{
 "nbformat": 4,
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 "metadata": {
  "kernelspec": {
   "name": "provsql-studio",
   "display_name": "ProvSQL (SQL)",
   "language": "sql"
  },
  "language_info": {
   "name": "sql"
  },
  "provsql": {
   "scheme": "semiring",
   "database": "cs6",
   "generated_from": "doc/source/user/casestudy6.rst"
  }
 },
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Case Study: City Air-Quality Sensor Network\n",
    "\n",
    "This case study demonstrates ProvSQL’s continuous-distribution surface (see [the chapter on continuous distributions](https://provsql.org/docs/user/continuous-distributions.html)) end-to-end through ProvSQL Studio (see [the Studio chapter](https://provsql.org/docs/user/studio.html)). It is the first case study driven primarily by Studio rather than `psql`: random variables benefit far more from interactive visualisation – PDFs, CDFs, mixture DAG layouts, conditional histograms, simplifier before-vs-after – than from text-mode output, and the workflow below makes the rewriter, the simplifier, the analytic and Monte-Carlo paths, and conditional inference all visible in the canvas.\n",
    "\n",
    "## The Scenario\n",
    "\n",
    "A municipal observatory operates a small air-quality sensor network. Sensors of three different vendors report a $\\mathit{PM}_{2.5}$ concentration (*fine particulate matter*, i.e. airborne particles with aerodynamic diameter at most 2.5 μm, expressed in micrograms per cubic metre) on a fixed schedule. The sensors differ in calibration and noise characteristics:\n",
    "\n",
    "- high-end units report `normal(μ, σ)` with small σ;\n",
    "- low-cost units report `uniform[μ−δ, μ+δ]` over a small window;\n",
    "- a drift-prone unit reports `exponential(λ)` while its internal hardware self-tests cycle;\n",
    "- a multi-pass aggregating unit reports `erlang(k, λ)` over the pass count.\n",
    "\n",
    "A reference station with a calibrated lab-grade instrument contributes deterministic readings.\n",
    "\n",
    "Regulatory categories partition the value axis: *Good* below 12, *Moderate* between 12.1 and 35, *Unhealthy* above 35.1 (loosely following the US EPA AQI breakpoints for PM2.5 in their pre-2024 form, simplified to three tiers). Each station has a Bernoulli probability of being in calibration on a given day. A separate batch table of *historical* readings carries the same shape so cross-batch queries via `UNION ALL` are meaningful.\n",
    "\n",
    "Your tasks:\n",
    "\n",
    "- inspect the per-row distributions and the rewriter’s effect on threshold queries;\n",
    "- compute the probability that each station’s reading exceeds an *Unhealthy* threshold, exercising the planner-hook rewrite for `WHERE reading > 35`;\n",
    "- model calibration uncertainty as a Bernoulli mixture and inspect the resulting `gate_mixture` shape;\n",
    "- aggregate per-district readings and watch the simplifier fold the mixture cascade;\n",
    "- run conditional inference (`E[reading | reading > 35]`) and see the closed-form truncated-distribution mean against the unconditional one;\n",
    "- filter on the expected value of an aggregated random variable, combine today’s and yesterday’s batches with `UNION ALL`, and compare probability methods (`'independent'` vs `'monte-carlo'` vs `'tree-decomposition'`) side by side;\n",
    "- find the *worst* reading per district with the order-statistic surface (`greatest` and the `max` aggregate);\n",
    "- read percentiles and conditional medians off [`quantile`](https://provsql.org/doxygen-sql/html/group__probability.html#ga82ca9fd9531cf491bcefde6850de8321);\n",
    "- clamp the exceedance over a threshold with `CASE` and take its expectation;\n",
    "- couple two sensors through a shared plume and measure it with [`covariance`](https://provsql.org/doxygen-sql/html/group__probability.html#ga3fda2caeb5e9ebcf8f3b9dc5c5fbb41b) / [`correlation`](https://provsql.org/doxygen-sql/html/group__probability.html#gab9108e85a4dbb83b43f606c0abcfa2e8);\n",
    "- model drift, lifetime, and episodic spikes with the Gamma, Weibull, and Pareto families, including a mixed-family probability computed by quadrature;\n",
    "- compute fleet statistics under sensor dropout with the SQL-standard statistic aggregates ([`stddev_pop`](https://provsql.org/doxygen-sql/html/group__random__variable__type.html#ga48fa5423c7cb473cafc71563fafce37e), [`corr`](https://provsql.org/doxygen-sql/html/group__random__variable__type.html#gac75785665e3f63091097c200a9ef7b3f), [`percentile_cont`](https://provsql.org/doxygen-sql/html/group__random__variable__type.html#ga42ddf88269d1d4b03c6a0b4be81f7292));\n",
    "- build a maintenance-triage headline with `CASE` over aggregates and read its exact expectation;\n",
    "- quantify the shared-plume coupling in nats with [`mutual_information`](https://provsql.org/doxygen-sql/html/group__probability.html#gac3a2e56e6753e67639fb9f43580924f4).\n",
    "\n",
    "## Setup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*The following cells set up the database with all the content this notebook requires; run them first, ideally on a fresh database.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "DROP TABLE IF EXISTS readings CASCADE;\n",
    "DROP TABLE IF EXISTS historical_readings CASCADE;\n",
    "DROP TABLE IF EXISTS stations CASCADE;\n",
    "DROP TABLE IF EXISTS calibration_status CASCADE;\n",
    "DROP TABLE IF EXISTS categories CASCADE;"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "-- Four monitoring stations across two districts.\n",
    "DROP TABLE IF EXISTS stations CASCADE;\n",
    "CREATE TABLE stations (\n",
    "  id        TEXT PRIMARY KEY,\n",
    "  name      TEXT NOT NULL,\n",
    "  district  TEXT NOT NULL\n",
    ");"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "TRUNCATE stations;\n",
    "INSERT INTO stations VALUES\n",
