MTP pricing

This study validates a longitudinal modified treatment policy on a price at 2, 3 and 5 nodes. The declared support of each price depends on the node’s engagement. LTMLE and SDR read each plan on the classifier route with an exact ratio, and the property cells read learned ratios. The modified treatment policies at a node section states the estimator and its conditions. The sequential pricing tutorial fits law P at 3 nodes with LTMLE on one draw.

Status of this page. The study is registered, and its Python phases have run. The evidence label is “Python property evidence, with comparator runs pending”. The study record sets pending_reference="lmtp", so no paired row is published. The publication policy is reporting. A red cell stays red on this page and in the red-cell ledger, with the owner row that names it.

part	what it is	status
Python property evidence	exact laws, interval calibration with two controls, one double-robustness pair and two necessity pairs	the tables below give the verdicts. It decides whether the method ships
lmtp comparator runs	the pinned lmtp 1.5.4 fits the same samples, and the study pairs the results	pending. The runs need R in the lmtp container, and step J of the X40 plan owns them. A finding in them does not block the method

What was compared

The primary phase draws 1,000 samples of 5,000 rows for each of six scenarios.

scenario	law and plans	pairing
pricing_ltmle_tau2, _tau3, _tau5	law P at 2, 3 and 5 nodes, LTMLE, plans natural_prices, price_up and price_down. Each node reads the exact ratio on the classifier route	exact under a gap of \(10^{-6}\)
pricing_sdr_tau2, _tau3, _tau5	the same draws, SDR	exact under a gap of \(10^{-6}\)

Law P has a price at each node. The declared support of the price is \([4 + 2 L_t, 8 + 2 L_t]\), where \(L_t\) is the engagement at the node. Engaged customers get and tolerate higher prices, and a high price lowers the next engagement. A truncated normal draws each price given its history, and a Bernoulli law draws the outcome. The revenue cell multiplies the outcome by the last price.

plan	map	what it tests
natural_prices	no change	the reference value
price_up	scale by 1.12, hold above \(u_t / 1.12\)	the upper boundary binds
price_down	hold below \(l_t / 0.85\), scale by 0.85 above	the lower boundary binds, with two pieces

setting	cleverly	R lmtp
datasets	1,000 samples of 5,000 rows per scenario	the identical rows
density ratio	the exact ratio, written beside the sample	the same ratio array, read from the replicate data
node regressions	a quasibinomial GLM at the last node and least squares at the others	SL.glm on the same columns
intervals	pointwise 95% Wald from the influence curve	the same, from lmtp’s influence curve

The truths at 2 nodes come from quadrature split at every break of the integrand. The truths at 3 and 5 nodes come from Monte Carlo draws of Definition 1 of Díaz, Williams, Hoffman and Schenck (2023), pinned with their Monte Carlo standard error (tests/studies/mtp_pricing_common.py).

The property cells draw the mean purchase under price_up at n = 2,000. They read a correctly specified final model, except in the pairs. The learned ratio is one of two readings. The learned density is a spline hazard with 320 bins on the position relative to the declared support. The screened classifier is a quadratic logistic classifier, chosen by a screen that the study declares before the pilot. The comparator adapters are tests/canonical/lmtp_mtp_adapter.R and tests/canonical/lmtp_sdr_adapter.R.

The first run of the primary phase used n = 2,000. It read the four tau = 5 price_down rows below the coverage that the exact ratio supports: coverage 0.903 to 0.912, an SE ratio of 0.93, and a bias under 0.04 standard deviations. A probe on fresh draws read an SE ratio of 0.887 at n = 2,000 and 0.968 at n = 4,000. The influence-curve variance runs short at the smaller size. The primary therefore moved to n = 5,000, and the property cells kept their declared sizes. tests/studies/canonical_mtp_pricing.py records the declaration.

