25  Homework 7: Bayesian Linear Regression

Your numbers will not match mine, and that is fine

Every result in this course depends on a random seed, and often also on your library versions, your hardware, and how many CPU threads your machine uses. Two people running identical, correct code will get different numbers — sometimes noticeably different.

What is graded is that you used the right model and the right procedure, and that your interpretation follows from the numbers you actually obtained. Quote your own output.

What is not graded is matching any particular value. Do not tune your way toward a number you saw on a slide or heard from a classmate — that is the one way to turn this into a wrong answer.

One caveat worth knowing. The broad ordering of models usually survives this variation. If yours comes out backwards from what the lecture found, say so in a sentence and offer a reason: either you have found something real about your run, or there is a bug worth hunting. Both are worth more marks than quietly reporting a number you do not believe.

25.1 Overview

Bayesian linear regression replaces OLS point estimates with full posterior distributions over every coefficient. Instead of asking “Is beta significant at p < 0.05?”, you ask “What is the probability that this effect is positive?” and “Is the effect large enough to matter for business decisions?”

This final homework fits a Bayesian regression model for store sales as a function of promotions, competition, and store characteristics. It also introduces Directed Acyclic Graphs (DAGs) as a tool for deciding which variables to include and how to interpret the coefficients causally.

Learning objectives:

  • Draw and interpret a DAG for a business forecasting model
  • Specify weakly informative priors for regression coefficients
  • Read posterior coefficient distributions (HDI, probability of direction)
  • Apply ROPE analysis to distinguish statistical from practical significance
  • Use posterior samples to simulate conditional scenarios

25.2 Dataset and Setup

The question for this assignment is a causal one: how much does demand respond to price? That is an elasticity, and it is the number a category manager would actually act on.

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import pymc as pm
import arviz as az
import statsmodels.api as sm
import warnings, logging
warnings.filterwarnings('ignore')
logging.getLogger('pymc').setLevel(logging.ERROR)

from pathlib import Path

def data_path(name):
    for base in (Path.cwd(), *Path.cwd().parents):
        p = base / "data" / "processed" / name
        if p.exists():
            return p
    raise FileNotFoundError(
        f"Could not find data/processed/{name}. "
        "Run `python scripts/prep_m5.py` from the repository root first."
    )

RANDOM_STATE = 42

weekly = (pd.read_csv(data_path("m5_weekly.csv"), parse_dates=["week_start"])
            .sort_values(["store_id", "cat_id", "week_start"]))
weekly["series"] = weekly.store_id + "_" + weekly.cat_id

df = weekly.dropna(subset=["units", "avg_price"]).copy()
df["ly"] = np.log(df.units.clip(lower=1))          # log units
df["lp"] = np.log(df.avg_price)                     # log price
df["t"]  = (df.week_start - df.week_start.min()).dt.days / 365.25
df["snap"]  = (df.snap_days >= 5).astype(float)     # high-SNAP week
df["event"] = (df.event_days > 0).astype(float)     # any event that week
codes, uniq = pd.factorize(df.series)
df["idx"] = codes
J = len(uniq)

# Center log price within series. NOT standardized -- see the note below.
df["lpc"] = df.lp - df.groupby("series").lp.transform("mean")

print(f"{len(df):,} rows | {J} series | "
      f"price range {df.avg_price.min():.2f}-{df.avg_price.max():.2f}")

Why logs on both sides. With ly and lp both logged, the price coefficient is an elasticity: the percentage change in units for a one percent change in price. That is directly interpretable, and it is what makes the scenario analysis in Q6 possible.

Do not standardize log price

It is a common reflex, and here it is a trap. Dividing lp by its standard deviation rescales the coefficient into “per standard deviation of log price” — which is no longer an elasticity, and every percentage claim you build on it in Q6 will be wrong by the factor you divided by. Leave lp in log units. Every predictor in this assignment is already on a scale a manager reads directly: log price in percent, snap and event as 0/1, t in years.

Centering within series (lpc) is different and is fine — it shifts the variable without rescaling it, leaves the elasticity untouched, and greatly improves sampling, because price levels differ across series and otherwise correlate with the intercepts.

25.3 Initial Prompt for Your AI Assistant

Any assistant works — ChatGPT, Claude, Gemini, or Copilot, in an editor or in a browser tab. Paste the block below as your first message, before you ask anything else.

Initial Prompt for Your AI Assistant

Paste the following as your first message to your assistant. Save it as INITIAL_PROMPT.md in your submission folder.