    "  ('s1', 'City Centre',         'centre'),\n",
    "  ('s2', 'Riverside Park',      'centre'),\n",
    "  ('s3', 'Industrial Estate',   'east'),\n",
    "  ('s4', 'Suburban Reference',  'east');"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT add_provenance('stations');"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "-- Per-station calibration probability (Bernoulli).  The high-end and\n",
    "-- multi-pass units are well calibrated; the low-cost and drift-prone\n",
    "-- ones less so; the reference station is always in-spec.\n",
    "DROP TABLE IF EXISTS calibration_status CASCADE;\n",
    "CREATE TABLE calibration_status (\n",
    "  station_id  TEXT PRIMARY KEY REFERENCES stations(id),\n",
    "  p           DOUBLE PRECISION NOT NULL\n",
    ");"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "TRUNCATE calibration_status;\n",
    "INSERT INTO calibration_status VALUES\n",
    "  ('s1', 0.95),\n",
    "  ('s2', 0.70),\n",
    "  ('s3', 0.60),\n",
    "  ('s4', 1.00);"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT add_provenance('calibration_status');"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "-- Regulatory categories (deterministic, no provenance).\n",
    "DROP TABLE IF EXISTS categories CASCADE;\n",
    "CREATE TABLE categories (\n",
    "  name  TEXT PRIMARY KEY,\n",
    "  lo    DOUBLE PRECISION NOT NULL,\n",
    "  hi    DOUBLE PRECISION NOT NULL\n",
    ");"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "TRUNCATE categories;\n",
    "INSERT INTO categories VALUES\n",
    "  ('Good',       0.0,    12.0),\n",
    "  ('Moderate',   12.1,   35.0),\n",
    "  ('Unhealthy',  35.1,   1000.0);"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "-- Today's readings.  Eight rows of pm25, one per (station, sample),\n",
    "-- exercising the normal, uniform, exponential, and Erlang noise models\n",
    "-- plus the implicit numeric cast for the reference station.\n",
    "DROP TABLE IF EXISTS readings CASCADE;\n",
    "CREATE TABLE readings (\n",
    "  id          INTEGER PRIMARY KEY,\n",
    "  station_id  TEXT NOT NULL REFERENCES stations(id),\n",
    "  ts          TIMESTAMP NOT NULL,\n",
    "  pm25        provsql.random_variable  -- NULL: sensor offline, no reading\n",
    ");"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "TRUNCATE readings;\n",
    "INSERT INTO readings (id, station_id, ts, pm25) VALUES\n",
    "  (1, 's1', '2026-05-12 08:00', provsql.normal(28.0, 2.0)),       -- high-end Gaussian\n",
    "  (2, 's2', '2026-05-12 08:00', provsql.uniform(10.0, 22.0)),     -- low-cost uniform window\n",
    "  (3, 's3', '2026-05-12 08:00', provsql.exponential(0.04)),       -- drift-prone, mean = 25\n",
    "  (4, 's4', '2026-05-12 08:00', 15.0),                            -- reference (implicit cast)\n",
    "  (5, 's1', '2026-05-12 09:00', provsql.normal(40.0, 4.0)),       -- high-end, into Unhealthy\n",
    "  (6, 's2', '2026-05-12 09:00', provsql.uniform(12.0, 24.0)),     -- low-cost, into Moderate\n",
    "  (7, 's3', '2026-05-12 09:00', provsql.erlang(3, 0.1)),          -- multi-pass Erlang, mean = 30\n",
    "  (8, 's4', '2026-05-12 09:00', 16.5);                            -- reference"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT add_provenance('readings');"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "-- Yesterday's batch, used for the UNION ALL step.  Same shape, slightly\n",
    "-- different distributions (a heatwave bumped the means).\n",
    "DROP TABLE IF EXISTS historical_readings CASCADE;\n",
    "CREATE TABLE historical_readings (\n",
    "  id          INTEGER PRIMARY KEY,\n",
    "  station_id  TEXT NOT NULL REFERENCES stations(id),\n",
    "  ts          TIMESTAMP NOT NULL,\n",
    "  pm25        provsql.random_variable NOT NULL\n",
    ");"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "TRUNCATE historical_readings;\n",
    "INSERT INTO historical_readings (id, station_id, ts, pm25) VALUES\n",
    "  (1, 's1', '2026-05-11 08:00', provsql.normal(34.0, 2.5)),\n",
    "  (2, 's2', '2026-05-11 08:00', provsql.uniform(15.0, 28.0)),\n",
    "  (3, 's3', '2026-05-11 08:00', provsql.exponential(0.03)),\n",
    "  (4, 's4', '2026-05-11 08:00', 18.0),\n",
    "  (5, 's1', '2026-05-11 09:00', provsql.normal(42.0, 3.0)),\n",
    "  (6, 's2', '2026-05-11 09:00', provsql.uniform(20.0, 35.0)),\n",
    "  (7, 's3', '2026-05-11 09:00', provsql.erlang(3, 0.08)),\n",
    "  (8, 's4', '2026-05-11 09:00', 19.5);"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT add_provenance('historical_readings');"
   ],
   "outputs": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "-- A provenance mapping so the Studio eval-strip's sr_formula and\n",
    "-- PROV-XML export can label leaves with station names rather than\n",
    "-- raw UUIDs.\n",
    "DROP TABLE IF EXISTS station_mapping;\n",
    "DROP TABLE IF EXISTS station_mapping CASCADE;\n",
    "CREATE TABLE station_mapping AS\n",
    "  SELECT s.name AS value, r.provsql AS provenance\n",
    "  FROM readings r JOIN stations s ON s.id = r.station_id;\n",
    "SELECT remove_provenance('station_mapping');\n",
    "CREATE INDEX ON station_mapping(provenance);"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The script creates the schema below and seeds the random-variable readings via the constructors documented in [the chapter on continuous distributions](https://provsql.org/docs/user/continuous-distributions.html). It is five tables:\n",
    "\n",
    "- `stations(id, name, district)` – four monitoring stations across two districts, provenance-tracked.\n",
    "- `readings(station_id, ts, pm25 random_variable)` – one `pm25` reading per station per timestamp; the `random_variable` carries the per-station noise model (normal, uniform, exponential, erlang, or a deterministic lifted from the reference station).\n",