Accuracy against known truth

law	estimand	what was tested	implementation	bias (99% interval)	coverage	SE ratio	result
2-node pricing law with engagement feedback on the declared price support, LTMLE	ate_regimen[price_down vs natural_prices]	difference in mean outcome between the plans “multiply each price by 0.85, holding a price within 15% of the support’s lower end” against “leave every observed price unchanged”	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000818 to 0.000549	0.9590	0.9772	pass
2-node pricing law with engagement feedback on the declared price support, LTMLE	ate_regimen[price_up vs natural_prices]	difference in mean outcome between the plans “multiply each price by 1.12, holding a price within 12% of the support’s upper end” against “leave every observed price unchanged”	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000628 to 0.000542	0.9570	1.0186	pass
2-node pricing law with engagement feedback on the declared price support, LTMLE	ey_regimen[natural_prices]	mean outcome under the plan leave every observed price unchanged	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000640 to 0.000410	0.9410	0.9784	pass
2-node pricing law with engagement feedback on the declared price support, LTMLE	ey_regimen[price_down]	mean outcome under the plan multiply each price by 0.85, holding a price within 15% of the support’s lower end	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.0010 to 0.000530	0.9460	0.9678	pass
2-node pricing law with engagement feedback on the declared price support, LTMLE	ey_regimen[price_up]	mean outcome under the plan multiply each price by 1.12, holding a price within 12% of the support’s upper end	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000980 to 0.000664	0.9490	1.0166	pass
3-node pricing law with engagement feedback on the declared price support, LTMLE	ate_regimen[price_down vs natural_prices]	difference in mean outcome between the plans “multiply each price by 0.85, holding a price within 15% of the support’s lower end” against “leave every observed price unchanged”	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.0015 to 0.000458	0.9360	0.9543	pass
3-node pricing law with engagement feedback on the declared price support, LTMLE	ate_regimen[price_up vs natural_prices]	difference in mean outcome between the plans “multiply each price by 1.12, holding a price within 12% of the support’s upper end” against “leave every observed price unchanged”	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000488 to 0.0013	0.9460	0.9542	pass
3-node pricing law with engagement feedback on the declared price support, LTMLE	ey_regimen[natural_prices]	mean outcome under the plan leave every observed price unchanged	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000316 to 0.000682	0.9420	1.0265	pass
3-node pricing law with engagement feedback on the declared price support, LTMLE	ey_regimen[price_down]	mean outcome under the plan multiply each price by 0.85, holding a price within 15% of the support’s lower end	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.0013 to 0.000663	0.9470	0.9817	pass
3-node pricing law with engagement feedback on the declared price support, LTMLE	ey_regimen[price_up]	mean outcome under the plan multiply each price by 1.12, holding a price within 12% of the support’s upper end	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000497 to 0.0016	0.9420	0.9622	pass
5-node pricing law with engagement feedback on the declared price support, LTMLE	ate_regimen[price_down vs natural_prices]	difference in mean outcome between the plans “multiply each price by 0.85, holding a price within 15% of the support’s lower end” against “leave every observed price unchanged”	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000974 to 0.0024	0.9160	0.9310	fail
5-node pricing law with engagement feedback on the declared price support, LTMLE	ate_regimen[price_up vs natural_prices]	difference in mean outcome between the plans “multiply each price by 1.12, holding a price within 12% of the support’s upper end” against “leave every observed price unchanged”	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000889 to 0.0019	0.9380	0.9841	pass
5-node pricing law with engagement feedback on the declared price support, LTMLE	ey_regimen[natural_prices]	mean outcome under the plan leave every observed price unchanged	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000536 to 0.000467	0.9550	1.0208	pass
5-node pricing law with engagement feedback on the declared price support, LTMLE	ey_regimen[price_down]	mean outcome under the plan multiply each price by 0.85, holding a price within 15% of the support’s lower end	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000992 to 0.0023	0.9170	0.9479	fail
5-node pricing law with engagement feedback on the declared price support, LTMLE	ey_regimen[price_up]	mean outcome under the plan multiply each price by 1.12, holding a price within 12% of the support’s upper end	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.0010 to 0.0020	0.9430	0.9885	pass
