fundamental limitation of BoW is that it discards order, so it cannot natively capture modifiers, phrase combinations, or polarity reversals caused by negation.
• Term frequency–inverse document frequency (TF-IDF) downweights common function words and gives greater weight to domain-specific information terms, producing continuous dense or sparse vectors that reflect term importance.

5.9 Offline LLM Research Ledger and Defenses Against Irreproducibility

Use of large language models (LLMs) in quantitative research should be governed by an offline-auditable ledger that verifies a cryptographic digest:

• Compute a SHA-256 cryptographic hash (digest) of the complete prompt (prompt template plus actual input text);
• A hash checks only whether a digest matches and requires an independently trusted expected digest. It is not encryption or immutable timestamped evidence. If both the input and the recorded digest change, a local recomputation may still match. An independently stored trusted digest can serve as a comparison, but this teaching code does not provide a trusted custody mechanism for that digest.
• Explicitly record the model call status as not called (model_status: not_run) and the model identifier as not_run / unknown; use the training-corpus cutoff only to test a hypothetical counterexample;
• Record reproducibility limits explicitly: temperature=0 and a fixed seed do not guarantee bit-for-bit reproducibility across hardware architectures or inference-engine versions; underlying model updates or floating-point drift can materially change text features.

All causal tables in this chapter are constructed association data. The generator does not observe potential outcomes and cannot test identification assumptions such as no unmeasured confounding. A standardized difference can be interpreted as an ATE only if those assumptions hold.

6. Derivations

6.1 Deriving the Beta-Binomial Conjugate Posterior and Predictive Covariance

Step 1: Prior density and likelihood. Let parameter $p \in [0, 1]$ follow a Beta prior:

After observing $y$ successes in $n$ Bernoulli trials that are conditionally independent and identically distributed given the shared $p$, the binomial likelihood is:

Step 2: Calculate the posterior density. By Bayes’ theorem, the posterior density is proportional to the product of the prior and likelihood:

Comparing with the kernel of a Beta distribution, we identify the posterior distribution directly:

Step 3: Posterior mean and variance. Using the first and second moments of the Beta distribution:

Step 4: Predictive probabilities and covariance. For a single future trial $\tilde{y} \in \{0, 1\}$:

For two future trials $(\tilde{y}_1, \tilde{y}_2)$:

Using the variance identity $E[p^2 \mid y] = \operatorname{Var}(p \mid y) + (E[p \mid y])^2$, substitute the second raw moment of the Beta distribution:

The covariance of success in the two periods is:

This proves the result. Under the finite, positive-parameter conditions above, the two trials are conditionally independent given $p$ but positively correlated after integration. If the first-period outcome is then observed to be a success, the posterior for $p$ updates and raises the conditional prediction of success in the second period. This is information updating, not a causal effect between the outcomes. The correlation in this example is $1/(10+4+1)=1/15$.

6.2 Deriving Shrinkage for a Normal Hierarchical Mean (Partial Pooling)

Assumptions: Retain the independent group priors and conditionally independent, homoskedastic within-group observations specified in §5.3. Hyperparameters $\mu, \tau^2, \sigma^2$ are known. Group $j$ has latent mean $\theta_j \sim \mathcal{N}(\mu, \tau^2)$, and within-group observations $y_{j, 1}, \dots, y_{j, n_j} \sim \mathcal{N}(\theta_j, \sigma^2)$. The sample mean $\bar{y}_j = \frac{1}{n_j}\sum_{i=1}^{n_j} y_{j, i}$ satisfies $\bar{y}_j \mid \theta_j \sim \mathcal{N}(\theta_j, \sigma^2 / n_j)$.

Step 1: Combine precisions under conjugate updating. The sampling precision of the group mean is $n_j / \sigma^2$, and the prior precision is $1 / \tau^2$. By conjugacy of the normal distribution, the log posterior density is quadratic:

Expand the coefficient of the quadratic term:

Read off the posterior precision $1/V_j$ and posterior variance $V_j$:

Step 2: Decompose the posterior mean into weighted terms. Dividing the linear coefficient by the quadratic coefficient gives the posterior mean $\hat{\theta}_j$:

Define the shrinkage weight $w_j$:

The posterior mean is then a convex combination of the prior mean and sample mean:

Step 3: Posterior predictive variance. For a new observation from the group, $\tilde{y}_j = \theta_j + \tilde{\epsilon}$, where $\tilde{\epsilon} \sim \mathcal{N}(0, \sigma^2)$ and is independent of $\theta_j$ and the old observations:

This proves the result.

