On the Limits of Strokes-Gained Inference for Individual Golfers
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- Open / Medium: Critique: Strokes Gained Limitations and Ecological Fallacy (
crit-strokes-gained-non-ergodic)
Strokes gained keeps score in a useful way. Predicting how practice will change that score takes more work.
The Total Still Adds Up
Against the same benchmark and starting conditions, gaining one total stroke means taking one fewer stroke. Different skill profiles do not break that arithmetic.
Categories Can Move Together
A closer approach can improve your score while your putting statistic looks worse, because you now face different putts. Add all the categories before reading the overall result.
Closer Helps by Different Amounts
The benefit depends on your chances from the old and new leaves. Being a worse putter does not guarantee a bigger benefit. Average distance also hides the spread of misses.
Abstract
Strokes gained provides an exact accounting of recorded strokes relative to a chosen benchmark. With a fixed benchmark and the same starting conditions, a one-stroke increase in total strokes gained is a one-stroke decrease in score. That identity survives differences between golfers, imperfect benchmark estimates, fatigue, and changes in strategy. An individual golfer does not have to resemble the benchmark player for the arithmetic to work.
The harder question is what an intervention will change. Better approach proximity, a new putting technique, or a faster swing can alter several shot distributions and several strokes-gained categories at once. A benchmark assigns a common value to a shot outcome; predicting the golfer’s subsequent score requires an appropriate continuation model. Neither a population table nor a personalized regression automatically identifies the effect of practice.
We distinguish accounting, prediction, and causal inference; derive how credit moves across a category boundary; and give counterexamples to the claim that weaker putters necessarily benefit more from closer leaves. We then connect swing mechanics, shot dispersion, strategy, and scoring without treating a mechanical simulation as a validated coaching prescription. All numerical examples below are constructed demonstrations, not estimates of professional or amateur performance.
1. Three Questions That Require Different Evidence
Consider the statement, “My putting improved by 0.3 strokes gained per round.” It could mean three things:
| Question | What Would Answer It? |
|---|---|
| How did the recorded putting contribution change? | A consistently calculated per-round category sum. |
| How well will I putt next month? | A validated forecast with uncertainty and relevant conditions. |
| How much did a particular practice program improve my score? | A comparison of outcomes under defined alternatives, supported by an experiment or justified causal assumptions. |
The first calculation is useful even when the other two questions remain unresolved. A negative category total describes performance relative to its benchmark; it does not by itself diagnose a mechanical fault or identify the most productive use of practice time.
This distinction matters for AffineDrift’s investigation of the swing. We want to understand how a golfer’s capabilities generate repeatable shot outcomes and how those outcomes affect play. Score decomposition can identify patterns worth investigating. Mechanics can help formulate explanations and interventions. Their connection requires evidence at each step.
2. The Accounting Identity Comes First
2.1 The Benchmark
Let \(s\) describe a recorded ball state: for example, distance to the hole and lie. Write \(B(s)\) for a fixed benchmark in strokes, with \(B(\text{holed})=0\). A common construction estimates the remaining score conditional on recorded state in a reference dataset:
\[ B(s)\approx \mathbb E_{\mathrm{ref}}[H\mid S=s]. \]
Here \(H\) is the remaining number of counted strokes. The dataset, weighting of players and shots, measurement procedure, course conditions, and smoothing method help define the reference. The benchmark can represent a chosen comparison group; it need not represent a Tour professional or a single physically realizable “average golfer.”
Broadie’s original analysis defines shot values from benchmark differences and establishes their additivity. Its discussion also recognizes that unrecorded conditions are averaged over. The accounting and the interpretation of the fitted benchmark should therefore be evaluated separately (Broadie 2012, sec. 2 and the opening of Section 3).