2-node pricing law with engagement feedback on the declared price support, SDR	ate_regimen[price_down vs natural_prices]	difference in mean outcome between the plans “multiply each price by 0.85, holding a price within 15% of the support’s lower end” against “leave every observed price unchanged”	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000819 to 0.000548	0.9570	0.9775	pass
2-node pricing law with engagement feedback on the declared price support, SDR	ate_regimen[price_up vs natural_prices]	difference in mean outcome between the plans “multiply each price by 1.12, holding a price within 12% of the support’s upper end” against “leave every observed price unchanged”	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000628 to 0.000543	0.9580	1.0185	pass
2-node pricing law with engagement feedback on the declared price support, SDR	ey_regimen[natural_prices]	mean outcome under the plan leave every observed price unchanged	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000640 to 0.000410	0.9410	0.9784	pass
2-node pricing law with engagement feedback on the declared price support, SDR	ey_regimen[price_down]	mean outcome under the plan multiply each price by 0.85, holding a price within 15% of the support’s lower end	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.0010 to 0.000529	0.9470	0.9687	pass
2-node pricing law with engagement feedback on the declared price support, SDR	ey_regimen[price_up]	mean outcome under the plan multiply each price by 1.12, holding a price within 12% of the support’s upper end	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000980 to 0.000664	0.9490	1.0166	pass
3-node pricing law with engagement feedback on the declared price support, SDR	ate_regimen[price_down vs natural_prices]	difference in mean outcome between the plans “multiply each price by 0.85, holding a price within 15% of the support’s lower end” against “leave every observed price unchanged”	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.0015 to 0.000443	0.9320	0.9545	pass
3-node pricing law with engagement feedback on the declared price support, SDR	ate_regimen[price_up vs natural_prices]	difference in mean outcome between the plans “multiply each price by 1.12, holding a price within 12% of the support’s upper end” against “leave every observed price unchanged”	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000499 to 0.0012	0.9470	0.9553	pass
3-node pricing law with engagement feedback on the declared price support, SDR	ey_regimen[natural_prices]	mean outcome under the plan leave every observed price unchanged	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000316 to 0.000682	0.9420	1.0265	pass
3-node pricing law with engagement feedback on the declared price support, SDR	ey_regimen[price_down]	mean outcome under the plan multiply each price by 0.85, holding a price within 15% of the support’s lower end	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.0013 to 0.000647	0.9490	0.9830	pass
3-node pricing law with engagement feedback on the declared price support, SDR	ey_regimen[price_up]	mean outcome under the plan multiply each price by 1.12, holding a price within 12% of the support’s upper end	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000507 to 0.0016	0.9400	0.9635	pass
5-node pricing law with engagement feedback on the declared price support, SDR	ate_regimen[price_down vs natural_prices]	difference in mean outcome between the plans “multiply each price by 0.85, holding a price within 15% of the support’s lower end” against “leave every observed price unchanged”	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000981 to 0.0024	0.9190	0.9339	fail
5-node pricing law with engagement feedback on the declared price support, SDR	ate_regimen[price_up vs natural_prices]	difference in mean outcome between the plans “multiply each price by 1.12, holding a price within 12% of the support’s upper end” against “leave every observed price unchanged”	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000906 to 0.0019	0.9400	0.9840	pass
5-node pricing law with engagement feedback on the declared price support, SDR	ey_regimen[natural_prices]	mean outcome under the plan leave every observed price unchanged	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000536 to 0.000467	0.9550	1.0208	pass
5-node pricing law with engagement feedback on the declared price support, SDR	ey_regimen[price_down]	mean outcome under the plan multiply each price by 0.85, holding a price within 15% of the support’s lower end	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.000997 to 0.0023	0.9230	0.9518	fail
5-node pricing law with engagement feedback on the declared price support, SDR	ey_regimen[price_up]	mean outcome under the plan multiply each price by 1.12, holding a price within 12% of the support’s upper end	cleverly LTMLE and SDR under price policies that bind at a declared support	-0.0011 to 0.0020	0.9440	0.9888	pass