6.3 Deriving Simpson’s Reversal and Causal Standardization

Let the binary confounder be $Z \in \{1, 2\}$, the binary treatment be $T \in \{0, 1\}$, and the binary outcome be $Y \in \{0, 1\}$.

Step 1: Unadjusted association difference. By the law of total probability, the unadjusted conditional expectation is:

Therefore, the unadjusted difference is:

Note that $P(Z=z \mid T=1) \ne P(Z=z \mid T=0)$; a severe imbalance in weights can produce Simpson’s reversal.

Step 2: Identification derivation for causal standardization. By consistency and conditional exchangeability $(Y(t) \perp T \mid Z)$:

Take the marginal expectation over the target population $P(Z)$:

Under the identification assumptions above and positivity in the target population, subtracting gives the causal average treatment effect:

In this equation, both terms use the same marginal target-population weights $P(Z=z)$ in every stratum. Under the stated identification assumptions, this removes the confounding bias caused by $Z$ in this DAG.

6.4 Formal TF-IDF Mathematics and a Step-by-Step Calculation for Two Documents

Step 1: Formal definition. Let the corpus contain $|D|$ documents and have a vocabulary of size $M$.

1. Term frequency (TF): Let $f_{t, d}$ denote the number of occurrences of term $t$ in document $d$. Standard term frequency uses the raw count, $\mathrm{tf}(t, d) = f_{t, d}$.

2. Smoothed inverse document frequency (IDF): To avoid division by zero and excessive weight on rare terms that occur only once, use the standard scikit-learn smoothed log formula:

   $$\mathrm{idf}(t, D) = \ln\left(\frac{1 + |D|}{1 + \mathrm{df}(t)}\right) + 1$$

Here, $\mathrm{df}(t)$ is the number of documents in the corpus that contain term $t$. 3. Weighted vector and L2 normalization:

To reduce differences in document-length scale, apply L2 unit normalization when $\|\mathbf x(d)\|_2>0$:

If a document is empty or all its terms are OOV, this implementation retains a zero vector and does not divide by zero; cosine similarity between a zero vector and another vector is undefined.

Step 2: Hand calculation for two teaching documents. Suppose the training corpus contains only two miniature documents:

• Document 1: $d_1 = \text{"growth profit"}$
• Document 2: $d_2 = \text{"growth loss"}$ The corpus contains $|D| = 2$ documents.

1. Extract the vocabulary and term frequencies: The alphabetically sorted vocabulary is: ["growth", "loss", "profit"] ($M=3$).

   • Term-frequency vector for $d_1$: $\mathrm{tf}_1 = [1, 0, 1]$
   • Term-frequency vector for $d_2$: $\mathrm{tf}_2 = [1, 1, 0]$

2. Calculate document frequency $\mathrm{df}$ and smoothed IDF $\mathrm{idf}$:

   • The term "growth" appears in both $d_1$ and $d_2$, so $\mathrm{df}=2$:

     $$\mathrm{idf}(\text{"growth"}) = \ln\left(\frac{1+2}{1+2}\right) + 1 = \ln(1) + 1 = 0.0 + 1.0 = 1.000000$$

   • The term "loss" appears only in $d_2$, so $\mathrm{df}=1$:

     $$\mathrm{idf}(\text{"loss"}) = \ln\left(\frac{1+2}{1+1}\right) + 1 = \ln(1.5) + 1 \approx 0.405465 + 1.0 = 1.405465$$

   • The term "profit" appears only in $d_1$, so $\mathrm{df}=1$:

     $$\mathrm{idf}(\text{"profit"}) = \ln\left(\frac{1+2}{1+1}\right) + 1 = \ln(1.5) + 1 \approx 0.405465 + 1.0 = 1.405465$$