2.2 Shots, Penalties, and Complete Holes
For a recorded transition from \(s_t\) to \(s_{t+1}\), let \(c_t\) be its counted stroke cost. Normally \(c_t=1\); penalties must also appear in the accounting, either as separate cost events or assigned consistently to the transition that includes them. Define
\[ g_t=B(s_t)-c_t-B(s_{t+1}). \]
For a complete hole with \(N\) recorded transitions and score \(C=\sum_t c_t\),
\[ G=\sum_{t=0}^{N-1}g_t=B(s_0)-C. \]
Every intermediate benchmark value cancels. For example, boundary values \((4.2,2.8,1.5,0)\) and costs \((1,2,1)\) give shot values \((0.4,-0.7,0.5)\), totaling \(0.2=4.2-4\). Omitting the penalty changes the apparent total to \(1.2\) and fails to reconcile with the scorecard.
The cancellation does not require \(B\) to be an accurate estimate, an optimal value function, or a Markov model. A poor benchmark can allocate credit poorly while still satisfying the identity. Missing shots, inconsistent penalties, unrecorded pick-ups, changing benchmark versions within a sequence, and incorrect terminal states can break the intended score reconciliation.
For a round, let \(K\) be the sum of benchmark values at the hole starts. Then \(G_{\mathrm{total}}=K-C_{\mathrm{round}}\). Defining \(\Delta\) as later minus earlier gives
\[ \Delta\mathbb E[G_{\mathrm{total}}] =\Delta\mathbb E[K]-\Delta\mathbb E[C_{\mathrm{round}}]. \]
If the start-state distribution and benchmark are fixed, \(\Delta\mathbb E[K]=0\): improving total strokes gained by \(0.5\) means scoring \(0.5\) strokes better on average. If categories form an exhaustive, nonoverlapping partition, their sums equal the total. Improving putting by \(0.5\) while every other category is unchanged has the same one-for-one implication. What is uncertain is whether a proposed intervention will produce that isolated category change.
Field or course adjustments can add further terms to a published statistic. Comparisons must use the provider’s actual definition and the same adjustment convention; the unadjusted identity alone does not establish equivalence between differently adjusted reports.
2.3 Policy Evaluation Is Different From Optimization
A predictive model needs more structure. Suppose player \(i\) follows policy \(\pi\), which assigns probabilities to actions such as club and target choices. Let \(P_i(ds',dc\mid s,a)\) describe the resulting transition and counted cost. For a sufficient state and a policy that completes the hole with finite expected cost, its value satisfies the policy evaluation equation
\[ V_i^\pi(s)=\sum_a\pi(a\mid s) \int [c+V_i^\pi(s')]\,P_i(ds',dc\mid s,a). \]
An expected-score optimal policy instead has value
\[ V_i^*(s)=\min_a\int[c+V_i^*(s')]\,P_i(ds',dc\mid s,a). \]
The minimization is an additional decision problem. A regression fitted to observed play does not become optimal merely because it predicts remaining strokes. Broadie’s dynamic-programming discussion explicitly motivates a near-optimal-play approximation for the professional data; it does not prove that every empirical benchmark or golfer satisfies that approximation. The distinction is also explicit in Sutton and Barto’s treatment (Sutton and Barto 2018, secs. 3.5–3.6).
A small absorbing model makes this visible. State 1 always moves to state 2 at cost one. From state 2, each shot costs one and has probability \(0.75\) of finishing and \(0.25\) of remaining there. Therefore \(V_2=1+0.25V_2=4/3\) and \(V_1=1+V_2=7/3\). These are correct policy values. If another available action finishes immediately from state 1 at cost one, the original policy is not optimal there.
Even policy evaluation needs a sufficient state. If player identity persists between shots, pooling players and then treating the next shot as a fresh draw from a generic player can describe a different process. A conditional remaining-score average can still exist when the recorded state is insufficient for a closed Bellman recursion.
3. Individual Continuation and Category Boundaries
For a specified player and continuation policy, write \(V_i(s)\) for expected remaining strokes. For this section, assume each approach costs one, the same approach start is used, and its end state \(X\) begins a putting sequence that stays on the green until the ball is holed. Putting skill and the continuation policy are unchanged. These assumptions isolate a proximity intervention.