Agreement with the canonical implementation

Python property evidence, with comparator runs pending. The lmtp runs have not run, so this study publishes no paired row, and equivalence.csv is empty. The change that registers the runs adds the paired table here.

Theory properties

property	cell	role	what was tested	what must hold	measured	result
double_robustness	up_tau3__both_wrong_control	control	the mean purchase under the 12% price rise at 3 nodes: the outcome is misspecified and the density ratio is one	bias interval must fall entirely outside the margin, with the reported standard error still on the scale of the empirical spread	bias 0.0951 to 0.0970, margin 0.0028, SE ratio 0.8773	pass
double_robustness	up_tau3__outcome_wrong_learned_classifier	positive	the mean purchase under the 12% price rise at 3 nodes: the outcome is misspecified and the declared classifier estimates the density ratio	bias interval inside the equivalence margin, with the reported standard error on the scale of the empirical spread	bias -0.000398 to 0.0029, margin 0.0050, SE ratio 1.0022	pass
interval_calibration	crossfit_ltmle_learned_classifier_tau3__correctly_specified	positive	the mean purchase under the 12% price rise at 3 nodes, LTMLE cross-fitted over five folds: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9437 to 0.9570, SE ratio 0.9849 to 1.0285	pass
interval_calibration	crossfit_ltmle_learned_classifier_tau3__noise_control	control	the mean purchase under the 12% price rise at 3 nodes, LTMLE cross-fitted over five folds: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8232 to 0.8458, SE ratio 0.6921 to 0.7226	pass
interval_calibration	crossfit_ltmle_learned_classifier_tau3__shrunken_se_control	control	the mean purchase under the 12% price rise at 3 nodes, LTMLE cross-fitted over five folds: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8180 to 0.8410, SE ratio 0.6902 to 0.7197	pass
interval_calibration	crossfit_sdr_learned_classifier_tau3__correctly_specified	positive	the mean purchase under the 12% price rise at 3 nodes, SDR cross-fitted over five folds: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9458 to 0.9589, SE ratio 0.9878 to 1.0317	pass
interval_calibration	crossfit_sdr_learned_classifier_tau3__noise_control	control	the mean purchase under the 12% price rise at 3 nodes, SDR cross-fitted over five folds: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8233 to 0.8460, SE ratio 0.6943 to 0.7249	pass
interval_calibration	crossfit_sdr_learned_classifier_tau3__shrunken_se_control	control	the mean purchase under the 12% price rise at 3 nodes, SDR cross-fitted over five folds: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8200 to 0.8429, SE ratio 0.6914 to 0.7224	pass
interval_calibration	revenue_learned_classifier_tau3__correctly_specified	positive	the mean revenue under the 12% price rise at 3 nodes, learned classifier: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9393 to 0.9531, SE ratio 0.9814 to 1.0242	pass
interval_calibration	revenue_learned_classifier_tau3__noise_control	control	the mean revenue under the 12% price rise at 3 nodes, learned classifier: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8179 to 0.8408, SE ratio 0.6894 to 0.7188	pass
interval_calibration	revenue_learned_classifier_tau3__shrunken_se_control	control	the mean revenue under the 12% price rise at 3 nodes, learned classifier: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8223 to 0.8450, SE ratio 0.6865 to 0.7174	pass
interval_calibration	up_learned_classifier_tau2__correctly_specified	positive	the mean purchase under the 12% price rise at 2 nodes, learned classifier: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9401 to 0.9538, SE ratio 0.9688 to 1.0111	pass