    "- `calibration_status(station_id, p)` – Bernoulli probability that each station is in calibration on the day of interest.\n",
    "- `categories(name, lo, hi)` – three regulatory categories (*Good* / *Moderate* / *Unhealthy*) keyed by their interval bounds.\n",
    "- `historical_readings(...)` – same shape as `readings`, populated from yesterday’s batch.\n",
    "\n",
    "The schema panel lists the fixture’s six relations: the four provenance-tracked tables (`stations`, `calibration_status`, `readings`, `historical_readings`) carry the purple `prov` pill, `categories` is plain, and `station_mapping` is tagged `mapping`. The `pm25` column on `readings` and `historical_readings` is flagged with a terracotta `rv` pill: a heads-up that comparison and arithmetic operators on this column are intercepted by the planner hook and lifted into provenance gates, so a query like `pm25 > 35` produces a circuit rather than a Boolean.\n",
    "\n",
    "## Step 1: Inspect a Noisy Reading\n",
    "\n",
    "In the Studio query box:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT id, ts, pm25\n",
    "FROM readings\n",
    "WHERE station_id = 's1'\n",
    "ORDER BY ts"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The result table renders `pm25` as a clickable `random_variable` cell carrying the underlying gate UUID. Click into a row’s `pm25`: Studio switches to Circuit mode and renders the `gate_rv` leaf with the distribution-kind initial in the circle (N for a Normal, U for Uniform, Exp for Exponential, Erl for Erlang). Pick *Distribution profile* from the *Distribution* group of the eval strip and click `Run`: the panel returns headline stats: support, mean $\\mu$, standard deviation $\\sigma$, and entropy $H$, as well as an inline histogram with a PDF/CDF toggle.\n",
    "\n",
    "The histogram is backed server-side by [`rv_histogram`](https://provsql.org/doxygen-sql/html/group__circuit__introspection.html#ga4e287ff30a597e203f43c80d12098c89); pinning `provsql.monte_carlo_seed` in the Config panel (under *Provenance*) makes the shape reproducible across re-runs.\n",
    "\n",
    "## Step 2: A First Probabilistic Threshold\n",
    "\n",
    "The $\\mathit{PM}_{2.5}$ *Unhealthy* category begins above 35.1. Find the rows whose reading might cross it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT id, station_id, ts\n",
    "FROM readings\n",
    "WHERE pm25 > 35"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because `pm25` is a random variable, the comparison is not a yes/no test: it stands for the *event* “this reading exceeds 35”, and ProvSQL attaches that event to each row’s provenance.\n",
    "\n",
    "Click into a result row’s auto-added `provsql` cell. Circuit mode shows the Boolean wrapper (a `gate_times` over the row’s input token and the `gate_cmp`); the cmp’s child link reaches into the `gate_rv` from Step 1.\n",
    "\n",
    "The eval strip’s *Marginal probability* entry exposes the full method catalogue (see [the chapter on probabilities](https://provsql.org/docs/user/probabilities.html)). Pick `monte-carlo` and set `n = 10000`; the panel returns the probability with a Hoeffding confidence band. Pin `provsql.monte_carlo_seed = 42` in the Config panel and re-run: the result is now identical across runs. Toggle the seed back to `-1` and re-run to see the band shift between runs.\n",
    "\n",
    "## Step 3: The Simplifier in Action\n",
    "\n",
    "The planner hook emits the comparator as a raw `gate_cmp` regardless of what its operands look like. A *simplifier* pass then folds comparators whose answer can be decided from the operand support alone, for example, `U(10, 22) > 35` is universally false because the uniform’s upper bound is below the threshold. The fold is controlled by `provsql.simplify_on_load` (default on), which the Config panel exposes under *Provenance*.\n",
    "\n",
    "Click into row 2’s auto-added `provsql` cell from the Step 2 result (station `s2`, `pm25 ~ U(10, 22)`). With `provsql.simplify_on_load` on, the canvas shows a single `𝟘` (zero) gate: the simplifier resolved the comparator to a constant-false leaf and dropped the whole subtree. Toggle the GUC off in the Config panel and click the cell again: the canvas now shows the raw construction shape, a `gate_times` (`⊗`) over the row’s input token `ι` and a `gate_cmp` (`>`) whose children are the `U(10, 22)` leaf and the constant `35`. Both views are semantically identical; the simplified view is what the semiring evaluators and the Monte-Carlo sampler actually consume.\n",
    "\n",
    "## Step 4: Calibration via Mixtures\n",
    "\n",
    "Each station has a probability of being mis-calibrated; a mis-calibrated unit over-reports by 20% (the *reading* it records is `1.2` times the true value). The corrected estimate of the true reading is therefore `pm25` with probability `p` (the station is in spec) and `pm25 / 1.2` with probability `1 - p` (the report needs to be scaled back). Express this as a Bernoulli mixture:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT r.id, r.station_id,\n",
    "       provsql.mixture(cs.p, r.pm25, r.pm25 / 1.2) AS pm25_calibrated\n",
    "FROM readings r JOIN calibration_status cs USING (station_id)\n",
    "WHERE r.station_id = 's1'"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Click into a result row’s `pm25_calibrated` cell. Circuit mode renders the `gate_mixture` as a `Mix` node with three labelled outgoing edges (`p` / `x` / `y`) matching the SQL constructor’s argument order: `p` points to the Bernoulli mixing probability, `x` to the in-spec arm, and `y` to the correction arm.\n",
    "\n",