Expected approach credit is \(B(s_0)-1-\mathbb E[B(X)]\). Expected putting credit is \(\mathbb E[B(X)-V_i(X)]\). Adding gives
\[ \mathbb E[G_{\mathrm{approach}}+G_{\mathrm{putting}}] =B(s_0)-1-\mathbb E[V_i(X)]. \]
The benchmark at the category boundary cancels. Let an intervention change the leave distribution from \(P_0\) to \(P_1\). Use positive \(b\) for expected strokes saved:
\[ \begin{aligned} b_B&=\mathbb E_{P_0}[B(X)]-\mathbb E_{P_1}[B(X)],\\ b_i&=\mathbb E_{P_0}[V_i(X)]-\mathbb E_{P_1}[V_i(X)]. \end{aligned} \]
Then
\[ \Delta G_{\mathrm{approach}}=b_B,\qquad \Delta G_{\mathrm{putting}}=b_i-b_B,\qquad \Delta G_{\mathrm{total}}=b_i. \]
An approach improvement can change putting strokes gained even though the golfer’s putting skill is unchanged. That is a consequence of applying a common benchmark to a different set of putting opportunities. It does not invalidate the total accounting or show that the metric secretly ignores the golfer’s actual putts.
If an approach holes out, leaves the green, incurs a penalty, or starts another non-putting sequence, use the full transition and actual category rules. A category consisting of disconnected shot segments has several entry and exit boundaries; it cannot always be represented by a single first-putt value.
When the Intervention Changes More Than Proximity
A change in technique or strategy can alter immediate counted cost, the next-state distribution, and subsequent performance together. Let \(\bar c_a\) be mean immediate cost, \(P_a\) the next-state distribution, and \(V_i^a\) the continuation value under alternative \(a\in\{0,1\}\). Define the expected cost of that alternative by
\[ F_a=\bar c_a+\mathbb E_{P_a}[V_i^a(X)]. \]
Its scoring benefit is \(b_i=F_0-F_1\). Adding and subtracting \(\mathbb E_{P_1}[V_i^0(X)]\) yields an exact decomposition:
\[ \begin{aligned} b_i={}&(\bar c_0-\bar c_1)\\ &+\bigl(\mathbb E_{P_0}[V_i^0]-\mathbb E_{P_1}[V_i^0]\bigr)\\ &+\mathbb E_{P_1}[V_i^0-V_i^1]. \end{aligned} \]
The three terms are the immediate-cost change, the distribution change evaluated using old continuation skill, and the continuation change evaluated over the new opportunities. This is an accounting decomposition of specified models, not an identification result. Evaluating the distribution change using new skill instead changes the component allocations when the effects interact, while preserving their total.
For example, take old/new mean costs \(1\) and \(1.1\), old/new probabilities over two next states \((0.5,0.5)\) and \((0.75,0.25)\), and old/new continuation values \((1.5,2.0)\) and \((1.3,1.9)\). Expected costs are \(2.75\) and \(2.55\), a benefit of \(0.20\). The decomposition above gives \(-0.10+0.125+0.175=0.20\). Reversing the attribution order gives \(-0.10+0.15+0.15=0.20\). The predicted improvement is well defined for the declared model; its division into component effects needs a stated convention.
4. Proximity Requires a Distribution
4.1 A Local Derivative Has a Limited Meaning
At a fixed lie and other conditions, a deterministic small distance change \(\delta<0\) gives the approximation
\[ b_i=V_i(x)-V_i(x+\delta)\approx -V_i'(x)\delta. \]
This requires local differentiability and a sufficiently small change. The derivative has units of strokes per unit distance. Matching benchmark and player levels is not necessary for matching this contrast: \(V_i(x)=B(x)+k\) gives exactly the same distance differences for any constant \(k\). Conversely, matching at one distance does not imply matching derivatives.
For a random old leave \(X\) and a specified coupled change \(\delta(X)\), a second-order expansion is
\[ b_i\approx-\mathbb E[V_i'(X)\delta(X)] -\tfrac12\mathbb E[V_i''(X)\delta(X)^2]. \]
This is generally not the derivative at the mean distance multiplied by the mean change. In several continuous state variables, gradients and Hessians replace the first and second derivatives. Discrete lie changes, hazards, and the point mass at “holed” are better handled by the full outcome distribution. A shorter mean leave alone says nothing about which parts of the distribution improved or worsened.