interval_calibration	up_learned_classifier_tau2__noise_control	control	the mean purchase under the 12% price rise at 2 nodes, learned classifier: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8136 to 0.8368, SE ratio 0.6783 to 0.7074	pass
interval_calibration	up_learned_classifier_tau2__shrunken_se_control	control	the mean purchase under the 12% price rise at 2 nodes, learned classifier: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8108 to 0.8341, SE ratio 0.6781 to 0.7080	pass
interval_calibration	up_learned_classifier_tau3__correctly_specified	positive	the mean purchase under the 12% price rise at 3 nodes, learned classifier: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9401 to 0.9538, SE ratio 0.9695 to 1.0120	pass
interval_calibration	up_learned_classifier_tau3__noise_control	control	the mean purchase under the 12% price rise at 3 nodes, learned classifier: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8230 to 0.8457, SE ratio 0.6953 to 0.7249	pass
interval_calibration	up_learned_classifier_tau3__shrunken_se_control	control	the mean purchase under the 12% price rise at 3 nodes, learned classifier: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8116 to 0.8349, SE ratio 0.6783 to 0.7084	pass
interval_calibration	up_learned_classifier_tau5__correctly_specified	positive	the mean purchase under the 12% price rise at 5 nodes, learned classifier: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9261 to 0.9413, SE ratio 0.9444 to 0.9878	pass
interval_calibration	up_learned_classifier_tau5__noise_control	control	the mean purchase under the 12% price rise at 5 nodes, learned classifier: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8049 to 0.8285, SE ratio 0.6764 to 0.7060	pass
interval_calibration	up_learned_classifier_tau5__shrunken_se_control	control	the mean purchase under the 12% price rise at 5 nodes, learned classifier: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.7944 to 0.8185, SE ratio 0.6611 to 0.6913	pass
interval_calibration	up_learned_density_tau2__correctly_specified	positive	the mean purchase under the 12% price rise at 2 nodes, learned density: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9530 to 0.9747, SE ratio 1.0265 to 1.1162	fail
interval_calibration	up_learned_density_tau2__noise_control	control	the mean purchase under the 12% price rise at 2 nodes, learned density: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8363 to 0.8770, SE ratio 0.7238 to 0.7845	pass
interval_calibration	up_learned_density_tau2__shrunken_se_control	control	the mean purchase under the 12% price rise at 2 nodes, learned density: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8257 to 0.8676, SE ratio 0.7187 to 0.7814	pass
interval_calibration	up_learned_density_tau3__correctly_specified	positive	the mean purchase under the 12% price rise at 3 nodes, learned density: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9473 to 0.9704, SE ratio 1.0493 to 1.1408	fail
interval_calibration	up_learned_density_tau3__noise_control	control	the mean purchase under the 12% price rise at 3 nodes, learned density: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8437 to 0.8836, SE ratio 0.7376 to 0.8016	pass
interval_calibration	up_learned_density_tau3__shrunken_se_control	control	the mean purchase under the 12% price rise at 3 nodes, learned density: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8415 to 0.8817, SE ratio 0.7352 to 0.7988	pass
interval_calibration	up_learned_density_tau5__correctly_specified	positive	the mean purchase under the 12% price rise at 5 nodes, learned density: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9485 to 0.9713, SE ratio 1.0428 to 1.1323	fail
interval_calibration	up_learned_density_tau5__noise_control	control	the mean purchase under the 12% price rise at 5 nodes, learned density: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8162 to 0.8591, SE ratio 0.7238 to 0.7847	pass
