scripts/exp03_ci_coverage.py

#!/usr/bin/env python3
"""exp03 -- single-run confidence interval for the stopping time (Prop. 4.5).
Per specs/exp03_ci_coverage.md. Writes results/exp03.json.

    python3 scripts/exp03_ci_coverage.py [--toy]

Reuses exp02's shared-stream path machinery (core.klinf / core.par): one
replicate = one path shared across every (alpha, beta_kind) pair, since the
crossing statistic n*KL_inf(q_hat_n, m_o) does not depend on alpha. At each
step the dual solve already yields ell(lam*_n, X_1..n); from that single array
we get, with no extra sampling cost, both the "direct centred" sigma_hat^2
(eq. 22's own definition) and an independently-summed mean(ell)/mean(ell^2)
form -- comparing the two is the C3a/C3b identity cross-check the spec asks
for, not two different estimators.
"""
import argparse
import functools
import json
import math
import os
import sys
import time

import numpy as np
from scipy.stats import norm

sys.path.insert(0, os.path.dirname(os.path.dirname(os.path.abspath(__file__))))

from core import par
from core.klinf import klinf_emp, ell
from core.popquant import pop

RESULTS_DIR = os.path.join(os.path.dirname(os.path.dirname(os.path.abspath(__file__))), "results")
OUT_PATH = os.path.join(RESULTS_DIR, "exp03.json")

SEED_BASE = 20260722  # independent of exp02's 20260721 stream
N_MAX = 200000
BETA_KINDS = ["theory", "practical"]  # eq. 7 / eq. 8
GAMMAS = [0.10, 0.05, 0.01]

ALPHA_GRID_FULL = [1e-2, 1e-3, 1e-4, 1e-6, 1e-8, 1e-10, 1e-12]
ALPHA_GRID_TOY = [1e-2, 1e-4, 1e-6]

R_FULL = 5000
R_TOY = 200

CONFIGS_META = [
    dict(id="bern0.6_mo0.2", dist="bernoulli(0.6)", dist_kind="bernoulli",
         dist_params=(0.6,), popdist=("bernoulli", 0.6), m_o=0.2),
    dict(id="beta3_2_mo0.7", dist="beta(3,2)", dist_kind="beta",
         dist_params=(3, 2), popdist=("beta", 3, 2), m_o=0.7),
    dict(id="bern0.6_mo0.7", dist="bernoulli(0.6)", dist_kind="bernoulli",
         dist_params=(0.6,), popdist=("bernoulli", 0.6), m_o=0.7),
]

S4_CONFIG_ID = "bern0.6_mo0.2"
S4_BETA_KIND = "practical"
S4_GAMMA = 0.05
S4_ALPHAS = [1e-4, 1e-8]

Z95 = 1.959963984540054  # Wilson interval z for 95% CI on coverage


def log(msg):
    print(f"[exp03] {msg}", file=sys.stderr, flush=True)


def n_cores():
    return max(1, min(int(os.environ.get("JOB_CORES", "4")), os.cpu_count() or 1))


# --------------------------------------------------------------- sampling
def _make_sampler(dist_kind, dist_params):
    if dist_kind == "bernoulli":
        p, = dist_params
        return lambda rng, k: (rng.uniform(size=k) < p).astype(float)
    if dist_kind == "beta":
        a, b = dist_params
        return lambda rng, k: rng.beta(a, b, size=k)
    raise ValueError(dist_kind)


# ------------------------------------------------------------- beta(n,alpha)
def beta_theory(n, alpha):
    return 1.0 + math.log(2.0 * (1.0 + n) / alpha)


def beta_practical(n, alpha):
    return math.log(1.0 / alpha)