4.2 Equal Means Can Hide Different Scores
Take illustrative remaining values \(V(1)=1.1\), \(V(2)=1.5\), and \(V(3)=2.1\), with distances in metres. One strategy always leaves two metres. Another leaves one or three metres with equal probability. Both average two metres, but their expected remaining scores are \(1.5\) and \(1.6\) respectively.
Thus “proximity stayed the same” can conceal a scoring change. A change in average driving distance can similarly conceal different rough, recovery, or penalty probabilities. For a convex value function, a mean-preserving spread cannot decrease its expectation. A larger variance alone does not establish that ordering between arbitrary distributions; outside the stated condition, calculate the full expectation.
One precise mean-preserving-spread condition is a coupling of old distance \(X\) and new distance \(Y\) with \(\mathbb E[Y\mid X]=X\). If \(V\) is convex on their support and the expectations exist, conditional Jensen’s inequality gives \(\mathbb E[V(Y)\mid X]\ge V(X)\), and averaging gives \(\mathbb E[V(Y)]\ge\mathbb E[V(X)]\). This condition compares distributions, not just two reported variance numbers.
5. Putting Examples With Complete Accounting
5.1 Probability Level Does Not Determine Slope
In a deliberately simplified model, a putt from positive distance \(x\) either holes immediately with probability \(p_i(x)\) or misses and is followed by one certain finishing putt. Then
\[ V_i(x)=p_i(x)+2[1-p_i(x)]=2-p_i(x),\qquad V_i'(x)=-p_i'(x). \]
The model excludes three-putts and does not derive the follow-up leave distribution. At the already-holed state the value is zero; the positive-distance model describes a ball that still requires a stroke. It is an illustration, not a calibrated putting model.
Suppose two constructed make curves are \(p_A(x)=e^{-0.3x}\) and \(p_B(x)=e^{-1.2x}\), with \(x\) in metres and the rate constants in inverse metres. Golfer B has a lower make probability at every positive distance. Yet the marginal proximity benefits switch order:
| Distance | Player A Slope \(V_A'(x)\) | Player B Slope \(V_B'(x)\) | Larger Local Benefit From Moving Closer |
|---|---|---|---|
| \(0.5\) m | \(0.2582\) strokes/m | \(0.6586\) strokes/m | B |
| \(3.0\) m | \(0.1220\) strokes/m | \(0.0328\) strokes/m | A |
The slopes cross at \(x=\log(4)/0.9\approx1.5403\) m. Moving from \(2.5\) to \(2.0\) m saves about \(0.0764\) strokes for A and \(0.0409\) for B. A lower make probability does not imply a steeper slope or a larger finite benefit. The original article’s unconditional “poor putters benefit more” claim therefore fails even in its own two-putt model class.
For a separate accounting example, let the reference remaining value fall from \(1.5\) to \(1.3\) when an approach leaves the ball nearer. Construct three golfers with the following continuation values:
| Golfer | Far Value | Near Value | Actual Benefit \(b_i\) | Approach SG Change | Putting SG Change |
|---|---|---|---|---|---|
| A | 1.20 | 1.10 | +0.10 | +0.20 | −0.10 |
| B | 1.70 | 1.40 | +0.30 | +0.20 | +0.10 |
| C | 1.80 | 1.75 | +0.05 | +0.20 | −0.15 |
These are internally valid values for the simplified one-or-two-putt continuation model. B and C are both worse putters than A at both locations. B benefits more from the approach improvement, while C benefits less. In every row, the two category changes sum to the actual scoring benefit. A’s putting statistic worsens while A’s expected score improves.
5.2 Lag Putting and Per-Round Units
If the same closer leave instead comes from a lag putt, both the first putt and its continuation belong to putting. The benchmark at the second-putt start cancels within that category. For a fixed initial putt with benchmark \(2.0\), let the first-putt end benchmark change from \(1.5\) to \(1.3\), while the golfer’s continuation value changes from \(1.7\) to \(1.4\).