3. Calculate the unnormalized feature vector $\mathbf{x} = \mathrm{tf} \times \mathrm{idf}$:

   • $\mathbf{x}_1 = [1 \times 1.0, 0 \times 1.405465, 1 \times 1.405465] = [1.0, 0.0, 1.405465]$

   • Magnitude $\|\mathbf{x}_1\|_2 = \sqrt{1.0^2 + 0^2 + 1.405465^2} = \sqrt{1.0 + 1.975332} = \sqrt{2.975332} \approx 1.724915$

   • Normalized vector $\mathbf{v}_1$:

     $$\mathbf{v}_1 = \left[ \frac{1.0}{1.724915}, 0.0, \frac{1.405465}{1.724915} \right] \approx [0.579739, 0.000000, 0.814802]$$

   • $\mathbf{x}_2 = [1 \times 1.0, 1 \times 1.405465, 0 \times 1.405465] = [1.0, 1.405465, 0.0]$

   • Magnitude $\|\mathbf{x}_2\|_2 \approx 1.724915$

   • Normalized vector $\mathbf{v}_2$:

     $$\mathbf{v}_2 = \left[ \frac{1.0}{1.724915}, \frac{1.405465}{1.724915}, 0.0 \right] \approx [0.579739, 0.814802, 0.000000]$$

4. Calculate cosine similarity between the two documents:

   $$\cos(\mathbf{v}_1, \mathbf{v}_2) = \mathbf{v}_1 \cdot \mathbf{v}_2 = 0.579739^2 + 0.0 \times 0.814802 + 0.814802 \times 0.0 = 0.579739^2 \approx 0.336097$$

Although the two documents share the common term "growth", the feature terms "profit" and "loss" receive higher IDF weights and are orthogonal to one another. This lowers the cosine similarity after normalization to approximately $0.3361$.

6.5 Deriving Dense Word-Embedding Geometry and Mean Pooling

Let each word map to a unit vector in a $K$-dimensional Euclidean space, $\mathbf{e}(w) \in \mathbb{R}^K$, with $\|\mathbf{e}(w)\|_2 = 1$.

1. Mean pooling for a phrase or sentence: For a text sequence $S = (w_1, w_2, \dots, w_L)$ containing $L$ terms, the unnormalized dense sentence representation is the simple arithmetic mean of its word vectors:

   $$\mathbf{s} = \frac{1}{L} \sum_{l=1}^L \mathbf{e}(w_l)$$

2. Projection onto the unit sphere (requires $L>0$ and $\|\mathbf s\|_2>0$):

   $$\mathbf{u}_S = \frac{\mathbf{s}}{\|\mathbf{s}\|_2}$$

Even unit word vectors can cancel: the mean of $[1,0]$ and $[-1,0]$ is zero and cannot be normalized. Empty text also requires an explicit policy. This example is nonzero; a general pipeline should reject a zero vector or retain it explicitly and mark its cosine as undefined. 3. Semantic similarity between two text passages (both normalized vectors must exist):

Compared with sparse, high-dimensional TF-IDF, dense word embeddings map synonyms (such as "growth" and "expansion") to nearby regions of a hypersphere, avoiding the complete orthogonality of unigram bag-of-words features. In financial applications, however, generic embeddings without domain-specific fine-tuning may incorrectly judge antonyms (such as "profit" and "loss") to be highly similar because they often occur in the same grammatical contexts. This is a key reason to audit embedding quality carefully in quantitative finance.

7. Financial Interpretation

7.1 How Does Bayesian Shrinkage Express Uncertainty in Portfolio Inputs?

In quantitative asset allocation (for example, the Black-Litterman model or multi-factor return forecasts), directly using historical sample means in mean-variance optimization is a well-known “error maximizer.” Historical sample means are heavily affected by short-term, market-specific shocks. The financial interpretation of a normal hierarchical model (partial pooling) is that all assets in the market share a baseline long-run risk-premium mean $\mu$, but are heterogeneous in the cross-section, with variance $\tau^2$. For a small-cap stock with few disclosed observations ($n_j=4$), the data contain little information, so the model shrinks its estimate strongly toward the market center (shrinkage weight $w_j=0.5$). For a large-cap stock with substantial data ($n_j=36$), the model gives more weight to its own sample ($w_j=0.9$). In posterior predictive variance $\operatorname{Var}_{\mathrm{pred}}=V_j+\sigma^2$, a small sample increases uncertainty about the group mean, $V_j$; the future observation noise $\sigma^2$ is the same for both groups. The hyperparameters here are known and group data are not used to estimate shared hyperparameters; this differs from empirical Bayes or a full hierarchical model with unknown hyperparameters. This teaching example does not demonstrate improved live returns or optimization stability.

