Why Condition on Discovery?
A locus enters a discovery report because its estimated effect crossed a threshold. Consequently, its observed estimate reflects both the underlying effect and the selection rule. A Bayesian analysis can represent this process by modeling the estimate conditional on having passed the rule, then combining that conditional likelihood with a prior distribution for effects before selection. The resulting posterior is not the same as ordinary Bayesian shrinkage: the selection probability varies with the true effect and changes the relative likelihood assigned to candidate effects.
The adjustment is only as credible as its model. It requires a specified selection event, a sampling distribution for the estimate, and an appropriate prior. It is also important to say what “conditioning on selection” means. A posterior built from a conditional likelihood and a preselection prior is a particular selection-adjusted analysis; it should not be confused with every possible coherent Bayesian treatment of the discovery process.
Model and Selection-Conditioned Posterior
Under the normal sampling model, the probability of selection is
where \(\Phi\) is the standard normal cumulative distribution function. The first term is the probability that \(Y/s<-c\); the second is the probability that \(Y/s>c\). The posterior density is
For a discrete prior, replace the integral by a sum. The division by \(q(\beta)\) removes the part of the sampling likelihood attributable to how likely that effect was to be selected. Since small effects are less likely to pass a stringent threshold, conditioning can give them a larger likelihood relative to the unconditioned likelihood. It does not guarantee that the posterior mean will move in a particular direction: the prior, observed estimate, threshold, and effect scale all matter.
Worked Calculations
Worked Example: Equal Prior Mass on Three Candidate Effects
Suppose the standard error is \(s=1\), the two-sided discovery cutoff is \(c=1.96\), and the observed estimate is \(y=2.1\). Consider a discrete preselection prior assigning probability \(1/3\) to each candidate effect \(\beta\in\{-1,0,1\}\). All effects and estimates are in the same units.
The observed estimate passes the rule because \(|2.1/1|=2.1>1.96\). The selection probabilities are approximately \(q(-1)=0.1701\), \(q(0)=0.0500\), and \(q(1)=0.1701\). For example, at \(\beta=1\),
The normal sampling density is \(f(y\mid\beta)=\phi(y-\beta)\), because \(s=1\). Using unrounded values in the division gives the following selection-conditioned likelihood values, rounded for display:
The prior masses are equal, so they cancel in normalizing. The total is \(0.01921+0.87975+1.28099=2.17995\). Thus the posterior probabilities are approximately \(0.01921/2.17995=0.00881\), \(0.87975/2.17995=0.4036\), and \(1.28099/2.17995=0.5876\), respectively. The posterior mean is
For comparison, ordinary Bayesian updating with the same prior uses the unconditioned densities. Their normalized probabilities are approximately \(0.0123,\ 0.1659,\ 0.8218\), giving posterior mean about \(0.8095\). The two analyses differ because the selected likelihood discounts candidate effects in proportion to their selection probabilities. Neither calculation says that the locus’s true effect is known; each is a posterior summary under its stated model.
Worked Example: The Prior Can Change the Selected Posterior
Keep the same estimate, standard error, threshold, candidate effects, and selection-conditioned likelihood values as above, but use a different preselection prior: approximately \(0.9735\) on \(\beta=-1\), \(0.01762\) on \(\beta=0\), and \(0.008886\) on \(\beta=1\). These probabilities sum to approximately one. This deliberately concentrated prior illustrates how prior information remains consequential after selection adjustment.
Multiply each prior probability by its conditional likelihood. The unnormalized weights are approximately
Their sum is approximately \(0.04559\). Dividing each weight by this sum gives posterior probabilities of about \(0.4103,\ 0.3399,\ 0.2497\) for \(-1,0,1\). The posterior mean is therefore
The negative posterior mean is not a consequence of the positive estimate alone; it arises from the highly asymmetric prior as well as the selected likelihood. This example is a reminder to report the prior and not to present a selection-adjusted estimate as if it were determined by the discovery data alone.
Worked Example: A One-Sided Discovery Rule
Suppose instead that a study reports a locus only when \(Y>1.96\), with \(s=1\), and observes \(y=2.1\). Consider candidate effects \(\beta=0\) and \(\beta=1\) with equal prior probabilities. The event is one-sided, so \(q(\beta)=P(Y>1.96\mid\beta)\), not the two-sided probability used above. The observed value satisfies the rule.
At \(\beta=0\), \(q(0)=1-\Phi(1.96)\approx0.02500\). At \(\beta=1\), \(q(1)=1-\Phi(0.96)\approx0.1685\). The sampling densities at \(y=2.1\) are approximately \(\phi(2.1)=0.04399\) and \(\phi(1.1)=0.2179\). Thus the conditional likelihoods are approximately \(0.04399/0.02500=1.760\) and \(0.2179/0.1685=1.293\). With equal prior masses, normalizing these two values gives posterior probabilities \(1.760/(1.760+1.293)\approx0.5765\) for \(\beta=0\) and \(0.4235\) for \(\beta=1\). Ordinary updating instead gives probability \(0.04399/(0.04399+0.2179)\approx0.1680\) to \(\beta=0\). The direction and definition of the selection rule therefore matter to the adjustment.