The first-putt SG changes from \(-0.5\) to \(-0.3\), a gain of \(0.2\). Expected continuation SG changes from \(1.5-1.7=-0.2\) to \(1.3-1.4=-0.1\), a gain of \(0.1\). The entire putting sequence gains \(0.3\), exactly matching the reduction in expected strokes from \(2.7\) to \(2.4\). Crediting only the first putt would miss part of the effect.
“SG putting per round” is the average of each round’s sum of putt values. It is not the average value per putt. If the number of putts or opportunities changes, the two denominators answer different questions. Comparisons should report the unit, round selection, category definition, benchmark version, and treatment of incomplete rounds.
6. Driving, Approaches, and the Short Game
6.1 What Is Ten More Yards Worth?
For a fixed lie, fixed direction, unchanged cost, and a hypothetical ten-yard reduction in remaining distance, \(B(d)-B(d-10)\) is a benchmark contrast when \(d\) is expressed in yards. The player’s corresponding continuation contrast is \(V_i(d)-V_i(d-10)\). Equality requires equality of those contrasts, not identical skill profiles at every distance.
A swing change that increases launch speed does not generally translate the entire drive distribution ten yards toward the hole. It can alter launch angle, spin, strike location, direction, carry, roll, and penalty risk. The relevant decision compares \(\mathbb E[c+V_i(S')]\) under the full distributions generated by the alternatives. The value of ten yards per drive also is not automatically its value per round: the number and type of affected opportunities matter.
Being good with mid-irons does not alone determine the marginal value of distance. What matters is the change in this player’s continuation cost between the actual possible next states, together with their probabilities. That can be estimated, but cannot be inferred just from a general skill label.
6.2 Lay-Up Targets Are Actions, Not Guaranteed Leaves
A preference for a 90-yard wedge over a 50-yard wedge does not by itself settle a lay-up decision. The shots intended to reach those distances have their own dispersions, lies, penalties, and costs. Compare the action values
\[ Q_i^\pi(s,a)=\mathbb E_i[c+V_i^\pi(S')\mid s,a] \]
for the available actions. Using \(V_i(90)\) and \(V_i(50)\) alone treats the target states as certain and ignores how the golfer gets there. A credible personal advantage at one distance is useful evidence; it still needs to be large and well established enough to affect the complete action comparison.
6.3 Short-Game Outcomes Include the Next Shot
Chip, pitch, bunker, and fringe-putting decisions combine the chance of holing out, the distribution of misses, subsequent putting, and the chance of another recovery shot. The preferred action may depend on a golfer’s putting strengths, but a common benchmark remains useful for reporting what happened. Using a putter from off the green need not count as “putting” under a provider’s lie-based category rule; equipment and statistical category are different classifications.
7. Heterogeneity, Selection, and Causal Inference
A pooled shot dataset generally weights players differently at different states. Let \(I\) denote player identity and \(V_i^{\mathrm{obs}}(s)=\mathbb E[H\mid S=s,I=i]\). The law of total expectation gives
\[ B(s)=\sum_i w_i(s)V_i^{\mathrm{obs}}(s),\qquad w_i(s)=\Pr(I=i\mid S=s), \]
for the population quantity represented by that sampling scheme. It is not generally an equally weighted average over players. For a scalar state coordinate with differentiable weights and values (and a finite player mixture),
\[ B'(s)=\sum_i w_i(s)(V_i^{\mathrm{obs}})'(s) +\sum_i w_i'(s)V_i^{\mathrm{obs}}(s). \]
The second term represents changing composition. For an illustrative scalar state \(d\in[1,2]\), let \(V_A=1+0.1d\), \(V_B=1.5+0.2d\), and \(w_A=0.8-0.2d\). At \(d=1.5\), the weighted individual slopes total \(0.15\), while the composition term is \(0.13\), giving pooled slope \(0.28\). A pooled derivative need not be the average individual derivative.
