Three noisy readings of one unknown, weighted by how well each has done
The anchored estimate treats consensus as the anchor and moves a fixed fraction of the way toward the model. This method treats all three readings symmetrically and lets each one’s own track record decide its weight.
One unknown: \(\theta\), the company’s revenue growth for the coming quarter. Three noisy readings of it, each with its own variance \(\sigma_j^2\) and precision \(\tau_j = 1/\sigma_j^2\). Symbols follow the notation page.
| Reading | Symbol | Precision | Where its variance comes from |
|---|---|---|---|
| Prior | \(\mu_0\) | \(\tau_0\) | the company’s own realised growth history |
| Model | \(x_M\) | \(\tau_M\) | the model’s error against realised growth |
| Consensus | \(x_C\) | \(\tau_C\) | consensus’s error, or analyst dispersion |
Assume each reading is an independent Gaussian observation of \(\theta\). The posterior is proportional to prior times likelihoods, and collecting the exponent as a quadratic in \(\theta\) gives a Gaussian again (Gelman et al. 2013). The posterior growth \(\hat\theta\) is the precision-weighted mean, and its variance \(\sigma_\theta^2\) is one over the total precision \(T\):
\[ \hat\theta = \frac{\mu_0\tau_0 + x_M\tau_M + x_C\tau_C}{T}, \qquad \sigma_\theta^2 = \operatorname{Var}(\theta \mid \text{data}) = \frac{1}{T}, \qquad T = \tau_0 + \tau_M + \tau_C \]
Two structural facts follow immediately:
The method works on growth over the previous actual \(y_{t-1}\), not on revenue levels: the model reading is \(x_M = m/y_{t-1} - 1\) and the consensus reading \(x_C = c/y_{t-1} - 1\). Two reasons:
history supplies the statistics, each read strictly before the moment being forecast.
AS_OF = pd.Timestamp("2026-05-10", tz="UTC")
estimates = pf.validate_estimates(examples.quarterly_history())
estimates.groupby("source").value.describe()[["count", "min", "max"]]| count | min | max | |
|---|---|---|---|
| source | |||
| actual | 8 | 568 | 762.9 |
| consensus | 8 | 585.1 | 798.8 |
| model | 8 | 598.6 | 845.3 |
prior = pf.growth_prior(estimates, AS_OF)
prior.tail(1)[
[
"previous_actual",
"prior_growth_mean",
"prior_growth_variance",
"realised_growth_count",
]
]| previous_actual | prior_growth_mean | prior_growth_variance | realised_growth_count | |
|---|---|---|---|---|
| 8 | 762.9 | 0.0430518 | 0.0000183461 | 7 |
The error record per source is the same idea: for each past quarter, what that source said before the selected actual became known, against that actual. Preparation uses the latest actual known at the cutoff; select first releases upstream if required.
errors = pf.growth_error_variance(estimates, AS_OF, "model").merge(
pf.growth_error_variance(estimates, AS_OF, "consensus"),
on=pf.subject_index(prior),
)
errors.tail(1)[["model_growth_error_variance", "consensus_growth_error_variance"]]| model_growth_error_variance | consensus_growth_error_variance | |
|---|---|---|
| 8 | 0.000161513 | 0.000202727 |
A source’s estimate for a past quarter has to be the one that stood before that quarter’s selected actual became knowable. First-release behavior requires selecting first actuals before preparation. On a panel that read has to be anchored, once per quarter. A test asserts that a revision published after the actual is never read.
subjects = pf.prepare_bayesian_subjects(
estimates,
AS_OF,
forecast_source="model",
settings=BAYESIAN_DEFAULTS,
)
method = BayesianPosterior(forecast_source="model", settings=BAYESIAN_DEFAULTS)
published = method.fit(subjects).apply(subjects)
row = published.dropna(subset=["posterior_level"]).iloc[-1]
pd.DataFrame(
[
{
"quantity": name,
"precision": row[f"precision_{name}"],
"weight": row[f"weight_{name}"],
}
for name in ("prior", "model", "consensus")
]
)| quantity | precision | weight | |
|---|---|---|---|
| 0 | prior | 54,507.5 | 0.830506 |
| 1 | model | 6,191.45 | 0.0943364 |
| 2 | consensus | 4,932.73 | 0.0751578 |
pd.DataFrame(
[
{"field": "posterior_growth", "value": f"{row.posterior_growth:+.4%}"},
{
"field": "posterior_growth_stdev",
"value": f"{row.posterior_growth_stdev:.6f}",
},
{"field": "posterior_level", "value": f"{row.posterior_level:,.4f}"},
{"field": "posterior_lower", "value": f"{row.posterior_lower:,.4f}"},
{"field": "posterior_upper", "value": f"{row.posterior_upper:,.4f}"},
{"field": "settings_version", "value": row.settings_version},
]
)| field | value | |
|---|---|---|
| 0 | posterior_growth | +4.9481% |
| 1 | posterior_growth_stdev | 0.003903 |
| 2 | posterior_level | 800.6488 |
| 3 | posterior_lower | 771.7952 |
| 4 | posterior_upper | 832.4804 |
| 5 | settings_version | bayesian-defaults |
The prior carries most of the weight here, because eight quarters of steady growth are a much sharper reading than either estimate’s error record. That is the method working as intended, not a bug: a company whose growth is stable is mostly predicted by its own history.