    "The same node-inspector panel exposes `Distribution profile` on the mixture root. Because station `s1` is in spec 95% of the time, the histogram is dominated by the `N(28, 2)` arm and the out-of-spec `N(23.33, 1.667)` contributes only a small left shoulder rather than a visually distinct second mode; the panel headline reflects this with a mixture mean slightly below 28. To see clear bimodality, re-run the query with a larger calibration error, e.g. replace `r.pm25 / 1.2` with `r.pm25 / 2.0` so the out-of-spec arm folds to `N(14, 1)`, well separated from the in-spec `N(28, 2)`; the two peaks then show up distinctly on the histogram even at the 95%/5% weighting.\n",
    "\n",
    "## Step 5: Aggregation Over Random Variables\n",
    "\n",
    "Compute average $\\mathit{PM}_{2.5}$ per district:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT s.district,\n",
    "       avg(r.pm25)  AS avg_pm25,\n",
    "       sum(r.pm25)  AS total_pm25\n",
    "FROM readings r JOIN stations s ON s.id = r.station_id\n",
    "GROUP BY s.district"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Click into a row’s `avg_pm25` cell. Circuit mode shows the [`avg`](https://provsql.org/doxygen-sql/html/group__random__variable__type.html#ga3e227bce63085af57849b6f021d3992e) lowering: a `gate_arith(DIV, num, denom)` over two `gate_arith(PLUS, …)` subtrees, each child a per-row `gate_mixture` produced by `rv_aggregate_semimod`. The right child of the outer division is the count of *included* rows under their per-row provenance: rows whose provenance is false contribute the additive identity to both numerator and denominator. Run *Distribution profile* on the root: the panel shows the per-district average as a tight distribution centred at the inclusion-weighted mean.\n",
    "\n",
    "## Step 6: Conditional Inference\n",
    "\n",
    "Re-open the filtered query from Step 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT id, station_id, ts, pm25\n",
    "FROM readings\n",
    "WHERE pm25 > 35\n",
    "  AND station_id = 's1'"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Click a result row’s `pm25` cell. The eval strip’s `Condition on` text input auto-presets to the row’s provenance UUID, and the `Conditioned by:` badge underneath the input is active. Pick *Distribution profile* and run: the histogram now shows the *truncated* shape, restricted to the tail above `35`. Pick *Moment* with `k = 1` and `raw`: the panel returns the closed-form Mills-ratio mean of the [truncated normal](https://en.wikipedia.org/wiki/Truncated_normal_distribution), exactly $\\mu + \\sigma \\cdot \\frac{\\phi(\\alpha)}{1 - \\Phi(\\alpha)}$ with $\\alpha = (35 - \\mu)/\\sigma$. Click the active badge to clear the conditioning; the panel reverts to the unconditional mean $\\mu$. Click the muted badge to restore the row provenance.\n",
    "\n",
    "The closed-form truncation table covers every family implementing truncated moments – Normal (Mills ratio), Uniform (intersected support), Exponential (memorylessness on a lower bound or finite-interval truncation), Log-normal, Weibull, Pareto (tail self-similarity), and Beta. For other shapes, the joint circuit between `pm25` and the row’s provenance is loaded with shared `gate_rv` leaves correctly coupled, and the conditional moment is estimated by rejection sampling at budget `provsql.rv_mc_samples`.\n",
    "\n",
    "## Step 7: Diagnostic Sampling\n",
    "\n",
    "For raw inspection or downstream analytics, draw samples from the conditional distribution. With the same row pinned and the *Conditioned by* badge active, pick *Sample* from the *Distribution* group; set `n = 200` and run. The result panel shows a six-value inline preview with a “show full list” expander; clicking it dumps all 200 samples.\n",
    "\n",
    "For shapes that fall outside the closed-form table the sampler falls back to rejection sampling at the `provsql.rv_mc_samples` budget; if the conditioning event is so unlikely that fewer than `n` samples land inside that budget, the panel surfaces a hint pointing at the GUC, e.g. *MC accepted 47/200. Raise* `provsql.rv_mc_samples` *in the Config panel to widen the rejection-sampling budget.* Re-running with a larger budget (set `rv_mc_samples = 50000` in the Config panel) recovers the full batch.\n",
    "\n",
    "## Step 8: Combining Batches via UNION\n",
    "\n",
    "Both batches share the same id space (rows `1` through `8`, one per `(station, timestamp)` slot), so a `UNION` (without `ALL`) over `(station_id, id)` deduplicates a slot to a single result row whose provenance combines today’s reading and yesterday’s reading via the semiring addition. With `WHERE pm25 > 35` lifted on each branch, each contributing row carries a `gate_cmp(pm25 > 35)` of its own:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "(SELECT station_id, id FROM readings            WHERE pm25 > 35)\n",
    "UNION\n",
    "(SELECT station_id, id FROM historical_readings WHERE pm25 > 35)\n",
    "ORDER BY station_id, id"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pick a result row’s `provsql` cell: Circuit mode shows a `gate_plus` (`⊕`) over the two contributing inputs (today’s row and yesterday’s row), each carrying its own `gate_cmp(pm25 > 35)` from the lifted `WHERE`. `probability_evaluate(provenance())` on the result gives the probability that *at least one* of the two days produced an Unhealthy reading for that slot. We deliberately keep the `random_variable` `pm25` column out of the `SELECT`: there is no duplicate-elimination semantics for `random_variable`, so a `UNION` over an RV column would have no well-defined meaning.\n",
    "\n",
    "## Step 9: Filtering Grouped Random Variables by Expected Value\n",
    "\n",
    "Filter the per-district aggregates from Step 5 by their expected average. Because [`avg`](https://provsql.org/doxygen-sql/html/group__random__variable__type.html#ga3e227bce63085af57849b6f021d3992e) over a `random_variable` column returns a `random_variable` (not an `agg_token`), and [`expected`](https://provsql.org/doxygen-sql/html/group__probability.html#ga7124b41224adc29ff5405d5ad6db277e) collapses it to a plain `double`, the HAVING qual is deterministic from the planner-hook’s perspective; the rewrite leaves it for PostgreSQL to evaluate natively while still adding a `delta(gate_agg)` wrapper to each surviving group’s provenance:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT s.district, avg(r.pm25) AS avg_pm25\n",
