Conditioning on selected records

Build a recording mechanism and distinguish source independence, selected-record dependence and evidence outside model support.

16 minutes · advanced · Original Poker4Quant lesson

What you will learn

Construct conditional populations under explicit selection rules and explain why source independence need not survive selection.

Prerequisites: Bayes with a frequency table · Sequential updates · Testing event independence

A record is evidence about its selection mechanism

Generate two independent fair bits A and B. The four source worlds00,01,10and11each have probability1/4. Now record a world only when at least one bit equals1. Call this recording event S=1. This OR rule removes00but leaves the other three worlds equally likely conditional on a record.

Within records, P(A=1|S=1)=2/3. Source independence still describes the unselected generator, but it does not license treating the selected population as unchanged. A dataset containing only records comes with information about how cases became visible. This exact finite example examines conditional populations, not the precision of a sample mean or an estimated real-world selection process.

One explanation can reduce the need for another

For an OR record, learning B=1 leaves01and11, so P(A=1|S=1,B=1)=1/2. Learning B=0 instead leaves only10, so P(A=1|S=1,B=0)=1. The information about B changes inference about A inside records, even though A and B were independent before selection. B=1 can explain the record without requiring A=1.

The specified generator has two inputs pointing into their shared recording gate: A→S←B. This is a common-effect or collider structure. Here the arrows are assumptions used to define the artificial generator, not a causal graph discovered from observed association. Conditioning is not an intervention. These calculations do not establish causal direction in a real observational dataset.

The truth table and its support matter

Change the rule to XOR: record exactly one1. Records are now01and10. Without B information, P(A=1|S=1)=1/2; with B=1it becomes0, and with B=0it becomes1. The zero posterior has a positive evidence denominator: one compatible world exists but none has A=1.

Under an AND rule, only11is recorded. Both variables are then constant1inside records and are trivially independent. A collider structure does not guarantee dependence for every distribution or gate. The combination recorded AND B=0has no compatible world at all; its conditional probability is undefined rather than zero. Observing it would challenge the deterministic gate or reported observations. Do not invent a posterior by dividing0by0, resetting to1/2or silently modifying the mechanism.

Worked example

Use the independent fair-bit source with an OR recording rule. Compare P(A=1) before recording, after a record, and after a record with B=1.

In the source, two of four worlds have A=1, so P(A=1)=1/2.
The OR record population is01,10,11. Two of these three have A=1, giving2/3.
Adding B=1retains01and11. One of these two has A=1, giving1/2.
For contrast, record plus B=0retains only10and gives1. Each query restarts from the record population rather than accumulating contradictory B observations.

Different conditioning populations give different posteriors without changing the original independent source law. The known recording rule is part of the evidence model.

Hands-on: Build recording gates and compatible evidence populations

Construct OR/XOR/AND truth tables, intersect three different evidence queries per gate, normalize only possible populations and flag zero-support evidence.

Reading selection workshop…

Common mistakes

Applying source independence inside a selected record population without checking it.
Treating successive comparison queries as cumulative B=1and B=0evidence.
Confusing a zero numerator with a zero denominator.
Claiming every collider parameterization must produce dependence.
Interpreting conditioning on a record as an intervention or proof of real-world causal direction.

Keep these ideas

Specify how observations become records.
Intersect the record set with all current evidence before normalizing.
Source independence can fail after selection, but dependence is not automatic in every model.
Positive-denominator zero probabilities and impossible conditioning evidence are different cases.