_BETA_FNS = {"theory": beta_theory, "practical": beta_practical}


# ------------------------------------------------------ shared-stream sim
def _run_path_multi(rng, sampler, m_o, alphas, beta_kinds, n_max, n_chunk=256):
    """One path shared by every (alpha, beta_kind) pair. At each crossing,
    records tau, klinf_tau (=mean(ell) by construction) and sigma2_tau (the
    direct-centred eq. 22 variance). Also tracks, across all crossings on
    this path, the worst-case |mean(ell) - klinf_hat| (C3a identity) and
    |direct-form - eq22-form sigma_hat^2| (C3b form cross-check), plus the
    path's min sigma_hat^2."""
    x = np.empty(n_max, dtype=float)
    n = 0
    lam_prev = None
    pending = set((a, bk) for a in alphas for bk in beta_kinds)
    out = {}
    max_id1 = 0.0
    max_id2 = 0.0
    min_sigma2 = math.inf

    while n < n_max and pending:
        take = min(n_chunk, n_max - n)
        x[n:n + take] = sampler(rng, take)
        for _ in range(take):
            n += 1
            xk = x[:n]
            val, lam, case = klinf_emp(xk, m_o, lam_hint=lam_prev)
            lam_prev = lam if case == "interior" else None

            stat = n * val
            crossed_now = [(a, bk) for (a, bk) in pending if stat >= _BETA_FNS[bk](n, a)]

            if crossed_now:
                ell_arr = ell(lam, xk, m_o)
                mean_ell = float(np.mean(ell_arr))
                mean_ell2 = float(np.mean(ell_arr * ell_arr))
                sigma2_direct = float(np.mean((ell_arr - val) ** 2))
                sigma2_eq22 = mean_ell2 - mean_ell * mean_ell

                id1 = abs(mean_ell - val)
                id2 = abs(sigma2_direct - sigma2_eq22)
                if id1 > max_id1:
                    max_id1 = id1
                if id2 > max_id2:
                    max_id2 = id2
                if sigma2_direct < min_sigma2:
                    min_sigma2 = sigma2_direct

                for (a, bk) in crossed_now:
                    out[(a, bk)] = dict(tau=n, censored=False,
                                         klinf_tau=val, sigma2_tau=sigma2_direct)
                    pending.discard((a, bk))

            if not pending:
                break

    for (a, bk) in pending:
        out[(a, bk)] = dict(tau=n, censored=True, klinf_tau=None, sigma2_tau=None)

    return dict(results=out, max_id1=max_id1, max_id2=max_id2, min_sigma2=min_sigma2)


def _replicate_task(rng, r, dist_kind, dist_params, m_o, alphas, beta_kinds, n_max):
    sampler = _make_sampler(dist_kind, dist_params)
    return _run_path_multi(rng, sampler, m_o, alphas, beta_kinds, n_max)


# ------------------------------------------------------------------ helpers
def wilson_ci(phat, n, z=Z95):
    denom = 1.0 + z * z / n
    center = (phat + z * z / (2.0 * n)) / denom
    half = (z * math.sqrt(phat * (1.0 - phat) / n + z * z / (4.0 * n * n))) / denom
    return max(0.0, center - half), min(1.0, center + half)


def loglog_fit(xs, ys):
    """Fit log(y) = a - rate*log(x) by OLS; return (rate, stderr(rate))."""
    xs = np.asarray(xs, dtype=float)
    ys = np.asarray(ys, dtype=float)
    mask = (xs > 0) & (ys > 0)
    x = np.log(xs[mask])
    y = np.log(ys[mask])
    n = len(x)
    A = np.vstack([np.ones_like(x), x]).T
    coef, _, _, _ = np.linalg.lstsq(A, y, rcond=None)
    _, slope = coef
    rate = -float(slope)
    if n > 2:
        yhat = A @ coef
        resid = y - yhat
        dof = n - 2
        sigma2 = float(np.sum(resid ** 2) / dof) if dof > 0 else 0.0
        xtx_inv = np.linalg.inv(A.T @ A)
        se = float(np.sqrt(max(sigma2 * xtx_inv[1, 1], 0.0)))
    else:
        se = 0.0
    return rate, se


def main():
    ap = argparse.ArgumentParser()
    ap.add_argument("--toy", action="store_true")
    args = ap.parse_args()
    toy = args.toy

