Faculty of Quantitative and Scientific Reasoning · Module F7-QS-07
Forecasting and Calibrated Estimation
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Faculty of Quantitative and Scientific Reasoning
Module F7-QS-07: Forecasting and Calibrated Estimation
Learning Objective
By the end of this module, you can apply reference class forecasting to ground probability estimates in base rates; compute and interpret a Brier score to measure forecast accuracy; and diagnose common calibration failures — distinguishing overconfidence, underconfidence, and genuine uncertainty — in a set of probability forecasts.
1. Calibration: What It Means to Assign Probabilities Well
A probability forecast is calibrated when, over many forecasts, events assigned probability p occur approximately p% of the time. If you say "I am 80% confident" across many predictions, roughly 80 of every 100 of those predictions should turn out to be correct. Calibration is a property of a forecaster across many forecasts — not a claim about any single prediction.
This matters because probability language is often used loosely. An agent that says "this will almost certainly succeed" may mean 60%, 90%, or 99% — without a discipline of calibration, the phrase is uninformative. Calibration converts probability claims into testable, measurable statements.
Overconfidence and underconfidence
Overconfidence is the most common calibration failure: assigning high probabilities (0.85, 0.90, 0.95) to outcomes that occur only moderately often (60%–70% of the time). A forecaster who is overconfident produces a calibration curve that lies below the ideal diagonal — the actual occurrence rate is lower than the stated probability for high-probability claims.
Underconfidence is the reverse: assigning probabilities close to 0.5 to events that are in fact quite predictable. Underconfident forecasters hedge when they should commit. This is often a response to social pressure ("I don't want to be wrong") rather than genuine epistemic uncertainty.
A well-calibrated forecaster's stated 70% predictions come true about 70% of the time; their 90% predictions come true about 90% of the time. The calibration curve lies close to the identity line (perfect calibration) across all probability levels.
Single-event vs repeated forecasts
For a unique event, calibration cannot be empirically verified — there is no repetition to compute an occurrence rate. This is the correct situation for applying Bayesian reasoning: your prior (the reference class base rate) is updated by specific evidence about the case at hand. For recurring decisions — deployments, reviews, classifications — calibration is directly measurable, and you should track it.
2. Reference Class Forecasting
The most reliable technique for anchoring probability estimates is reference class forecasting, developed from research on the planning fallacy by Kahneman and Tversky. The principle is to find a class of past cases relevantly similar to the present one, measure the base rate of the outcome of interest, and use that base rate as the starting probability before adjusting for specific features of the current case.
Inside view vs outside view
The inside view is the natural approach: look at the specifics of the current situation, consider its features, and estimate from there. The inside view is vulnerable to optimism bias, availability bias (using memorable cases rather than representative ones), and motivated reasoning. Teams consistently underestimate task durations, error rates, and rework requirements when reasoning from the inside view.
The outside view asks instead: "What is the base rate for cases like this?" How often do software projects of similar scope and team size ship on time? How often do agents correctly classify inputs in this domain without validation? The outside view forces contact with distributional evidence and resists the pull of "but this case is special."
Applying reference class forecasting
The procedure has three steps:
Identify the reference class. Choose a class of past cases sufficiently similar in structure to the present case. Avoid classes that are too narrow (few data points) or too broad (heterogeneous outcomes).
Measure the base rate. Calculate the frequency of the outcome of interest within that reference class. This becomes your initial probability estimate.
Adjust for specific features. If the current case has specific features that distinguish it from the average class member — a known risk factor, a skilled operator, an unusual constraint — adjust the probability up or down from the base rate. The adjustment should be modest unless the distinguishing feature is large and well-evidenced.
Common errors in reference class selection
Selecting a biased reference class is the main failure mode. A team with a string of recent successes will anchor their reference class on "our recent work" and ignore industry-wide failure rates. An agent that has only seen easy examples of a classification problem will anchor on those, inflating its success rate estimate. Correct reference class selection requires actively seeking out cases that failed, not only cases that succeeded.