7.2 Confounding and Adverse Selection in Market-Maker Strategies

Consider a possible market-microstructure mechanism: the association between active order flow $T$ and subsequent price change $Y$ may be affected by trader information and order size, among other factors $Z$. The overall correlation cannot be interpreted directly as the causal price impact of a buy order. Associations stratified by informed and uninformed traders also require real data and an identification model. This chapter does not estimate actual price impact for either group and cannot support a claim that one group has a smaller impact.

7.3 Alternative-Data Delays and Backtest Vanity

When reporting alternative-data backtests to management, quantitative teams can easily show a Sharpe ratio that is too high. The source is often “timestamp fabrication”: although news or text appeared at 16:05, vendor parsing may queue during network disruptions, or internal data-engineering ETL may face an after-hours concurrency bottleneck. Requiring both availability and ingestion times to be no later than the signal is necessary for the text to be available; trade timing, labels, sample selection, and model validity still require separate checks.

7.4 News Updates and Earnings-Call Revisions: Two Timing Scenarios

Constructed news item CONSTRUCTED_NEWS_001 is published at 09:12 EDT on 2022-03-14, available from the vendor at 09:13, and ingested locally at 09:14. A 09:15 signal may use v1. Version v2, published at 09:16 and ingested at 09:18, cannot overwrite that historical signal even if it is more accurate. In UTC these times are 13:12, 13:13, 13:14, 13:15, 13:16, and 13:18 on the same date; this differs from the after-hours, cross-day earnings-call scenario in §8. The standalone code below reuses the complete field structure and changes only the constructed news record; it does not add the record to the frozen main experiment:

import pandas as pd
from quant_course.bayesian_text import (
    generate_constructed_text_dataset,
    filter_pit_eligible_documents,
)
base, _ = generate_constructed_text_dataset()
v1 = base.iloc[[0]].copy()
v1["source_id"] = "CONSTRUCTED_NEWS_001"
times = {
    'event_time': '2022-03-14T13:12:00Z',
    'revision_time': '2022-03-14T13:12:00Z',
    'available_at': '2022-03-14T13:13:00Z',
    'ingested_at': '2022-03-14T13:14:00Z',
    'signal_time': '2022-03-14T13:15:00Z',
    'trade_time': '2022-03-14T13:30:00Z',
    'label_start': '2022-03-14T13:30:00Z',
    'label_end': '2022-03-15T13:30:00Z',
    'label_available_at': '2022-03-15T13:30:00Z',
}
for field, stamp in times.items():
    v1[field] = pd.Timestamp(stamp)
v1["text"] = "constructed preliminary revenue growth report"
v2 = v1.copy()
v2["version_id"] = 2
v2["revision_time"] = pd.Timestamp("2022-03-14T13:16:00Z")
v2["available_at"] = pd.Timestamp("2022-03-14T13:17:00Z")
v2["ingested_at"] = pd.Timestamp("2022-03-14T13:18:00Z")
v2["text"] = "constructed corrected revenue decline report"
selected = filter_pit_eligible_documents(pd.concat([v1, v2], ignore_index=True))
assert selected["version_id"].tolist() == [1]
print(selected[["source_id", "version_id", "text"]])

Whether a news vendor preserves the original version and whether an earnings-call transcript is retrospectively corrected must be verified with the specific source. PIT filtering cannot recover an old version that was never archived. Training-window TF-IDF also cannot fix the error of using a future revision as historical input.

The Bayesian example also has model boundaries: the binomial update assumes trials are conditionally independent given $p$ and that “success” is defined consistently. Dependent trades cannot simply be treated as independent coin flips. Prior sensitivity can be calculated by hand: for the same 8/10 observations, the posterior mean is 9/12=0.75 under Beta(1,1) and 28/50=0.56 under Beta(20,20). A stronger prior is harder to move with ten observations, but is not necessarily more accurate.