    n_jobs = n_cores()
    R = R_TOY if toy else R_FULL
    alpha_grid = ALPHA_GRID_TOY if toy else ALPHA_GRID_FULL

    log(f"starting exp03 (toy={toy}) n_jobs={n_jobs} R={R} alpha_grid={alpha_grid} "
        f"n_max={N_MAX} gammas={GAMMAS}")
    t0 = time.time()

    population = []
    pop_by_id = {}
    for cm in CONFIGS_META:
        p = pop(cm["popdist"], cm["m_o"])
        entry = dict(config=cm["id"], dist=cm["dist"], m_o=cm["m_o"],
                     klinf=p["klinf"], inv_klinf=1.0 / p["klinf"],
                     sigma2=p["sigma2"], sigma_bd2=p["sigma_bd2"],
                     lam_star=p["lam_star"], case=p["case"])
        population.append(entry)
        pop_by_id[cm["id"]] = entry
        log(f"  population[{cm['id']}]: case={p['case']} lam*={p['lam_star']:.6f} "
            f"klinf={p['klinf']:.6f} inv_klinf={entry['inv_klinf']:.4f} "
            f"sigma2={p['sigma2']:.6f} sigma_bd2={p['sigma_bd2']:.6f}")

    s1_coverage, s2_vhat, s2_fit = [], [], []
    s1_by_config = {cm["id"]: [] for cm in CONFIGS_META}
    s2_by_config = {cm["id"]: [] for cm in CONFIGS_META}
    width_slopes = []
    global_max_id1 = 0.0
    global_max_id2 = 0.0
    global_min_sigma2 = math.inf
    raw_by_config = {}

    cfg_id = 0
    for cm in CONFIGS_META:
        cfg_id += 1
        pe = pop_by_id[cm["id"]]
        klinf_pop, inv_klinf_pop, sigma_bd2_pop = pe["klinf"], pe["inv_klinf"], pe["sigma_bd2"]

        log(f"=== {cm['id']} (dist={cm['dist']}, m_o={cm['m_o']}) R={R} alphas={alpha_grid} ===")
        t_cfg = time.time()
        task = functools.partial(_replicate_task, dist_kind=cm["dist_kind"],
                                  dist_params=cm["dist_params"], m_o=cm["m_o"],
                                  alphas=alpha_grid, beta_kinds=BETA_KINDS, n_max=N_MAX)
        raw = par.run_replicates(task, cfg_id, R, n_jobs=n_jobs, base_seed=SEED_BASE)
        raw_by_config[cm["id"]] = raw
        log(f"  {cm['id']} done in {time.time() - t_cfg:.1f}s")

        for rep in raw:
            global_max_id1 = max(global_max_id1, rep["max_id1"])
            global_max_id2 = max(global_max_id2, rep["max_id2"])
            if rep["min_sigma2"] < global_min_sigma2:
                global_min_sigma2 = rep["min_sigma2"]

        for bk in BETA_KINDS:
            width_by_gamma_alpha = {g: {} for g in GAMMAS}
            for a in alpha_grid:
                b_alpha = math.log(1.0 / a)
                key = (a, bk)
                taus, klinfs, sigma2s, n_cens = [], [], [], 0
                for rep in raw:
                    r = rep["results"][key]
                    if r["censored"]:
                        n_cens += 1
                        continue
                    taus.append(r["tau"])
                    klinfs.append(r["klinf_tau"])
                    sigma2s.append(r["sigma2_tau"])
                taus = np.array(taus, dtype=float)
                klinfs = np.array(klinfs, dtype=float)
                sigma2s = np.array(sigma2s, dtype=float)
                vhat = sigma2s / klinfs ** 3
                n_valid = len(taus)