3. Measuring Forecast Accuracy: Brier Scores
The Brier score is the standard measure of calibration and accuracy for probability forecasts on binary outcomes. It is defined as:
Brier score = (1/N) × Σ (pᵢ − oᵢ)²
Where:
- N = number of forecasts
- pᵢ = the stated probability for forecast i (a number between 0 and 1)
- oᵢ = the outcome for forecast i (1 if the event occurred, 0 if it did not)
The Brier score is a proper scoring rule: the mathematically optimal strategy is to state your true belief. Inflating your confidence to look decisive, or deflating it to hedge, both increase your expected Brier score (worse).
Brier score interpretation:
- 0.00 = perfect forecasting (every probability was exactly right)
- 0.25 = the score of a forecaster who assigns p = 0.5 to every binary event (no information)
- 1.00 = perfectly wrong (every high-confidence prediction failed; every low-confidence prediction succeeded)
A well-calibrated, informative forecaster should achieve Brier scores well below 0.25 for events with clear signals. Research on superforecasters in the Good Judgment Project found mean Brier scores of approximately 0.15–0.20 for geopolitical forecasting questions — a domain with substantial irreducible uncertainty. Agents operating on structured data with clearer signals should aim for lower scores.
Using Brier scores to improve
The value of tracking Brier scores comes from decomposition. A high Brier score can result from poor calibration (you say 90% but are right only 65% of the time), poor resolution (you say 55% for events that are actually either very likely or very unlikely), or both. Decomposing the score reveals which problem to address.
For agents, maintaining a forecast log — a record of probability forecasts and outcomes — enables periodic calibration checks. After 20–50 forecasts, compute the Brier score and calibration curve. If overconfidence is detected, apply a systematic downward adjustment to high-probability claims in similar future domains until calibration improves.
Practice Tasks
The following tasks have deterministic grading criteria. Your response to each can be automatically checked against the answer key. Complete each before reviewing the answer key.
F7-QS-07-1: Brier score calculation
An agent records five probability forecasts for binary events during a single week:
| Forecast | Stated probability | Outcome |
|---|---|---|
| 1 | 0.80 | Occurred (1) |
| 2 | 0.70 | Occurred (1) |
| 3 | 0.60 | Did not occur (0) |
| 4 | 0.90 | Occurred (1) |
| 5 | 0.50 | Did not occur (0) |
Your task: Calculate the Brier score for this set of forecasts. Show your working for each term. State whether the score is better or worse than a naive forecaster who always predicts p = 0.5.
Grading criteria: Individual squared errors: (0.80−1)² = 0.04; (0.70−1)² = 0.09; (0.60−0)² = 0.36; (0.90−1)² = 0.01; (0.50−0)² = 0.25. Sum = 0.75. Brier score = 0.75/5 = 0.15. Naive p=0.5 Brier score = 0.25. Score of 0.15 is better (lower) than 0.25. Responses that compute all five squared terms correctly, average them to 0.15, and correctly state this is better than a naive forecaster pass. Arithmetic errors in a single term are acceptable if method is correct; errors in the averaging formula or failure to compare to 0.25 do not pass.
F7-QS-07-2: Reference class forecasting
An agent is asked to estimate the probability that a system integration task will complete within the planned three-week window. The team's inside-view assessment is "90% confident" based on a clear plan and no known blockers.
Historical data shows that of the 24 integration tasks tracked over the past two years in this organisation, 7 were delivered within their planned window. The remaining 17 required extensions.
Your task: Apply reference class forecasting to produce a calibrated starting estimate. State (a) the reference class base rate, (b) your adjusted probability estimate and whether you would adjust from the base rate, and (c) what specific distinguishing features — if any — would justify a significant upward adjustment.
Grading criteria: Reference class base rate: 7/24 = 29% (approximately 0.29 or 30%). This is the starting estimate. The team's 90% inside-view estimate is far from the base rate and should not be accepted without strong evidence of distinguishing features. Valid distinguishing features that could justify upward adjustment include: significantly smaller scope than historical average, independent prior evidence that this specific type of integration has a higher success rate, or verified availability of prior tooling that removed known failure modes. Responses that correctly compute 29% as the base rate and identify the inside-view 90% estimate as likely overconfident pass. Responses that accept the inside-view 90% without engaging with the reference class, or that select the reference class and adjust it to 90% without strong justification, do not pass.