8. Numerical Example

This section inherits constructed results from the historical A8-C frozen protocols qa/a8/ch42_ch43_batch/EXPERIMENT_PROTOCOL.json and EXPERIMENT_ADDENDUM_v1.json in the same directory. Historical hand/code checks and supplemental demonstrations retain their original status; BIL20 did not rerun them.

8.1 Hand Calculation: Beta-Binomial Example

• Input: Prior $\alpha = 2, \beta = 2$; $n = 10$ trials with $y = 8$ successes.

• Posterior parameters:

  $$\alpha_{\mathrm{post}} = 2 + 8 = 10, \quad \beta_{\mathrm{post}} = 2 + 10 - 8 = 4$$

• Posterior mean and variance:

  $$E[p \mid y] = \frac{10}{10 + 4} = \frac{10}{14} = \frac{5}{7} \approx 0.714286$$
  $$\operatorname{Var}(p \mid y) = \frac{10 \times 4}{14^2 \times (14 + 1)} = \frac{40}{196 \times 15} = \frac{40}{2940} = \frac{2}{147} \approx 0.013605$$

• Prediction for future trials:

  • Predicted probability of success on one trial: $P(\tilde{y} = 1 \mid y) = 10/14 = 5/7 \approx 0.714286$.

  • Joint probability of success in two trials:

    $$E[p^2 \mid y] = \frac{10 \times (10 + 1)}{14 \times (14 + 1)} = \frac{110}{210} = \frac{11}{21} \approx 0.523810$$

  • Squared posterior mean: $(E[p \mid y])^2 = (10/14)^2 = 100/196 = 25/49 \approx 0.510204$.

  • Check the difference:

    $$E[p^2 \mid y] - (E[p \mid y])^2 = \frac{11}{21} - \frac{25}{49} = \frac{77 - 75}{147} = \frac{2}{147} \equiv \operatorname{Var}(p \mid y)$$

The two values agree exactly.

8.2 Hand Calculation: Normal Hierarchical Shrinkage

Returns in this example are expressed as decimals: $0.02=2\%$ and $0.03=3\%$. Means and standard deviations use decimal return units; posterior variance $V_j$ and predictive variance use squared units. For example, group 1 has $V_1=0.0002$, the uncertainty about its mean parameter; its square root is $0.014142$, or about 1.4142 percentage points on the return scale. A future individual return also has observation noise, so predictive variance is $0.0002+0.0016=0.0018$, with square root about $0.042426$ (4.2426 percentage points). If returns are expressed in percentage points instead, means and standard deviations are multiplied by 100, while variances must be multiplied by $100^2$; the shrinkage weights do not change. Hyperparameters in this example remain known constants and are not estimated here.

• Known hyperparameters: Prior mean $\mu = 0.0$, prior standard deviation $\tau = 0.02$, and observation standard deviation $\sigma = 0.04$.

• Precision constants:

  $$\tau^{-2} = \frac{1}{0.02^2} = 2500, \quad \sigma^{-2} = \frac{1}{0.04^2} = 625, \quad \frac{\sigma^2}{\tau^2} = \frac{0.0016}{0.0004} = 4$$

• Group 1: $n_1 = 4$, sample mean $\bar{y}_1 = 0.03$.

  • Shrinkage weight: $w_1 = \frac{4}{4 + 4} = 0.5$.
  • Posterior mean: $\hat{\theta}_1 = (1 - 0.5) \times 0.0 + 0.5 \times 0.03 = 0.015$.
  • Posterior precision: $1/V_1 = 2500 + 4 \times 625 = 5000 \implies V_1 = 1/5000 = 0.0002$.
  • Posterior standard deviation: $\sqrt{0.0002} \approx 0.014142$.
  • Posterior predictive variance: $\operatorname{Var}_{\mathrm{pred}, 1} = V_1 + \sigma^2 = 0.0002 + 0.0016 = 0.0018$.
  • Predictive standard deviation: $\sqrt{0.0018} \approx 0.042426$.

• Group 2: $n_2 = 36$, sample mean $\bar{y}_2 = 0.01$.