readings = pd.DataFrame(
[
{
"source": "prior",
"growth": row.prior_growth_mean,
"weight": row.weight_prior,
},
{
"source": "model",
"growth": row.model / row.previous_actual - 1,
"weight": row.weight_model,
},
{
"source": "consensus",
"growth": row.consensus / row.previous_actual - 1,
"weight": row.weight_consensus,
},
]
)
figure, axes = plt.subplots(1, 2, figsize=(11, 3.6))
sns.barplot(data=readings, x="source", y="weight", ax=axes[0], color=examples.ACCENT)
axes[0].set(title="Weight is precision share", ylabel="weight", xlabel="")
sns.scatterplot(
data=readings, x="growth", y=0, hue="source", s=150, ax=axes[1], zorder=3
)
axes[1].axvline(
row.posterior_growth, color=examples.WARM, lw=1.6, ls="--", label="posterior"
)
axes[1].set(ylim=(-0.5, 0.5), yticks=[], ylabel="", xlabel="growth")
axes[1].set_title("Where the posterior lands")
axes[1].legend(loc="upper left", fontsize=8)
sns.despine(ax=axes[1], left=True)
figure.tight_layout()
The band is a Student-\(t\) interval, not a Gaussian one:
pd.Series(
{
"interval_width": BAYESIAN_DEFAULTS.interval_width,
"interval_df": BAYESIAN_DEFAULTS.interval_df,
"interval_loc": BAYESIAN_DEFAULTS.interval_loc,
"interval_scale": BAYESIAN_DEFAULTS.interval_scale,
}
)interval_width 0.9
interval_df 1.5
interval_loc 0.5
interval_scale 2.75
dtype: float64\(\nu = 1.5\) degrees of freedom is far below the Gaussian limit, \(\nu\to\infty\). The posterior variance formula would give a Gaussian band, and a Gaussian band on financial forecast errors is famously too narrow (Mandelbrot 1963).
from scipy import stats
grid = np.linspace(-6, 6, 600)
heavy, normal = stats.t(df=BAYESIAN_DEFAULTS.interval_df), stats.norm()
scale_t, scale_n = heavy.ppf(0.75), normal.ppf(0.75)
figure, axis = plt.subplots(figsize=(7.5, 3.4))
axis.plot(
grid,
heavy.pdf(grid * scale_t) * scale_t,
color=examples.ACCENT,
label=r"Student-$t$, $\nu$ = 1.5",
)
axis.plot(
grid,
normal.pdf(grid * scale_n) * scale_n,
color=examples.GREY,
ls="--",
label="normal",
)
for distribution, scale, colour in (
(heavy, scale_t, examples.ACCENT),
(normal, scale_n, examples.GREY),
):
quantile = distribution.ppf(0.95) / scale
axis.axvline(quantile, color=colour, lw=1, ls=":")
axis.annotate(
f"95th pct {quantile:.2f}", (quantile, 0.24), fontsize=8.5, color=colour
)
axis.set(xlabel="standardised forecast error", ylabel="density")
axis.set_title("Why the interval uses a heavy-tailed distribution")
axis.legend()
figure.tight_layout()
The two are different, and the distinction is the interesting part of the port.
short_record = subjects.assign(model_growth_error_variance=float("nan"))
degraded = method.fit(short_record).apply(short_record)
degraded_row = degraded.dropna(subset=["posterior_level"]).iloc[-1]
pd.DataFrame(
[
{
"case": "full record",
"model contributes": True,
"published": True,
"growth": f"{row.posterior_growth:+.4%}",
},
{
"case": "model track record too short",
"model contributes": False,
"published": bool(degraded_row[pf.ELIGIBLE_COLUMN]),
"growth": f"{degraded_row.posterior_growth:+.4%}",
},
]
)| case | model contributes | published | growth | |
|---|---|---|---|---|
| 0 | full record | True | True | +4.9481% |
| 1 | model track record too short | False | True | +4.3384% |
A model whose record is too short to estimate a precision drops out of the blend and the subject is still published on prior and consensus, with an interval. Its model precision and weight remain missing. A model whose value is missing refuses the subject outright: there is nothing to weigh.