Observed state contrasts also mix conditions and choices: golfers may reach the same distance with different lies, abilities, preceding shots, or strategic objectives. Calling \(B(d)-B(d-10)\) an “average treatment effect” does not make it one. A treatment effect requires a defined intervention and an identified comparison of its alternatives. Heterogeneity, selection, insufficient conditioning, and transport to a particular golfer are more precise concerns here than labeling every population-to-player use an ecological fallacy.
7.1 Personalization Helps Prediction; Identification Remains Separate
One predictive model is \(V_i(s)=B(s)+\delta_i(s)\), with the deviations partially pooled across players. This can reduce unstable estimates where individual data are sparse. It does not guarantee calibration, sensible extrapolation, correct derivatives, or a causal interpretation. A model must retain an appropriate terminal value and nonnegative expected counted costs, and should be assessed on future rounds or held-out players appropriate to its intended use. Gelman’s discussion of partial pooling and causal interpretation makes this distinction explicit (Gelman 2006).
For a specified intervention \(a\), let \(C^a\) be the score that would result under that intervention over a defined period and course distribution. A causal contrast might be \(\mathbb E[C^0-C^1\mid I=i]\). It concerns repeated possible outcomes for that player, not an observable pair of outcomes from the same played round.
Estimating such a contrast from observational data requires assumptions such as consistency of the intervention definition, exchangeability after relevant adjustment, and sufficient overlap of alternatives. Randomized comparisons can strengthen identification, but practice has learning and carryover effects: a simple before/after or alternating-session design does not automatically separate treatment from time. Hierarchical modeling addresses estimation and pooling; it cannot supply missing intervention information. See Hernán and Robins for the identification framework (Hernán and Robins 2020, 3).
7.2 Time Variation Is a Modeling Question
There is no requirement that one golfer’s repeated-play average converge to the average of a heterogeneous population. Even two stationary players can have different expectations. Conversely, independent repetitions from one fixed player’s distribution with finite mean have a sample average that converges to that player’s expectation; heterogeneous population membership does not invalidate this elementary result.
Fatigue, learning, equipment changes, and changing courses can make a fixed-distribution forecast unsuitable. Temporal dependence changes uncertainty and effective sample size. Neither dependence alone proves non-ergodicity nor time variation proves that expectations are meaningless. Ergodicity concerns convergence under a specified stochastic process; nonstationarity concerns how its distribution changes over time. These distinctions do not alter telescoping accounting.
For prediction, examine residual patterns over time, use time-ordered validation, and compare stable and evolving skill models. A dynamic model such as \(\theta_{t+1}=\theta_t+\eta_t\) is a possible assumption to test, not a demonstrated universal law of golfer development. The earlier appeal to economic ergodicity theory did not establish a golf-specific failure of SG.
9. A Defensible Coaching Workflow
9.1 Reconcile and Validate the Description
Start by reconciling shot costs and penalties with the scorecard. Record the benchmark and category conventions. Inspect category totals alongside the distributions of opportunities, not only the rankings. Report how much data support an apparent weakness and how sensitive it is to unusual rounds or conditions.
A single round losing two strokes in putting does not establish stable putting skill, let alone show that putting is weaker than every other part of that golfer’s game. Compare relevant categories and uncertainty. There is no universal number of shots that makes an individual value surface reliable: it depends on coverage of states, dependence, noise, model structure, and the precision required for the decision.
9.2 Specify What Would Change
Define the intervention, comparator, time horizon, outcome, and practice cost. “Improve putting” is an objective; “use this practice protocol for six weeks” is closer to an intervention. “Gain ten yards” is an outcome description that could arise through very different mechanical changes and risk tradeoffs.
Before prescribing, test whether the proposed mechanism changes the predicted shot distribution in repeatable measurements. Compare forecasts against subsequent outcomes, including misses and penalties. Separate uncertainty about physical parameters, trial-to-trial execution, course conditions, and the estimated score surface. A narrow confidence interval from a misspecified model does not settle the decision.