  • Shrinkage weight: $w_2 = \frac{36}{36 + 4} = \frac{36}{40} = 0.9$.
  • Posterior mean: $\hat{\theta}_2 = (1 - 0.9) \times 0.0 + 0.9 \times 0.01 = 0.009$.
  • Posterior precision: $1/V_2 = 2500 + 36 \times 625 = 2500 + 22500 = 25000 \implies V_2 = 1/25000 = 0.00004$.
  • Posterior standard deviation: $\sqrt{0.00004} \approx 0.006325$.
  • Posterior predictive variance: $\operatorname{Var}_{\mathrm{pred}, 2} = V_2 + \sigma^2 = 0.00004 + 0.0016 = 0.00164$.
  • Predictive standard deviation: $\sqrt{0.00164} \approx 0.040497$.

8.3 Hand Calculation: Simpson’s Paradox and Stratification for Causal Inference

• Definition of the evaluation metric: The outcome $Y \in \{0, 1\}$ in this example indicates whether the next-day opening excess return reaches a benchmark threshold. Its mean $E[Y]$ is a dimensionless success probability; the difference between two probabilities multiplied by 100 is a difference in percentage points (pp), not a percentage return.

• Contingency-table data (total sample $N=200$):

  • High-attention stratum $Z=\mathrm{high}$ (industry leaders):
    • Treatment group $T=1$ (call held): $n=90$, with $63$ successes, so $E[Y \mid T=1, Z=\mathrm{high}] = 63/90 = 0.70$ (70%);
    • Control group $T=0$ (no call): $n=10$, with $8$ successes, so $E[Y \mid T=0, Z=\mathrm{high}] = 8/10 = 0.80$ (80%);
    • Within-stratum contrast: $0.70 - 0.80 = -0.10$ (a 10-percentage-point reduction in the success rate).
  • Low-attention stratum $Z=\mathrm{low}$ (micro-cap tail):
    • Treatment group $T=1$ (call held): $n=10$, with $1$ success, so $E[Y \mid T=1, Z=\mathrm{low}] = 1/10 = 0.10$ (10%);
    • Control group $T=0$ (no call): $n=90$, with $18$ successes, so $E[Y \mid T=0, Z=\mathrm{low}] = 18/90 = 0.20$ (20%);
    • Within-stratum contrast: $0.10 - 0.20 = -0.10$ (a 10-percentage-point reduction in the success rate).

• Unadjusted crude contrast:

  • Treatment group total: $n=100$, with $63 + 1 = 64$ successes, so $E[Y \mid T=1] = 64/100 = 0.64$ (64%);
  • Control group total: $n=100$, with $8 + 18 = 26$ successes, so $E[Y \mid T=0] = 26/100 = 0.26$ (26%);
  • Unadjusted crude difference: $0.64 - 0.26 = +0.38$ (the apparent success rate is 38 percentage points higher).

• Standardized observed difference; interpreted as the ATE only under the identification assumptions in §5.5 (target-population distribution $P(Z=\mathrm{high})=0.5, P(Z=\mathrm{low})=0.5$):

  $$\operatorname{ATE} = 0.5 \times (-0.10) + 0.5 \times (-0.10) = -0.10$$

The unadjusted difference is +38 percentage points and the standardized difference is −10 percentage points; the latter is interpreted as the target-population ATE only if the identification conditions assumed in this example hold. The table demonstrates an association reversal caused by weights; it does not itself prove the true causal direction of treatment.

8.4 Side-by-Side Hand Calculation: TF-IDF for Two Documents

Based on the step-by-step derivation in §6.4, the term frequencies, smoothed inverse document frequencies, and feature vectors for $d_1 = \text{"growth profit"}$ and $d_2 = \text{"growth loss"}$ are shown below:

• Vector magnitudes: $\|\mathbf{x}_1\|_2 = \|\mathbf{x}_2\|_2 = \sqrt{1.0^2 + 1.405465^2} \approx 1.724915$
• Cosine similarity: $\cos(\mathbf{v}_1, \mathbf{v}_2) = 0.579739^2 + 0 + 0 = 0.336097$

8.5 Strict Time-Series Split and Constructed Results for Text Prediction

Under the frozen historical protocol, 120 constructed events are generated by repeating four fixed text templates; they are not 120 independent new texts. The 59 mature training, 29 validation, and 30 test documents each contain the same four templates. This tests extrapolation across time blocks only; it does not test generalization to new texts, templates, or company language. The OOV negative case checks only that the vocabulary and IDF remain unchanged; it is not a test of prediction ability on new text.