    "FROM readings r JOIN stations s ON s.id = r.station_id\n",
    "GROUP BY s.district\n",
    "HAVING expected(avg(r.pm25)) > 20"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The inner [`avg`](https://provsql.org/doxygen-sql/html/group__random__variable__type.html#ga3e227bce63085af57849b6f021d3992e) is recognised as a `random_variable` aggregate (gate_arith DIV over per-row gate_mixture children, as in Step 5); [`expected`](https://provsql.org/doxygen-sql/html/group__probability.html#ga7124b41224adc29ff5405d5ad6db277e) collapses the distribution to its mean (Monte Carlo here, since the DIV gate has no closed-form evaluator); the `> 20` is a plain comparison on a `double`, so the row survives iff its expected average exceeds the threshold. For the case-study fixture both districts pass (centre at ≈ 25.5, east at ≈ 21.6); clicking either result row’s `provsql` cell shows the `delta(gate_agg)` shape, identical to the no-HAVING aggregate from Step 5 but filtered to the surviving groups.\n",
    "\n",
    "## Step 10: Independent vs Monte Carlo\n",
    "\n",
    "For threshold queries whose contributing rows have structurally independent provenance, the `'independent'` probability method (see [the chapter on probabilities](https://provsql.org/docs/user/probabilities.html)) is *exact* and far cheaper than Monte Carlo. Compare two exact methods against `monte-carlo` on the Step 2 query:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT id,\n",
    "  probability_evaluate(provenance(), 'independent')         AS p_ind,\n",
    "  probability_evaluate(provenance(), 'monte-carlo', '10000') AS p_mc,\n",
    "  probability_evaluate(provenance(), 'tree-decomposition')  AS p_td\n",
    "FROM readings WHERE pm25 > 35"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Studio’s eval strip exposes these methods directly; running each method against the same pinned subnode shows the analytic `independent` and `tree-decomposition` returning the same value to full precision, while `monte-carlo` returns a Hoeffding-bounded estimate that tightens as `n` grows.\n",
    "\n",
    "## Step 11: The Worst Reading in a District\n",
    "\n",
    "Which district had the worst air this morning? The question asks for the distribution of the *maximum* over each district’s readings – an order statistic, not a sum. The `max` aggregate over a `random_variable` column builds exactly that:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT s.district, max(r.pm25) AS worst,\n",
    "       expected(max(r.pm25)) AS worst_mean\n",
    "FROM readings r JOIN stations s ON s.id = r.station_id\n",
    "GROUP BY s.district\n",
    "ORDER BY s.district"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The *centre* district’s worst reading averages ≈ 40.0 – its `N(40, 4)` reading dominates the maximum – and *east* comes out at ≈ 39.2, driven by the heavy right tails of `Exp(0.04)` and `erlang(3, 0.1)` even though both means sit near 25–30. Note how the extremum tells a different story from Step 5’s averages (≈ 25.5 and ≈ 21.6): the east district looks fine on average and just as alarming at the extreme. `worst_mean` itself is a plain number – `expected` has already collapsed the distribution to its mean – which is why the query also projects the unreduced aggregate: the `worst` cell is a clickable `random_variable` token whose circuit is a `gate_arith` `MAX` root over the per-row mixtures; a row absent in a world contributes `-∞`, the order-statistic identity, so it never perturbs the maximum. Because the children here are mixtures (not bare leaves), the expectation is estimated by Monte Carlo.\n",
    "\n",
    "The same-row form is `greatest` / `least` – the SQL keywords, lifted over `random_variable` arguments by the planner hook; in a query that involves no provenance-tracked relation, reach them through the schema-qualified `provsql.greatest(…)` / `provsql.least(…)` constructors instead, as Step 15 does:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT greatest(a.pm25, b.pm25) AS worse,\n",
    "       expected(greatest(a.pm25, b.pm25)) AS worse_of_two\n",
    "FROM readings a, readings b\n",
    "WHERE a.id = 1 AND b.id = 2"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Clicking the `worse` cell shows the same `gate_arith` `MAX` root, but here the children *are* independent bare leaves (`N(28, 2)` and `U(10, 22)`), so the mean is computed analytically by the layer-cake integral – ≈ 28.00003, barely above the normal’s mean, since the uniform almost never wins.\n",
    "\n",
    "## Step 12: Percentiles and Exceedance Margins\n",
    "\n",
    "[`quantile`](https://provsql.org/doxygen-sql/html/group__probability.html#ga82ca9fd9531cf491bcefde6850de8321) reads the inverse CDF off any reading – the natural regulatory question is “what level is only exceeded 5% of the time?”:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT id, station_id, quantile(pm25, 0.95) AS p95\n",
    "FROM readings\n",
    "WHERE id IN (1, 3, 5, 7)\n",
    "ORDER BY id"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each family answers through its own inverse CDF: the normals give `μ + 1.645·σ` (≈ 31.29 for `N(28, 2)`, ≈ 46.58 for `N(40, 4)`), the exponential gives `−ln(0.05)/λ ≈ 74.89`, and the Erlang – which has no elementary inverse – is bisected on its closed-form CDF to ≈ 62.96. No sampling is involved in any of these. The same readout is available interactively: with a `pm25` cell pinned, pick *Quantile* from the *Distribution* group of the eval strip, set the fraction `p`, and run.\n",
    "\n",
