Bayesian Causal Analysis — Immunological Conditions, Allergies & Food Intolerances

Pearl Structural Causal Model · Backdoor adjustment · Cross-correlation eigenvalue correction · Dose-response saturation · Monte Carlo (n=10,000) · E-value · PNS/PN/PS · PAF · Pareto minimum-effective-set
Judea Pearl SCM v3.2 Do-calculus + Backdoor Criterion 7 categories · 42 conditions · 67 interventions Educational / Research — Not Medical Advice
⚠ Educational tool only. Hazard ratios are pooled from published meta-analyses and large RCTs; effects are population-averaged and may not transfer to an individual. Discuss any intervention with a licensed allergist/immunologist or rheumatologist. Anaphylaxis, severe immunodeficiency, biologic-eligible disease, and pregnancy require specialist supervision.

① Condition & patient profile

Toggle interventions below. Each one is rated for the currently-selected condition. Sliders adjust dose/intensity; the displayed hazard ratio follows the published dose-response curve with diminishing returns (saturation modelled as HR(d) = 1 − (1−HRmax)·d/(d+ED₅₀)).
Risk-reduction target
Greedy Pareto search will auto-select the minimum set of interventions to reach this reduction.
50% target HR ≤ 0.500
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Force-directed causal DAG. Blue nodes=active interventions; purple=mediators; amber=confounders on backdoor paths; red=outcome. Solid edges=direct causal effects; dashed amber=backdoor paths requiring adjustment (Pearl backdoor criterion).
Intervention
Mediator
Confounder
Outcome

Adjusted combined hazard ratio (do-calculus)

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No interventions selected
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Relative risk reduction
Naive product HR
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After ρ-correction
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95% CI (lower)
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95% CI (upper)
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E-value (combined)
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PNS
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PN (necessity)
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PS (sufficiency)
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PAF (population-attrib.)
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NNT (1-year, scaled)
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Per-intervention contribution

InterventionHRWtΔ%

Cumulative HR as interventions are added (greedy ordering by marginal effect)

Pairwise correlation between intervention effects (shared mechanisms, overlapping pathways, confounded exposures). The eigenvalue-corrected adjustment factor avoids double-counting. Cells: white = ρ<0.10 (independent); amber = 0.10–0.40 (moderate overlap); red = >0.40 (strong overlap, large correction).

Eigenvalue diagnostics

λmax (dominant eigenvalue)
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Condition number κ
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Effective # independent IVs
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Mean off-diagonal ρ̄
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When κ>30 or neff drops below ~60% of n, the naive product of HRs materially overstates the joint effect. The system below applies a Cholesky-decomposed correction.

Per-intervention dose-response curves. Effects saturate following HR(d) = 1 − (1 − HRmax) · d / (d + ED₅₀) (Michaelis-Menten form). Vertical line = current dose setting; horizontal dashed = HRmax asymptote.
Pareto frontier: minimum-cardinality intervention sets achieving each risk-reduction target. Algorithm: greedy marginal-benefit selection with cross-correlation penalty, then 2-opt local search.
Threshold — target risk reduction
Drag to any reduction level; the minimum effective set that reaches it is computed live. Use “Apply this set” to load those interventions.
50% target HR ≤ 0.500
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Pareto frontier — risk reduction vs # of interventions

Minimum effective sets

TargetnAdj HRMembers
One-at-a-time tornado: removing each active intervention and recomputing the adjusted HR. Wider bars = greater dependence of the joint estimate on that single intervention. E-value column quantifies how strong an unmeasured confounder would have to be to nullify the effect.
Monte Carlo propagation: each intervention HR sampled from log-normal centred on its point estimate with SD derived from the published confidence interval. 10,000 iterations. Cross-correlations imposed via Cholesky-factored covariance.

Posterior distribution of adjusted HR

Quantiles

2.5%ile
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25%ile
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Median
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75%ile
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97.5%ile
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P(HR < 0.5)
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Scientific honesty: every result above is subject to limits. Read these counter-arguments before acting.