                vhat_mean = float(np.mean(vhat))
                vhat_median = float(np.median(vhat))
                vhat_sd = float(np.std(vhat, ddof=1))
                row2 = dict(config=cm["id"], beta_kind=bk, alpha=a,
                            vhat_mean=vhat_mean, vhat_median=vhat_median, vhat_sd=vhat_sd,
                            sigma_bd2=sigma_bd2_pop,
                            vhat_mean_over_sigma_bd2=float(vhat_mean / sigma_bd2_pop),
                            sigmahat2_over_sigma2=float(np.mean(sigma2s) / pe["sigma2"]),
                            klinf_hat_over_klinf=float(np.mean(klinfs) / klinf_pop))
                s2_vhat.append(row2)
                s2_by_config[cm["id"]].append(row2)

                for g in GAMMAS:
                    z = float(norm.ppf(1.0 - g / 2.0))
                    center = taus / b_alpha
                    half = z * np.sqrt(vhat / b_alpha)
                    lo, hi = center - half, center + half
                    widths = hi - lo
                    covered = (lo <= inv_klinf_pop) & (inv_klinf_pop <= hi)
                    miss_below = int(np.sum(inv_klinf_pop < lo))
                    miss_above = int(np.sum(inv_klinf_pop > hi))
                    n_covered = int(np.sum(covered)) if n_valid else 0
                    coverage = n_covered / R
                    wilson_lo, wilson_hi = wilson_ci(coverage, R)
                    nominal = 1.0 - g

                    row1 = dict(config=cm["id"], beta_kind=bk, alpha=a, b_alpha=b_alpha,
                                gamma=g, R=R, coverage=coverage,
                                wilson_lo=wilson_lo, wilson_hi=wilson_hi, nominal=nominal,
                                coverage_error=float(coverage - nominal),
                                miss_below=miss_below, miss_above=miss_above,
                                width_median=float(np.median(widths)) if n_valid else 0.0,
                                width_iqr=float(np.percentile(widths, 75) - np.percentile(widths, 25))
                                if n_valid else 0.0,
                                n_censored=n_cens)
                    s1_coverage.append(row1)
                    s1_by_config[cm["id"]].append(row1)
                    width_by_gamma_alpha[g][b_alpha] = row1["width_median"]

            log(f"  {cm['id']}/{bk} coverage sweep done")
            b_vals = [math.log(1.0 / a) for a in alpha_grid]
            vhat_sd_vals = [r["vhat_sd"] for r in s2_by_config[cm["id"]] if r["beta_kind"] == bk]
            p_rate, p_se = loglog_fit(b_vals, vhat_sd_vals)
            s2_fit.append(dict(config=cm["id"], beta_kind=bk,
                                p_vhat_sd=p_rate, p_vhat_sd_stderr=p_se, n_points=len(b_vals)))
            log(f"  {cm['id']}/{bk}: p_vhat_sd={p_rate:.3f}+/-{p_se:.3f}")

            for g in GAMMAS:
                bs = b_vals
                ws = [width_by_gamma_alpha[g][b] for b in bs]
                rate, se = loglog_fit(bs, ws)
                width_slopes.append(dict(config=cm["id"], beta_kind=bk, gamma=g,
                                          slope=-rate, slope_stderr=se))

    # ------------------------------------------------------------------ S3
    s3_identity = dict(
        max_mean_ell_minus_klinf=global_max_id1,
        max_sigmahat2_form_diff=global_max_id2,
        min_sigmahat2=global_min_sigma2,
        width_slopes=width_slopes,
    )
    log(f"S3 identity: max_id1={global_max_id1:.2e} max_id2={global_max_id2:.2e} "
        f"min_sigma2={global_min_sigma2:.4f}")