    "Quantiles compose with conditioning: the median of a reading *given* that it crossed the Unhealthy line is the `p = 0.5` quantile of the truncated distribution, again in closed form:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT id, quantile(pm25, 0.5, provenance()) AS median_given_high\n",
    "FROM readings\n",
    "WHERE pm25 > 35 AND station_id = 's1'\n",
    "ORDER BY id"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Row 1 (`N(28, 2)`, for which the event is a far tail) gives ≈ 35.36, just above the threshold; row 5 (`N(40, 4)`) gives ≈ 40.53. In the eval strip, the same conditioning rides on the *Conditioned by* badge from Step 6: with it active, *Quantile* returns the truncated-distribution quantile.\n",
    "\n",
    "## Step 13: Expected Excess via CASE\n",
    "\n",
    "*How far* above the threshold does a station land, on average? The excess `max(pm25 − 35, 0)` is a piecewise transform, written as a searched `CASE` (which the planner lowers into a `gate_case` guarded selection, see [the continuous distributions chapter](https://provsql.org/docs/user/continuous-distributions.html)):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT id,\n",
    "       expected(CASE WHEN pm25 > 35 THEN pm25 - 35\n",
    "                     ELSE as_random(0) END) AS expected_excess\n",
    "FROM readings\n",
    "WHERE station_id = 's1'\n",
    "ORDER BY id"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The guard and the branches read the same `pm25`, and a piecewise function of one random variable is among the `CASE` shapes whose moments come back in closed form: the expectation integrates each branch over its guard’s region, so the correlation between “the reading exceeded 35” and “by how much” is preserved exactly, with no sampling – the answers hold even under `SET provsql.rv_mc_samples = 0`. Row 1 returns ≈ 0.0001 – `N(28, 2)` essentially never crosses – while row 5 returns the closed-form partial expectation $\\sigma\\\\,\\phi(\\alpha) + (\\mu - 35)\\\\,\\Phi(-\\alpha)$ with $\\alpha = (35 - \\mu)/\\sigma$, ≈ 5.2023 for `N(40, 4)`, to full precision. Note the explicit `as_random(0)` (or equivalently `0::random_variable`) on the `ELSE` branch: `CASE` type resolution needs the branches in a single type category, so a bare numeric literal will not lift on its own.\n",
    "\n",
    "## Step 14: Correlated Sensors\n",
    "\n",
    "The two centre-district stations sit a few hundred metres apart; on a windy day both are shifted by the same dust plume. Model the plume once and add it to both readings – the shared leaf is what makes them correlated – then measure it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "WITH plume AS (SELECT provsql.normal(0, 3) AS dust)\n",
    "SELECT covariance(a.pm25 + p.dust, b.pm25 + p.dust)  AS cov_shared,\n",
    "       correlation(a.pm25 + p.dust, b.pm25 + p.dust) AS corr_shared,\n",
    "       covariance(a.pm25, b.pm25)                    AS cov_indep,\n",
    "       stddev(a.pm25)                                AS sd_a\n",
    "FROM readings a, readings b, plume p\n",
    "WHERE a.id = 1 AND b.id = 2"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The two raw readings are structurally independent – their base-RV footprints are disjoint – so `cov_indep` is *exactly* `0` (no sampling), and `sd_a` is exactly `2`. The plume-shifted pair shares the `dust` leaf: in theory `Cov = Var(dust) = 9` and `ρ = 9 / √((4+9)·(12+9)) ≈ 0.545`; the readouts land close (≈ 9.06 and ≈ 0.544 at seed 42). A shared-leaf cross-moment has no closed form, so both statistics come from a single coupled Monte-Carlo pass: each iteration draws the plume once, evaluates both shifted readings against it, and the sample covariance / correlation over those pairs is reported – which is what keeps the estimates tight around the theory values.\n",
    "\n",
    "## Step 15: Drift, Lifetime, and Spikes: More Families\n",
    "\n",
    "The vendor spec sheets bring in three more families, all available as constructors (see [the chapter on continuous distributions](https://provsql.org/docs/user/continuous-distributions.html)). The multi-pass unit’s zero drift accumulates as a Gamma; its per-batch drift check is a chi-squared test statistic:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT expected(provsql.gamma(2.5, 0.5))       AS drift_mean,\n",
    "       variance(provsql.gamma(2.5, 0.5))       AS drift_var,\n",
    "       quantile(provsql.chi_squared(4), 0.95)  AS chi2_crit"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`gamma(k = 2.5, λ = 0.5)` has mean `k/λ = 5` and variance `k/λ² = 10`, both exact; the χ²₄ critical value comes out ≈ 9.4877, bisected on the regularised-incomplete-gamma CDF.\n",
    "\n",
    "Sensor lifetimes are Weibull. A station carrying two redundant units fails over to the second when the first dies; the time to *first* failure is the minimum of the two lifetimes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT expected(provsql.least(provsql.weibull(1.5, 4.0),\n",
    "                              provsql.weibull(1.5, 4.0))) AS first_failure,\n",
    "       expected(provsql.weibull(1.5, 4.0))                AS single_lifetime"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A single `weibull(k = 1.5, λ = 4)` unit lasts `λ·Γ(1 + 1/k) ≈ 3.61` years on average; the minimum of two i.i.d. Weibulls is again Weibull (min-stability), so the first failure at `≈ 2.27` years is computed in closed form.\n",
    "\n",
    "Construction-season episodic spikes are heavy-tailed – a Pareto over a 30 μg/m³ floor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT expected(provsql.pareto(30, 2.5))       AS spike_mean,\n",
    "       quantile(provsql.pareto(30, 2.5), 0.95) AS spike_p95,\n",