interval_calibration	up_learned_density_tau5__shrunken_se_control	control	the mean purchase under the 12% price rise at 5 nodes, learned density: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8431 to 0.8831, SE ratio 0.7310 to 0.7926	pass
interval_calibration	up_oracle_tau2__correctly_specified	positive	the mean purchase under the 12% price rise at 2 nodes, exact ratio: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9389 to 0.9527, SE ratio 0.9810 to 1.0232	pass
interval_calibration	up_oracle_tau2__noise_control	control	the mean purchase under the 12% price rise at 2 nodes, exact ratio: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8270 to 0.8495, SE ratio 0.6976 to 0.7292	pass
interval_calibration	up_oracle_tau2__shrunken_se_control	control	the mean purchase under the 12% price rise at 2 nodes, exact ratio: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8129 to 0.8361, SE ratio 0.6863 to 0.7165	pass
interval_calibration	up_oracle_tau3__correctly_specified	positive	the mean purchase under the 12% price rise at 3 nodes, exact ratio: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9404 to 0.9540, SE ratio 0.9759 to 1.0183	pass
interval_calibration	up_oracle_tau3__noise_control	control	the mean purchase under the 12% price rise at 3 nodes, exact ratio: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8244 to 0.8471, SE ratio 0.6944 to 0.7253	pass
interval_calibration	up_oracle_tau3__shrunken_se_control	control	the mean purchase under the 12% price rise at 3 nodes, exact ratio: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8096 to 0.8330, SE ratio 0.6830 to 0.7133	pass
interval_calibration	up_oracle_tau5__correctly_specified	positive	the mean purchase under the 12% price rise at 5 nodes, exact ratio: both nuisances are correctly specified	SE ratio and coverage intervals both inside their calibration bands	coverage 0.9301 to 0.9449, SE ratio 0.9671 to 1.0092	pass
interval_calibration	up_oracle_tau5__noise_control	control	the mean purchase under the 12% price rise at 5 nodes, exact ratio: a declared scale of independent noise is added to each estimate	the SE-ratio interval must fall below the calibration band	coverage 0.8186 to 0.8415, SE ratio 0.6910 to 0.7215	pass
interval_calibration	up_oracle_tau5__shrunken_se_control	control	the mean purchase under the 12% price rise at 5 nodes, exact ratio: the reported standard errors are multiplied by a declared factor below one	the SE-ratio interval must fall below the calibration band	coverage 0.8044 to 0.8280, SE ratio 0.6768 to 0.7067	pass
ratio_necessity	up_tau3__learned_classifier	positive	the mean purchase under the 12% price rise at 3 nodes: the declared classifier estimates the density ratio, with the outcome misspecified	bias interval inside the equivalence margin	bias -0.000013 to 0.0032, margin 0.0049	pass
ratio_necessity	up_tau3__unit_ratio_control	control	the mean purchase under the 12% price rise at 3 nodes: the same fit replaces the density ratio with one, with the outcome misspecified	bias interval must fall entirely outside the margin	bias 0.0957 to 0.0975, margin 0.0028	pass
ratio_necessity	up_tau5__learned_classifier	positive	the mean purchase under the 12% price rise at 5 nodes: the declared classifier estimates the density ratio, with the outcome misspecified	bias interval inside the equivalence margin	bias -0.0011 to 0.0023, margin 0.0053	pass
ratio_necessity	up_tau5__unit_ratio_control	control	the mean purchase under the 12% price rise at 5 nodes: the same fit replaces the density ratio with one, with the outcome misspecified	bias interval must fall entirely outside the margin	bias 0.1273 to 0.1286, margin 0.0020	pass