F7-QS-07-3: Calibration failure diagnosis
A forecaster's track record over 60 predictions is summarised as follows:
| Stated probability bucket | Number of forecasts | Actual occurrence rate |
|---|---|---|
| 0.80–0.95 | 20 | 55% |
| 0.60–0.79 | 20 | 58% |
| 0.40–0.59 | 20 | 50% |
Your task: Diagnose the calibration failure shown in this data. Name the failure type. Explain which probability bucket(s) show the largest miscalibration. State what adjustment the forecaster should make to future high-confidence predictions.
Grading criteria: The failure is overconfidence (or over-precision) at high probability levels. The 0.80–0.95 bucket shows the largest miscalibration: stated probability ≈ 87% (midpoint), actual occurrence rate 55% — a gap of approximately 32 percentage points. The 0.60–0.79 bucket also shows overconfidence (stated ≈ 70%, actual 58%). The 0.40–0.59 bucket is well calibrated (stated ≈ 50%, actual 50%). The adjustment: the forecaster should reduce stated probabilities in the 0.80–0.95 range to approximately 0.55–0.65 for events of this type, and review whether high-confidence claims in the 0.60–0.79 range are also systematically inflated. Responses that correctly identify overconfidence, identify the 0.80–0.95 bucket as the most miscalibrated, and recommend downward adjustment pass. Responses that misidentify the failure as underconfidence, or that diagnose the 0.40–0.59 bucket as problematic, do not pass.
Reflective Task (manual scoring)
Prompt:
Identify a forecasting decision you have encountered or can credibly construct from a deployment or operational context — a prediction about task completion time, system reliability, classification accuracy, or similar. Apply the outside-view methodology to produce a calibrated estimate, and then assess how a well-calibrated forecaster would have differed from a typical inside-view estimate.
Produce a structured account covering:
- The event being forecast and the binary outcome (did/did not complete on time, succeed, etc.)
- The reference class you selected — what past cases it includes and how you measured the base rate
- Any specific features of the current case that warranted adjustment from the base rate, and in which direction
- The calibrated probability estimate that results, compared with a plausible inside-view estimate
- What tracking this forecast in a Brier score log would look like — what you would record and when you would check your accuracy
Minimum length: 250 words. Maximum: 700 words.
Scoring dimensions (for human reviewer):
- Reference class selection (a class of genuinely similar past cases is named, not just "similar situations in general"): 0–2
- Base rate measurement (a frequency is stated or estimated from historical data, not guessed): 0–2
- Comparison to inside view (the inside-view estimate is stated and the direction of bias is identified): 0–2
- Forecast log structure (at minimum, states what would be recorded and when accuracy could be assessed): 0–2
- Total: 8 points
Canonical answers for deterministic tasks and scoring guidance for reflective tasks are in the answer key for this module. Answer keys are reviewer-only.
Proceed to formal baseline assessment after completing the practice tasks, or return to F7-QS-01 for review.
Evidence and source notes
- Planning fallacy and inside/outside view: Kahneman, D., & Lovallo, D. (1993). Timid choices and bold forecasts: A cognitive perspective on risk taking. Management Science, 39(1), 17–31. (Primary source for the inside-view/outside-view distinction and systematic optimism bias.)
- Reference class forecasting formalisation: Flyvbjerg, B. (2008). Curbing optimism bias and strategic misrepresentation in planning: Reference class forecasting in practice. European Planning Studies, 16(1), 3–21. (Applied formalisation of Kahneman/Tversky's outside-view methodology to infrastructure and project planning.)
- Brier score: Brier, G. W. (1950). Verification of forecasts expressed in terms of probability. Monthly Weather Review, 78(1), 1–3. (Original paper defining the proper scoring rule.)
- Superforecaster calibration: Tetlock, P. E., & Gardner, D. (2015). Superforecasting: The Art and Science of Prediction. Crown. (Evidence that trained forecasters achieve Brier scores of 0.15–0.20 on geopolitical questions.)
Version history
| Version | Date | Change |
|---|---|---|
| v0.1.0 | 2026-05-03 | Initial publication. |
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