I am working on Homework 7 for ECON 8310: Business Forecasting at the
University of Nebraska at Omaha.

Dataset: M5/Walmart — weekly aggregated, 30 store-category series, at
`data/processed/m5_weekly.csv`. Columns: store_id, cat_id, state_id,
week_start, units, avg_price, snap_days, event_days. I work with log units
(ly) on log price (lp), plus a trend and series identifiers.

This homework covers Bayesian linear regression using PyMC (v4+),
including DAGs, posterior coefficient interpretation, ROPE analysis,
and conditional scenario simulation. I will ask you questions one at
a time. For each:
1. Write clean, well-commented PyMC code that runs end-to-end
2. Explain what the output means so that I understand it — but do NOT draft the written
   answers I submit. The interpretation and business-recommendation sections are mine to
   write, in my own words.

My prompt budget is 16 (this context prompt is excluded). I will label
each question before asking.


Repository rules — these apply to every prompt in this thread:
- If you cannot write files or run git — a browser chat cannot — then the file and commit
  rules below are mine to carry out, not yours. Tell me what to write and what to commit,
  and I will do it. Everything else in this list still applies to you.
- My submission folder is `submissions/hw07_<my_name>/`. Put everything there.
- Keep a `PROMPT_LOG.md` in that folder. For every prompt I send, append a block:
      ### Prompt <n> — YYYY-MM-DD HH:MM
      <my prompt, verbatim>
  Numbered from 1, in order. Never edit or delete an earlier entry. If your tool cannot write
  files, I will paste them in myself.
- After finishing the work for each prompt, make exactly one git commit:
  `hw07 prompt <prompt_id>: <short summary>`
- If a prompt would take me past my budget of 16, do not do the work. Reply exactly
  PROMPT_LIMIT_REACHED, and still log that prompt.
- Never modify `AGENTS.md`, anything in `assignments/`, or anything in `policy/`.

Confirm by responding with: HW07_CONTEXT_ACK

25.4 Questions

How the code cells work

A cell that is empty except for a comment is where you write. The comment states the task; your code goes underneath it.

Some cells already contain setup code — a train/test split, a rename, a cutoff date — and end with a comment marking where your part begins. Leave the setup as it is and write below the comment. It is there so that everyone’s results are computed on the same basis.

Every cell runs as the document stands, so you can render at any point, finished or not. Doing that early and often is how you avoid finding a problem at the deadline.

The Your response blocks are written answers, not code. They are yours to write, in your own words — your assistant explains output to you, it does not draft these.

25.4.1 Q1 — Draw a DAG (Suggested prompts: 1)

Before fitting anything, think about the causal structure. This question determines what the rest of the assignment means, so do it first and do it honestly.

Draw a DAG for weekly demand including at least: units, avg_price, series (which store-category this is), time (trend), snap_days, and event_days.

No code needed — a Markdown table or ASCII diagram is fine. Justify every arrow in one sentence.

series  → avg_price   (categories differ systematically in price level)
series  → units       (categories differ systematically in volume)
time    → avg_price   (prices drift over five years)
time    → units       (demand grows over the sample)
avg_price → units     (the effect we want to estimate)
...

Then answer the question this exists to answer: is there a confounder between avg_price and units? Name it, explain the mechanism in business terms, and state what you must do about it in the regression.

Your response:

25.4.2 Q2 — OLS Benchmark, Two Ways (Suggested prompts: 2)

Fit two OLS regressions with statsmodels, and note that the difference between them is the whole point of Q1.

(a) Naive — regress ly on lp, snap, event.

(b) Controlled — regress ly on lpc, snap, event, plus a trend (t) and series fixed effects (pd.get_dummies(df.series)).

Report the coefficient table for each, and put the two price elasticities side by side.

# (a) naive OLS
# (b) OLS with trend and series fixed effects
# Compare the two price coefficients

Your response: Report both elasticities. They will differ substantially. Using your DAG from Q1, explain which one estimates the quantity you actually want, and why the other is biased in the direction it is.

Then look at the standard errors: the naive estimate is not merely different, it is at least as precise. Explain why precision is no defense here.

25.4.3 Q3 — Bayesian Linear Regression (Suggested prompts: 3)

Fit the controlled specification from Q2(b) as a Bayesian model — price, SNAP, events, trend, and a per-series intercept. Use weakly informative priors:

  • beta_j ~ Normal(0, 1) for each predictor (weakly informative on these scales)
  • alpha ~ Normal(0, 1) for the intercept
  • sigma ~ HalfNormal(1) for observation noise

Give the per-series intercepts a hierarchical prior, written non-centered, as in Homework 6.