The truths come from the exact law and its closed forms (tests/studies/mtp_pricing_common.py).

family	cells	replications and size
interval_calibration	twelve positives: the exact ratio, the learned density and the screened classifier at 2, 3 and 5 nodes, the revenue outcome at 3 nodes, and LTMLE and SDR cross-fitted over five folds at 3 nodes. Each positive has a shrunken-SE control and a noise control	the replications that the pooled rule sets, at n = 2,000
double_robustness	3 nodes, wrong outcome GLM. The screened classifier against the wrong GLM with a ratio of one	1,000 pairs at n = 2,000
ratio_necessity	3 and 5 nodes, wrong outcome GLM. The screened classifier against a ratio of one	1,000 pairs at n = 2,000

Each pair removes one term that can vanish at the truth, so each needs a nonzero witness. With a wrong outcome regression the targeting step carries the whole correction, and a wrong ratio then moves the estimate. The learned-density and classifier cells read a correct outcome model. They measure interval calibration with a learned ratio and give no evidence that the ratio is consistent. The double-robustness pair reads the ratio of the classifier alone.

The cross-fold pair classifier_split is not registered. Its pilot read a necessity power of 0.0002, so the pair cannot tell a leak from no leak. In tests/unit/test_classifier_ratio_controls.py, test_t_c1_no_unit_trains_the_model_that_predicts_it shows no leak under unit folds, and test_m_c1_folds_over_stacked_rows_leak shows a leak under row folds.

Measured values and declared margins

Names beginning margin: are thresholds declared before the run. Everything else is measured from the committed results and checked at the precision printed.

quantity	value	source
replicates	1000	paired replications
n	5000	observations per paired replication
independent_tests_passed	26	truth tests passing
independent_tests_total	30	truth tests reported
paired_tests_passed	0	paired comparisons passing
paired_tests_total	0	paired comparisons reported
property_cells_passed	39	property cells passing
property_cells_total	42	property cells reported
max_standardized_bias	0.0444	largest primary standardized bias
min_coverage	0.9160	lowest primary coverage
margin:confidence_level	0.9900	Monte Carlo confidence level
margin:alpha	0.0500	test size
margin:nominal_coverage	0.9500	nominal interval coverage
margin:bootstrap_replicates	10000	bootstrap replications
margin:standardized_bias	0.2500	standardized-bias margin
margin:coverage_floor	0.9000	primary coverage floor
margin:over_coverage_ceiling	0.9900	descriptive overcoverage threshold
margin:se_ratio_sanity_lower	0.8000	primary SE-ratio lower screen
margin:se_ratio_sanity_upper	1.2000	primary SE-ratio upper screen
margin:calibration_se_ratio_lower	0.9300	calibration SE-ratio lower bound
margin:calibration_se_ratio_upper	1.0700	calibration SE-ratio upper bound
margin:calibration_coverage_lower	0.9200	calibration coverage lower bound
margin:calibration_coverage_upper	0.9800	calibration coverage upper bound
margin:type_i_ceiling	0.1000	type-I upper bound
margin:paired_difference	0.1500	paired similarity margin in pooled SDs
margin:rmse_noninferiority	1.1000	RMSE-ratio noninferiority bound
margin:coverage_noninferiority	-0.0250	coverage-difference noninferiority bound
margin:calibration_noninferiority	0.0500	calibration-excess noninferiority bound
margin:minimum_power	0.8000	power lower bound
margin:root_n_slope	-0.5000	expected root-n slope
margin:root_n_slope_lower	-0.6250	root-n slope lower bound
margin:root_n_slope_upper	-0.3750	root-n slope upper bound
margin:excluded_slope	-0.2500	slower rate the interval must exclude
margin:union_model_se_lower	0.1000	union-model SE-ratio screen, lower limit
margin:union_model_se_upper	10	union-model SE-ratio screen, upper limit
margin:shrunken_se_factor	0.7000	negative-control SE multiplier
margin:necessity_displacement	0.1000	minimum ratio necessity displacement
margin:subject_failure_bound	0.0100	declared share of replications a cell may lose to a recordable fitting failure
margin:subject_failures_allowed	10	the same bound, in primary replications

Limits

limit	what it means for use
the four tau = 5 price_down primary rows are experimental	the exact ratio gives coverage 0.916 to 0.923 and an SE ratio of 0.931 to 0.952 for LTMLE and SDR at the primary size. The package labels these fits experimental. The roadmap item mtp-pricing-calibration owns the question
the learned-density cells publish as reporting	the screen read SE ratios above 1.05: 1.052, 1.105 and 1.079 at 2, 3 and 5 nodes. The interval is conservative, with coverage 0.953 to 0.975. The row mtp-pricing-learned-density of the red-cell ledger owns it
the screened ratio learner is the quadratic logistic classifier	the study uses it at every node count, with ratio="classifier"
the cross-fold leak pair classifier_split is not registered	the pilot cannot discriminate. The fast tier shows the effect of the leak only
a continuous outcome needs Targeting(q_bounds=(lower, upper))	the revenue cell declares q_bounds=(0, 10). The fit never clips an outcome
the lmtp comparison is pending	the agreement table is empty until the reference runs register. The Python cells carry the claim until then