    "       expected(provsql.pareto(30, 1.0))       AS undefined_mean"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `α = 2.5` spike averages `α·xₘ/(α−1) = 50` with a 95th percentile near 99.4 – the heavy tail at work. The last column shows moment honesty: `pareto(30, 1)` has *no* finite mean, and ProvSQL reports `Infinity` rather than estimating a divergent integral.\n",
    "\n",
    "Finally, does a spike beat the noisy high reading? A Pareto vs Normal comparison has no registered closed form, so ProvSQL integrates the pair by quadrature – exact-grade, no sampling:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT probability(x > y) AS p_spike_beats_normal\n",
    "FROM (SELECT provsql.pareto(30, 2.5) AS x,\n",
    "             provsql.normal(45, 5)   AS y) t"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which returns ≈ 0.384: even though the spike’s *mean* (50) is above the normal’s (45), the spike spends most of its mass near its 30 floor and only wins through its tail.\n",
    "\n",
    "## Step 16: Fleet Statistics Under Sensor Dropout\n",
    "\n",
    "Every reading so far contributed unconditionally. Today’s maintenance log says otherwise: a station out of calibration silently drops out of the feed. Pin each reading’s provenance probability to its station’s calibration probability, so a row is *present* only when its station is in calibration:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT set_prob(provenance(),\n",
    "                (SELECT p FROM calibration_status cs\n",
    "                  WHERE cs.station_id = r.station_id))\n",
    "FROM readings r"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The calibration probability is fetched through a scalar subquery so that `provenance()` stays the `readings` row’s own input token: joining the two tracked tables instead would make it the *joint* row provenance (a `times` gate), which `set_prob` cannot pin. ProvSQL warns that the subquery’s data is read outside provenance tracking (“treated as certain”) – exactly the intent here: `p` parameterises the presence model, it is not uncertain data itself.\n",
    "\n",
    "The SQL-standard statistic aggregates ([`stddev_pop`](https://provsql.org/doxygen-sql/html/group__random__variable__type.html#ga48fa5423c7cb473cafc71563fafce37e) / [`stddev_samp`](https://provsql.org/doxygen-sql/html/group__random__variable__type.html#gad15d1502387e11fbf7f4e508c89772bb), [`covar_pop`](https://provsql.org/doxygen-sql/html/group__random__variable__type.html#ga4feb1f67e46285e9f30df43fd4ef6e80) / [`covar_samp`](https://provsql.org/doxygen-sql/html/group__random__variable__type.html#gac29ea8812e647c91cbc957d7b3f04665) / [`corr`](https://provsql.org/doxygen-sql/html/group__random__variable__type.html#gac75785665e3f63091097c200a9ef7b3f), and the ordered-set [`percentile_cont`](https://provsql.org/doxygen-sql/html/group__random__variable__type.html#ga42ddf88269d1d4b03c6a0b4be81f7292)) are lifted to `random_variable` rows with exactly these semantics: a row absent in a world drops out of every sum, the count, and the percentile’s member set. Start where the answer is checkable by hand – the reference station is always in calibration (`p = 1`) and its readings are lab-grade constants `15.0` and `16.5`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT expected(stddev_pop(pm25))     AS sd_ref,\n",
    "       expected(corr(pm25, pm25 * 2)) AS corr_ref\n",
    "FROM readings\n",
    "WHERE station_id = 's4'"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Both statistics are Dirac – every Monte-Carlo draw is the same world and the same constants – so the answers are exact: `sd_ref = 0.75` (the population stddev of `{15, 16.5}`) and `corr_ref = 1` (a reading is perfectly correlated with its own rescaling, drawn once and observed by both argument positions).\n",
    "\n",
    "Now the probabilistic fleet: the *median* reading per district, where dropout changes the member set itself – when Riverside Park is out of calibration (30% of days), both of its rows leave the group:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT s.district,\n",
    "       expected(percentile_cont(0.5)\n",
    "                WITHIN GROUP (ORDER BY r.pm25)) AS median_pm25\n",
    "FROM readings r JOIN stations s ON s.id = r.station_id\n",
    "GROUP BY s.district"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `percentile_cont` gate carries every row’s presence indicator alongside its value (pin the result node and look for the `p50` circle in the canvas): per draw, the sampler keeps the values whose station is in calibration, sorts them, and interpolates. Under dropout the centre district’s expected median lands around 26 – pulled between the two Normal City-Centre readings and the two lower Riverside uniforms that are only present 70% of the time.\n",
    "\n",
    "## Step 17: A Maintenance-Triage Headline via CASE over Aggregates\n",
    "\n",
    "Which district needs the maintenance crew first? The desk wants a single *headline* number per district: if even the weakest station’s calibration confidence clears 0.8, report the district’s total confidence weight; otherwise flag the weakest station (the crew’s target). That is a `CASE` whose guards **and** branches are aggregates – the aggregate-carrier `CASE` (see [the aggregation chapter](https://provsql.org/docs/user/aggregation.html)), lowered to an `agg_case` guarded selection. (The exact machinery covers `sum` / `count` / `min` / `max` guards and branches; an `avg` branch inside a `CASE` takes the Monte-Carlo path, its exact arm being unconditional-only.) First make the maintenance log itself uncertain (each record is confirmed with a probability) and the station registry certain:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "DO $$ BEGIN\n",
    "  PERFORM set_prob(provenance(), 1.0) FROM stations;\n",
    "  PERFORM set_prob(provenance(),\n",
    "                   CASE station_id WHEN 's1' THEN 1.0\n",
    "                                   WHEN 's2' THEN 0.8\n",