missing_model = subjects.assign(model=float("nan"))
refused = method.fit(missing_model).apply(missing_model)
refused[[pf.ELIGIBLE_COLUMN, pf.ELIGIBILITY_REASON_COLUMN]].tail(1)| eligible | eligibility_reason | |
|---|---|---|
| 8 | False | missing_prediction |
Every refusal reason:
pd.DataFrame([{"reason": reason.value} for reason in PosteriorReason])| reason | |
|---|---|
| 0 | publishable |
| 1 | missing_input |
| 2 | insufficient_history |
| 3 | no_consensus_precision |
| 4 | missing_prediction |
| 5 | nonfinite_result |
Neither branch is a fallback in the forbidden sense — no unpublishable subject ever gets a substituted number. Both are declared method behaviour, and both carry tests.
BAYESIAN_DEFAULTS.model_dump(mode="json"){'minimum_observations': 3,
'include_model': True,
'consensus_stdev_variance_multiplier': 1.0,
'consensus_count_exponent': None,
'consensus_bias_lambda': 0.0,
'consensus_bias_process_noise': 1.0,
'consensus_bias_measurement_noise': 0.5,
'winsorize_fraction': None,
'interval_width': 0.9,
'interval_df': 1.5,
'interval_loc': 0.5,
'interval_scale': 2.75,
'dispersion_weighted': False,
'preset': None,
'fitted_through': None,
'version': 'bayesian-defaults'}BAYESIAN_DEFAULTS preserves the source signal’s chosen compatibility settings, with fitted_through=None.patient = BayesianSettings(
**{
**BAYESIAN_DEFAULTS.model_dump(),
"minimum_observations": 6,
"version": "patient-demo",
}
)
pd.DataFrame(
[
{
"settings": settings.version,
"minimum_observations": settings.minimum_observations,
"declared minimum_history": BayesianPosterior(
forecast_source="model", settings=settings
).requirements.minimum_history,
}
for settings in (BAYESIAN_DEFAULTS, patient)
]
)| settings | minimum_observations | declared minimum_history | |
|---|---|---|---|
| 0 | bayesian-defaults | 3 | 3 |
| 1 | patient-demo | 6 | 6 |
The declared requirement follows the setting, so a pipeline assembled with patient settings advertises what it actually needs.
Drop the prior from the blend. The remaining two-source update can be written as:
\[ \hat\theta = x_C + \underbrace{\frac{\sigma_C^2}{\sigma_M^2 + \sigma_C^2}}_{w_M}\,(x_M - x_C) \]
For comparison, also set the anchored intercept and revision contribution to zero. The simplified anchored level is \(\hat y = c + b_\Delta (m - c)\). Only under these simplifications do the two share a functional form, with
\[ b_\Delta \longleftrightarrow w_M = \frac{1}{1 + \sigma_M^2/\sigma_C^2} \]
Same functional form, different estimand. The regression estimates that ratio from the joint distribution of gap and surprise, across companies. The Bayesian computes it from each source’s own error history, per company.
ratio = np.logspace(-1, 1.6, 400)
curve = pd.DataFrame({"ratio": ratio, "weight": 1 / (1 + ratio)})
figure, axis = plt.subplots(figsize=(7.5, 3.8))
sns.lineplot(data=curve, x="ratio", y="weight", ax=axis, color=examples.ACCENT)
fitted = pf.ANCHORING_2025_01.gap_weight
axis.scatter([(1 - fitted) / fitted], [fitted], s=60, color=examples.WARM, zorder=3)
axis.annotate(
f"the anchored estimate's fitted\ngap weight, {fitted}",
((1 - fitted) / fitted, fitted),
xytext=(12, 22),
textcoords="offset points",
fontsize=8.5,
color=examples.WARM,
)
axis.set_xscale("log")
axis.set(xlabel=r"$\sigma_M^2/\sigma_C^2$, log scale", ylabel="weight on the model")
axis.set_title("Two routes to the same functional form")
figure.tight_layout()
A measured median error of 2.99% for the model against 1.54% for consensus implies a weight near 0.21 if the two errors were independent. The regression says 0.129. Error correlation can contribute to this difference: both track the same company, so their errors move together, and a naive precision weighting over-weights the model (Clemen and Winkler 1985). The independent variant is the default anyway, because covariance estimates are materially less stable than scalar variances when two sources are strongly collinear (Smith and Wallis 2009).