Sample with pm.sample(1000, tune=1500, chains=4, target_accept=0.9) and report the posterior mean, sd and 94% HDI for each coefficient — plus \(\hat{R}\), ESS and divergences.

Ask ArviZ for the HDI explicitly

az.summary(idata) reports an 89% equal-tailed interval by default, not the HDI. For the highest-density interval this course uses, call az.summary(idata, ci_kind="hdi", ci_prob=0.94), or az.hdi(samples, prob=0.94) on raw draws.

An equal-tailed interval cuts 5.5% off each tail; an HDI is the shortest interval holding 94% of the mass. They coincide for a symmetric posterior and diverge for a skewed one.

# Build and sample Bayesian linear regression

Your response: Compare the posterior means to the controlled OLS coefficients from Q2(b). With this much data they should agree closely — say whether they do, and explain why agreement is the expected result rather than a disappointing one.

Then: what does a wide HDI tell you that a narrow one does not, and which coefficient here has the widest relative to its mean?

25.4.4 Q4 — Posterior Visualization (Suggested prompts: 2)

Plot the posterior distributions for the three coefficients that matter most for a business decision — justify the choice from your DAG.

For each, report: - Posterior mean - 94% HDI - P(beta > 0) — probability the effect is positive - Whether the HDI excludes zero (the Bayesian analog of significance)

# Plot posterior distributions and compute P(beta > 0)

Your response: For the price coefficient, report \(P(\beta_{\text{price}} < 0)\) — the probability that raising price reduces demand. Contrast that statement with what the OLS p-value in Q2 actually claims, and say which one you could put in front of a manager without having to correct it afterwards.

25.4.5 Q5 — ROPE Analysis (Suggested prompts: 2)

“Credible” and “big enough to act on” are different questions. A ROPE — region of practical equivalence — makes the second one explicit.

For each predictor, compute the fraction of posterior mass inside |beta| < 0.05. Because nothing here is standardized, that threshold reads directly: for price, an elasticity between \(-0.05\) and \(0.05\) means a 1% price change moves units by less than 0.05% — no one would act on it. For snap and event it means a weekly lift under 5%.

Present results in a table:

Predictor Posterior Mean 94% HDI % in ROPE Practically significant?

A predictor is “practically significant” if less than 5% of its posterior falls in the ROPE.

# Compute % of posterior in ROPE for each predictor

Your response: Name any predictor that is credible but negligible — HDI excluding zero while most of its posterior sits inside the ROPE. That combination is common in large samples and is exactly what a p-value cannot express.

What should a manager do with such a finding, and what would have to change for it to become worth acting on?

25.4.6 Q6 — Conditional Scenario Simulation (Suggested prompts: 3)

A category manager is considering a 5% price increase across the board and wants to know what happens to volume — and how confident anyone can be about it.

Push the scenario through your posterior rather than through a point estimate.

# b = idata.posterior["b_price"].values.ravel()
# pct_change = 100 * (np.exp(b * np.log(1.05)) - 1)
# report the mean and the 94% interval of pct_change

(a) Report the mean percentage change in units and its 94% interval.

(b) Revenue rises if the percentage volume loss is smaller than the 5% price gain. Compute the probability that revenue increases, directly from the posterior samples.

(c) Repeat using the naive elasticity from Q2(a). Report how the recommendation changes.

Your response: Give the manager a two-sentence recommendation with a probability attached. Then state the caveat honestly: this is a causal claim about intervening on price, and it is only as good as the DAG in Q1. Name one arrow that, if wrong, would invalidate the answer.

25.4.7 Q7 — Course Capstone Reflection (Suggested prompts: 0 — write yourself)

No code required. In 200–250 words: Look back at the full arc of this course — from exponential smoothing in Homework 1 to Bayesian regression in this homework.

Answer the following:

  1. Which method from the course would you use first when you encounter a new business forecasting problem, and why? (Consider ease of implementation, interpretability, and typical performance.)

  2. Where does Bayesian inference add the most value compared to the classical and machine learning methods you used earlier? Give a specific example from your homework results.

  3. What is one limitation of every method in this course — something none of them handle well — and what would you do in practice to address it?

Your response:

25.5 Submission Checklist

Not graded, but worth doing: commit locally once per prompt, hw07 prompt <id>: <summary>. Nobody sees these — they are your own record of which prompt produced what, and they make it far easier to retrace your steps when something breaks. - [ ] REPORT.md with a clear business recommendation