9.3 Evaluate Interactions and Opportunity Cost
The largest current SG deficit need not offer the largest attainable improvement per hour of practice. An intervention can have diminishing returns, change several categories, or alter the frequency of opportunities. Define a joint expected-score model \(F(a,b)\) for two interventions. Their interaction benefit, relative to adding the two separate benefits, is
\[ \mathcal I=F(1,0)+F(0,1)-F(0,0)-F(1,1). \]
Positive \(\mathcal I\) means the combined score reduction exceeds the sum of the isolated reductions. For approach and putting interventions, this can occur because the approach changes where the improved putting skill is used. The sign and size are empirical or model-dependent questions. Shared time costs and adverse changes elsewhere also belong in the comparison.
10. What the Corrected Argument Establishes
Strokes gained retains its exact accounting meaning with a consistent benchmark and complete score records. Individual heterogeneity affects the interpretation of shot credit and the prediction of interventions; it does not invalidate additivity. Whole-category changes, single-shot benchmark changes, and causal effects of training are distinct quantities.
The strongest individual analysis uses common benchmarks for comparison, calibrated player-specific models where they improve predictions, and explicit evidence for causal claims. It evaluates complete distributions and continuation policies, includes cross-category effects, and tests whether the proposed improvement is feasible and worth its cost.
Connection to the AffineDrift Framework
The mechanics-to-score chain begins with the golfer’s state, capabilities, and chosen action. A model such as \(\dot z=f(z)+G(z)u\) describes a control-affine system only for specified states and inputs. Its zero-input drift need not be biologically passive: activation, elastic states, contact forces, and input conventions matter. Identifying parameters from observed motion and decomposing acceleration do not, by themselves, identify what an untried coaching intervention will do.
The next stages are impact and launch, flight and ground interaction, the distribution of resulting ball states, and the golfer’s continuation policy. A mechanics model may constrain which launch conditions are possible; it does not directly supply a calibrated probability distribution over a round’s lies, penalties, decisions, and scores. Uncertainty and model discrepancy must pass through each stage.
The useful connection is testable: specify a physical mechanism, predict how a feasible change alters a shot distribution, measure that change, propagate it through a validated continuation model, and compare the predicted scoring effect with outcomes. Statistical estimation is necessary for the mechanics as well as the scoring model. A convincing explanation of joint torque or clubhead speed becomes coaching evidence only when the relevant later links are also supported.
Sources and Reading Boundaries
- Broadie (2012), “Assessing Golfer Performance on the PGA TOUR,” Interfaces 42(2), 146–165. Published record; author’s April 2011 preprint. The relevant reading here is Section 2 and the opening benchmark discussion in Section 3. Historical data and near-optimal-play assumptions are not treated as current universal facts.
- Sutton and Barto (2018), Reinforcement Learning: An Introduction, second edition. Publisher; 2018 draft hosted by CMU. Sections 3.5 and the opening of 3.6 distinguish policy values and optimal values, including a simplified golf example. The draft contains production artifacts; its examples are not empirical golf estimates.
- Gelman (2006), “Multilevel (Hierarchical) Modeling: What It Can and Cannot Do,” Technometrics 48(3), 432–435. Author’s full paper. Its prediction and causal-inference distinction supports the limited role assigned to partial pooling here.
- Hernán and Robins (2020), Causal Inference: What If. Author’s book page. The intervention and identification framework is relevant; it is not a study of a golf practice intervention.
Earlier links to Broadie’s popular book and Peters’ economics article are retained as historical destinations. Neither is used as evidence for the corrected quantitative examples or for a golf-specific non-ergodicity claim. The linked governed critique remains visible with a correction explaining which of its premises require revision.
References
Citation
@online{olson2025,
author = {Olson, Dieter},
title = {On the {Limits} of {Strokes-Gained} {Inference} for
{Individual} {Golfers}},
date = {2025-11-28},
url = {https://affinedrift.com/articles/strokes-gained-limitations.html},
langid = {en}
}
The following are editorial objections and responses, not quotations attributed to named researchers.
A Common Benchmark Is Often Useful Enough
A player does not need a perfect personal value function to learn from category totals.
Individual Data Are Sparse
Detailed player surfaces may be dominated by noise.
Closer Skill Groups Reduce Mismatch
A reference group closer to the golfer’s ability may predict better than a professional baseline.