The 120 constructed earnings-call events (CONSTRUCTED_CALL_000 through CONSTRUCTED_CALL_119) use a strict time-series split. A training label must be available strictly before the first signal time in the validation set (label_available_at < next_signal_time, <); a validation label must be available strictly before the first signal time in the test set.

• Split sizes:
  • Training: 59 eligible documents (indices 0..58; the label for index 59 matures at 09:30 on $D_{61}$, after the first validation signal at 09:00 on $D_{61}$, so it is strictly purged);
  • Validation: 29 eligible documents (indices 60..88; the label for index 89 matures at 09:30 on $D_{91}$, after the first test signal at 09:00 on $D_{91}$, so it is strictly purged);
  • Test: 30 documents (indices 90..119).
• Training vocabulary size: Fit strictly on the 59 training documents, yielding 42 unique terms.
• Ridge hyperparameter tuning on validation:
  • $\alpha = 0.1 \implies \mathrm{MSE}_{\mathrm{val}} = 0.00021835$
  • $\alpha = 1.0 \implies \mathrm{MSE}_{\mathrm{val}} = 0.00021697$
  • $\alpha = 10.0 \implies \mathrm{MSE}_{\mathrm{val}} = 0.00021133$ (best hyperparameter: $\alpha^* = 10.0$)
• Final test-set comparison:

8.6 Hand Examples: Negation Sentiment and Word Embeddings

Each row of the hand-specified matrix corresponds to growth, profit, loss, decline, stable, and steady. Its three columns are teaching axes (strength, momentum, and uncertainty), not estimated financial factors:

First divide each row by its own norm. For example, the norms of growth and profit are $\sqrt{1.46}$ and $\sqrt{1.31}$, respectively, and their cosine similarity is:

The sentence representation is the mean of its unit word vectors followed by normalization; this example ignores the connector “and.” The code below returns all vectors and similarities:

from quant_course.bayesian_text import hand_embedding_matrix, negation_sentiment_scorer
vectors, similarities = hand_embedding_matrix()
print(vectors, similarities)
print(negation_sentiment_scorer("earnings performance did not improve showing no growth"))

• Negation reversal: Sentence: “earnings performance did not improve showing no growth despite aggressive restructuring”
  • Naive unigram scoring with a fixed lexicon (BoW itself does not specify sentiment weights): it recognizes "improve" (+1) and "growth" (+1), giving a score of +2 (a severe error that classifies the sentence as highly positive);
  • Negation-window algorithm: "not" activates a three-word reversal window, flipping "improve" to −1; "no" activates the reversal window and flips "growth" to −1, for a total score of −2 (matching the intended negative meaning of this constructed sentence). This three-word window rule is not guaranteed to handle complex negation, double negation, sarcasm, or general financial text accurately.
• Cosine similarities in the embedding space (using a $6 \times 3$ manually specified, normalized embedding matrix; no real corpus was used for training):
  • Cosine similarity between growth and profit: $+0.9834$;
  • Cosine similarity between growth and loss: $-0.8595$;
  • Cosine similarity between the mean-pooled phrases $S_1$ ("growth and profit") and $S_2$ ("loss and decline"): $-0.8836$.

9. Python Implementation

The core module is bayesian_text.py under src/quant_course. The standalone script is ch43_bayesian_text.py under code; tests are in test_bayesian_text.py under tests. Historical execution counts and results are in the applicable A8-C/C19 QA records; BIL20 revises teaching only.

9.1 Execution Command and Run Contract

• Execution environment: the course-standard virtual environment, invoked from the course source root as .venv/bin/python.
• Standalone CLI invocation:

  .venv/bin/python code/ch43_bayesian_text.py --out build/batches/A8-C/reader/runs/ch43_trial_01
  

• Safety guard: if the specified --out directory already exists, the script immediately raises FileExistsError and refuses to overwrite historical experimental evidence.