    "                                   WHEN 's3' THEN 0.7\n",
    "                                   ELSE 1.0 END)\n",
    "    FROM calibration_status;\n",
    "END $$"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One record per district is kept certain, so each district’s group exists in every world (an all-absent group would produce no headline row at all)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT s.district,\n",
    "       CASE WHEN min(cs.p) > 0.8 THEN sum(cs.p)\n",
    "            ELSE min(cs.p) END AS confidence_headline\n",
    "FROM calibration_status cs JOIN stations s ON s.id = cs.station_id\n",
    "GROUP BY s.district"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The result cells display as `0.7 (*)` and `0.6 (*)` – the `CASE` evaluated on the actual data (with every record confirmed, both districts have a weak link below the bar, so each headlines its worst station), with the `(*)` marking a probabilistic aggregate whose value is world-dependent. The headline’s distribution is evaluated *exactly* – possible-worlds decomposition over the first-match regions, no Monte Carlo, correct even under `SET provsql.rv_mc_samples = 0`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "SELECT s.district,\n",
    "       expected(CASE WHEN min(cs.p) > 0.8 THEN sum(cs.p)\n",
    "                     ELSE min(cs.p) END) AS expected_headline\n",
    "FROM calibration_status cs JOIN stations s ON s.id = cs.station_id\n",
    "GROUP BY s.district"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the centre district there are exactly two worlds: with both records confirmed (probability `0.8`) the weakest link `0.70` misses the bar and is the headline; with only City Centre’s record confirmed (`0.2`) the sole confidence `0.95` clears it and the total `0.95` is reported. So $E = 0.8 \\cdot 0.70 + 0.2 \\cdot 0.95 = 0.75$ exactly – and that is the value the query returns. The east district works out to $0.7 \\cdot 0.60 + 0.3 \\cdot 1.00 = 0.72$.\n",
    "\n",
    "## Step 18: Shared Information in Nats\n",
    "\n",
    "Step 14 measured the plume coupling with [`correlation`](https://provsql.org/doxygen-sql/html/group__probability.html#gab9108e85a4dbb83b43f606c0abcfa2e8); [`mutual_information`](https://provsql.org/doxygen-sql/html/group__probability.html#gac3a2e56e6753e67639fb9f43580924f4) is its information-theoretic counterpart – symmetric, in nats, and zero *exactly* when the two variables share no randomness:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "WITH plume AS (SELECT provsql.normal(0, 3) AS dust)\n",
    "SELECT mutual_information(a.pm25, b.pm25)  AS mi_indep,\n",
    "       mutual_information(a.pm25 + p.dust,\n",
    "                          b.pm25 + p.dust) AS mi_shared\n",
    "FROM readings a, readings b, plume p\n",
    "WHERE a.id = 1 AND b.id = 2"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`mi_indep` is **exactly 0** with no sampling at all: the two raw readings have disjoint stochastic-leaf footprints, the same structural-independence test the moment evaluators use. The plume-shifted pair shares the `dust` leaf, and the readout estimates their mutual information by a 2-D histogram over *coupled* joint draws (each iteration draws the shared plume once and both sums observe it) – about `0.21` nats at seed 42. For a jointly-Gaussian pair with the Step 14 correlation $\\rho \\approx 0.545$ the closed form would give $-\\tfrac12\\ln(1-\\rho^2) \\approx 0.176$; the estimate lands in that neighbourhood, a little above: the pair is not jointly Gaussian (Riverside’s noise is uniform), and the plug-in histogram estimator carries a small upward bias.\n",
    "\n",
    "## Step 19: An Offline Sensor Reports NULL\n",
    "\n",
    "Sensors go offline. When Riverside Park misses its 10:00 sample, the natural encoding is a row whose `pm25` is **SQL NULL** – there is no reading, not even an uncertain one:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "INSERT INTO readings (id, station_id, ts, pm25)\n",
    "VALUES (9, 's2', '2026-05-12 10:00', NULL);"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Three things follow, each matching plain SQL:\n",
    "\n",
    "- **Threshold events are unknown, never true.** `WHERE pm25 > 35` over the offline row is *unknown* under SQL’s three-valued logic in every possible world, so the lifted comparison annotates the row with the circuit’s zero – [`probability_evaluate`](https://provsql.org/doxygen-sql/html/group__probability.html#gabffa40fef0e37a75d39d4c39c4a1ec0f) returns exactly 0, with no sampling. (A NULL reading is not “certainly exceeding”, and not “possibly exceeding” either: it is no evidence at all.)\n",
    "- **Null tests are ordinary and deterministic.** `WHERE pm25 IS NULL` finds the offline row like any other predicate; PostgreSQL evaluates it and provenance simply carries the row’s token through.\n",
    "- **Aggregates skip the offline row.** `expected(avg(pm25))` over Riverside Park is the same with or without row 9: the NULL reading contributes to neither the sum nor the count. (Internally `avg` uses a per-row presence indicator that is NULL exactly when the value is, so the offline row drops out of the denominator too.)\n",
    "\n",
    "Delete the placeholder row to return to the running dataset:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "provsql": {}
   },
   "source": [
    "DELETE FROM readings WHERE id = 9;"
   ],
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The general rules – which predicates treat NULLs as unknown and what that means for the circuits – are in [the NULL semantics chapter](https://provsql.org/docs/user/nulls.html).\n",
    "\n",
    "See [the chapter on continuous distributions](https://provsql.org/docs/user/continuous-distributions.html) for the full surface and [the Studio chapter](https://provsql.org/docs/user/studio.html) for the Studio reference."
   ]
  }
 ]
}