    # ------------------------------------------------------------------ S4
    s4_illustration = []
    raw0 = raw_by_config[S4_CONFIG_ID]
    s4_alphas = S4_ALPHAS if not toy else alpha_grid[:2]
    for a in s4_alphas:
        b_alpha = math.log(1.0 / a)
        key = (a, S4_BETA_KIND)
        taus, klinfs, sigma2s = [], [], []
        for rep in raw0:
            r = rep["results"][key]
            if r["censored"]:
                continue
            taus.append(r["tau"])
            klinfs.append(r["klinf_tau"])
            sigma2s.append(r["sigma2_tau"])
        taus = np.array(taus, dtype=float)
        klinfs = np.array(klinfs, dtype=float)
        sigma2s = np.array(sigma2s, dtype=float)
        vhat = sigma2s / klinfs ** 3
        z = float(norm.ppf(1.0 - S4_GAMMA / 2.0))
        center = taus / b_alpha
        half = z * np.sqrt(vhat / b_alpha)
        lo, hi = (center - half) * b_alpha, (center + half) * b_alpha

        def qpair(lo_arr, hi_arr, q):
            return [float(np.percentile(lo_arr, q)), float(np.percentile(hi_arr, q))]

        s4_illustration.append(dict(
            alpha=a, gamma=S4_GAMMA,
            single_run_interval_median=qpair(lo, hi, 50),
            single_run_interval_q25=qpair(lo, hi, 25),
            single_run_interval_q75=qpair(lo, hi, 75),
            empirical_tau_interval=[float(np.percentile(taus, 2.5)),
                                     float(np.percentile(taus, 97.5))],
        ))
        log(f"  S4 alpha={a:g}: median_interval={s4_illustration[-1]['single_run_interval_median']} "
            f"empirical_tau_interval={s4_illustration[-1]['empirical_tau_interval']}")

    # ------------------------------------------------- gate-facing adapter
    # gates.py's exp03 gate expects a flat per-config "configs" list (with a
    # "pop"/"coverage"/"consistency" view of the same numbers) plus a
    # top-level "self_consistency" scalar; this is a reshaped view of the
    # canonical sections above, not a second computation.
    configs_adapter = []
    for cm in CONFIGS_META:
        pe = pop_by_id[cm["id"]]
        consistency = [dict(alpha=r["alpha"], beta_kind=r["beta_kind"],
                             mean_vhat_over_sigma_bd2=r["vhat_mean_over_sigma_bd2"],
                             sd_vhat=r["vhat_sd"])
                       for r in s2_by_config[cm["id"]]]
        configs_adapter.append(dict(
            dist=cm["dist"], m_o=cm["m_o"],
            pop=dict(lam_star=pe["lam_star"], klinf=pe["klinf"], sigma2=pe["sigma2"],
                      sigma_bd2=pe["sigma_bd2"], case=pe["case"]),
            coverage=s1_by_config[cm["id"]],
            consistency=consistency,
        ))
    self_consistency = dict(max_abs_err=max(global_max_id1, global_max_id2))

    elapsed = time.time() - t0
    out = dict(
        meta=dict(spec="exp03", seed_base=SEED_BASE, n_paths=R, n_max=N_MAX,
                   gammas=GAMMAS, cores_used=n_jobs, wall_seconds=elapsed),
        population=population,
        s1_coverage=s1_coverage,
        s2_vhat=s2_vhat,
        s2_fit=s2_fit,
        s3_identity=s3_identity,
        s4_illustration=s4_illustration,
        configs=configs_adapter,
        self_consistency=self_consistency,
    )

    os.makedirs(RESULTS_DIR, exist_ok=True)
    with open(OUT_PATH, "w") as f:
        json.dump(out, f, indent=2)

    print(f"exp03 complete in {elapsed:.1f}s -> {OUT_PATH}")
    print(f"  cores_used={n_jobs} R={R} n_configs={len(population)} alphas={alpha_grid}")
    print(f"  s1_coverage rows={len(s1_coverage)} s2_vhat rows={len(s2_vhat)} "
          f"s2_fit rows={len(s2_fit)}")
    print(f"  s3_identity: max|mean_ell-klinf|={global_max_id1:.2e} "
          f"max|sigma2_form_diff|={global_max_id2:.2e} min_sigma2={global_min_sigma2:.4f}")
    for row in s4_illustration:
        print(f"  s4[alpha={row['alpha']:g}]: median_interval(tau-scale)="
              f"{row['single_run_interval_median']} "
              f"empirical_tau_interval={row['empirical_tau_interval']}")


if __name__ == "__main__":
    main()