Justification and Bayesian Interpretation
The conditional likelihood follows directly from the definition of conditional density. For a fixed \(\beta\), and any observed \(y\) in the selection region \(S\),
because once \(y\in S\) is specified, the event \(S\) is determined and \(f(y,S\mid\beta)=f(y\mid\beta)\). The denominator is essential: integrating \(f(y\mid\beta)/q(\beta)\) over \(y\in S\) gives one. Bayes’ rule then multiplies this normalized conditional sampling density by the stated prior and divides by its integral (or sum) to produce a posterior distribution.
There is a subtle distinction between this conditional-likelihood procedure and conditioning a fully joint Bayesian model on the selection event. If \(\pi(\beta)\) is a preselection prior in a joint model and \(S\) is determined by the observed \(Y\), then the prior among selected loci is \(\pi(\beta\mid S)\propto q(\beta)\pi(\beta)\). Combining that selected-locus prior with \(f(y\mid\beta,S)\) cancels \(q(\beta)\), recovering a posterior proportional to \(\pi(\beta)f(y\mid\beta)\) when the exact \(y\) is observed. Thus a posterior using the conditional likelihood together with the original preselection prior is a distinct selection-adjusted analysis, not an automatic consequence of ordinary Bayesian conditioning. State which prior population and inferential target are intended.
Assumptions, Diagnostics, and Alternatives
The normal model assumes that the estimate is unbiased before selection, its standard error is known or adequately treated as fixed, and its sampling distribution is approximately normal conditional on \(\beta\). It also assumes that the locus was selected by the specified rule. For a two-sided \(Z\)-threshold, \(S=\{|Y/s|>c\}\); for a one-sided scan, the event must retain the correct direction. If selection depends on several statistics, multiple stages, or other loci, the event and its probability must represent that process rather than an oversimplified threshold.
Useful checks include confirming that the reported estimate actually lies in the stated selection region, verifying that \(q(\beta)\) is calculated on the same effect and standard-error scale as the likelihood, and checking that \(0<q(\beta)\leq1\). Recompute the posterior under plausible alternative priors and thresholds. A large change signals sensitivity, not a computational error by itself. For continuous priors, inspect the posterior numerically or graphically; a narrow region of high density or unstable integration can make a numerical posterior mean unreliable.
The main alternatives answer different questions. Ordinary Bayesian shrinkage uses the unconditioned likelihood \(f(y\mid\beta)\) and is often convenient, including with a conjugate normal prior, but does not explicitly adjust that likelihood for the selection event. Truncated-normal correction and conditional-likelihood methods also account for thresholding, but are not automatically Bayesian: they need not combine a conditional likelihood with a prior. Resampling corrections instead approximate the selection process by repeatedly sampling and applying the discovery rule. These methods can complement one another, but they rely on different assumptions and should not be treated as interchangeable estimates.
A conditional selection model does not repair a misspecified prior, winner selection across an incorrectly represented set of loci, or uncertainty in estimated standard errors. It also does not establish that an association is causal. When a genuinely independent replication estimate is available, it supplies additional information under a different sampling stage; combining it with discovery data requires modeling that relationship explicitly rather than silently treating the two estimates as interchangeable.
Common Mistakes and Practical Limits
- Using the wrong event probability. Dividing by a one-sided probability when discovery was two-sided misrepresents selection. Write down \(S\) first and calculate \(P(S\mid\beta)\) from that event.
- Rounding too early. Small errors in \(q(\beta)\) can alter normalized posterior weights. Retain adequate precision through the divisions and normalization, then round reported results.
- Calling the result a universal Bayesian correction. The conditional likelihood with a preselection prior is a specified analysis. Explain the prior’s population and distinguish it from the fully joint posterior conditional on exact data.
- Assuming every posterior must move toward zero. The adjustment depends on the threshold, prior, and observed estimate; it can change direction or magnitude. Calculate the posterior rather than asserting a direction.
- Ignoring uncertainty in the standard error or thresholding procedure. If those quantities are uncertain or the selection mechanism is more complex than the model, the stated posterior may understate uncertainty or adjust for the wrong event.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For \(Y\mid\beta\sim N(\beta,s^2)\) and \(S=\{|Y/s|>c\}\), write the two-sided selection probability \(q(\beta)\).
- If a locus is selected by \(Y/s>c\), should the selection probability include a lower-tail term? Explain.
- With equal prior probabilities, selected likelihood values \(0.2\) and \(0.8\) correspond to two candidate effects. Find their posterior probabilities.
- Why is the posterior based on a conditional likelihood and preselection prior not automatically identical in interpretation to the fully joint Bayesian posterior conditional on exact \(Y=y\)?
Solutions. (1) \(q(\beta)=\Phi(-c-\beta/s)+1-\Phi(c-\beta/s)\), the sum of the lower and upper tail probabilities. (2) No. The event is \(Y/s>c\), so \(q(\beta)=1-\Phi(c-\beta/s)\); adding a lower-tail term would model a different discovery rule. (3) Normalize the weights: \(0.2+0.8=1.0\), so the probabilities are \(0.2\) and \(0.8\). (4) In a fully joint model the prior among selected loci is proportional to \(q(\beta)\pi(\beta)\); multiplying by the conditional likelihood \(f(y\mid\beta)/q(\beta)\) cancels the selection factor. Using the original preselection prior with the conditional likelihood is a distinct specified selection-adjusted procedure.