9.2 Core Function Interfaces and Standalone Example

The code first passes in the raw 60/30/30 time slices; train_text_prediction_models checks PIT eligibility again internally and filters training and validation labels using label_available_at < the next block’s first signal_time. Index 59 has a label that matures at 09:30 on $D_{61}$, while the first validation signal is at 09:00 that day; index 89 is similarly later than the first test signal at 09:00 on $D_{91}$. The two rows are therefore purged separately, yielding 59/29/30 as in §8.5 and preventing immature labels from entering fitting or tuning. The test block has no following-fold fitting boundary, so its labels are used for retrospective evaluation. Equality does not satisfy the strict < condition.

The text-prediction path actually run in this chapter is TF-IDF + Ridge on constructed text. The $6\times3$ hand-specified embedding matrix in §8.6 is used only for a geometric calculation and was not a training feature for that predictor. The LLM section is an offline ledger and review exercise marked not_run / unknown. These are three distinct training and evaluation statuses.

# 1. Bayesian conjugate update and prediction
from quant_course.bayesian_text import beta_binomial_posterior_predictive
res_bayes = beta_binomial_posterior_predictive(
    alpha_prior=2.0, beta_prior=2.0, successes=8, trials=10
)

# 2. Normal hierarchical partial pooling
from quant_course.bayesian_text import normal_hierarchical_partial_pooling
res_normal = normal_hierarchical_partial_pooling(
    prior_mean=0.0, prior_sd=0.02, observation_sd=0.04,
    groups=[{"n": 4, "sample_mean": 0.03}, {"n": 36, "sample_mean": 0.01}]
)

# 3. Causal stratification and strict positivity defense
from quant_course.bayesian_text import (
    compute_stratified_ate,
    PositivityViolationError,
    generate_causal_dataset,
)
df_causal = generate_causal_dataset()
ate_res = compute_stratified_ate(
    df_causal, z_col="Z", t_col="T", y_col="Y",
    target_p_z={"high": 0.5, "low": 0.5}
)

# 4. PIT text filtering and training-window TF-IDF
from quant_course.bayesian_text import (
    filter_pit_eligible_documents,
    TextFeaturePipeline,
    train_text_prediction_models,
)
from quant_course.bayesian_text import generate_constructed_text_dataset
raw_docs_df, _ = generate_constructed_text_dataset()
eligible_docs = filter_pit_eligible_documents(raw_docs_df, cutoff_col="signal_time")
train_df, val_df, test_df = (
    eligible_docs.iloc[:60], eligible_docs.iloc[60:90], eligible_docs.iloc[90:]
)
model_res = train_text_prediction_models(train_df, val_df, test_df, alpha_grid=[0.1, 1.0, 10.0])

10. Visualization

The historical frozen CONSTRUCTED teaching run generated and saved five figures; this round reuses the original figures from course/zh/chapters/ch43/figures/root_r3/.

Figure 43-1: Bayesian Conjugate Updating and Normal Hierarchical Shrinkage

Figure 43-1 description:

• Left panel (Beta-Binomial conjugate update): The horizontal axis is the binomial success probability $p \in [0, 1]$; the vertical axis is probability density (the scaled likelihood is used only to compare shapes and is not a probability density over the parameter). The gold dashed line is the prior $\operatorname{Beta}(2, 2)$, the blue dotted line is the scaled likelihood (8 successes in 10 trials), and the red solid line is the posterior $\operatorname{Beta}(10, 4)$. The posterior mean (red dash-dot line at $0.714$) lies between the prior mean of $0.5$ and the maximum-likelihood estimate of $0.8$, clearly showing the Bayesian compromise with the data.
• Right panel (normal hierarchical partial-pooling shrinkage): The horizontal axis is group-mean return and the vertical axis lists asset groups. The black dashed line is the whole-market prior mean $\mu=0$. Blue circles are group sample means $\bar{y}$, and red diamonds are the shrunken posterior means $\hat{\theta}$. Group 1 (sample size $n=4$) is strongly shrunk by 50% (arrow from $0.03$ to $0.015$); Group 2 (sample size $n=36$) is shrunk by only 10% (from $0.01$ to $0.009$).