How Easley–O’Hara PIN Algorithms Estimate Informed-Trading Risk: Poisson Buy/Sell Counts, Information Events, Likelihood Mixtures, Numerical Underflow and Identification Failure

Reader question: The Glosten–Milgrom model explains how adverse selection can widen spreads, but how can we estimate from trade counts how much order flow may be information-driven?

The Easley–O’Hara Probability of Informed Trading (PIN) framework models daily buy and sell arrivals as a mixture of Poisson processes. Some days have no private information event; some have good-news information; some have bad-news information. Uninformed trades occur every day, while informed trades arrive only on information-event days and only on the side consistent with the signal.

Maximum likelihood estimates the event probability and trade-arrival rates. PIN then summarizes the unconditional share of expected order arrivals attributed to informed trading.

This article owns one precise computational job: likelihood-based estimation of the Easley–O’Hara PIN structural order-flow model. It does not own Glosten–Milgrom Bayesian quote updating, Kyle price impact, Lee–Ready trade signing, Hawkes order-flow clustering, or general market-making inventory control.

This is public mathematical and computational education. It is not a method for identifying actual insiders, a trading strategy, or personalized financial advice.

1. The daily latent-state model

Each day begins with a latent information state.

With probability:

1−α

there is no private information event.

With probability:

α

an information event occurs.

Conditional on an information event:

• bad news occurs with probability δ;
• good news occurs with probability 1−δ.

These states are not directly observed. They are inferred from the day’s buy and sell counts.

2. Uninformed order arrivals

Let:

ε_b

be the Poisson arrival rate of uninformed buys, and:

ε_s

the arrival rate of uninformed sells.

On a no-information day:

B ~ Poisson(ε_b),

S ~ Poisson(ε_s),

typically conditionally independent in the baseline model.

3. Informed order arrivals

Let:

μ

be the Poisson arrival rate of informed orders on information-event days.

On a good-news day, informed traders buy, so:

B ~ Poisson(ε_b+μ),

S ~ Poisson(ε_s).

On a bad-news day, informed traders sell:

B ~ Poisson(ε_b),

S ~ Poisson(ε_s+μ).

The daily observed counts are therefore generated by a three-component mixture.

4. One-day likelihood

Let:

p(k;λ)=e^−λλ^k/k!

be the Poisson probability.

For observed daily counts (B,S), the baseline likelihood is:

L(B,S|θ) = (1−α)p(B;ε_b)p(S;ε_s)

+ αδ p(B;ε_b)p(S;ε_s+μ)

+ α(1−δ)p(B;ε_b+μ)p(S;ε_s).

The parameter vector is:

θ=(α,δ,μ,ε_b,ε_s).

5. Sample likelihood

For T trading days:

L(θ)=∏_t=1^TL(B_t,S_t|θ).

In computation, maximize the log likelihood:

ℓ(θ)=Σ log L(B_t,S_t|θ).

Direct multiplication of daily probabilities is numerically unsafe because the product can underflow to zero even for ordinary sample sizes.

6. The PIN formula

The expected informed-order arrival rate is:

αμ.

The expected total order-arrival rate is:

αμ + ε_b + ε_s.

Therefore:

PIN = αμ / (αμ + ε_b + ε_s).

Under the symmetric simplification ε_b=ε_s=ε:

PIN = αμ/(αμ+2ε).

This is a model-implied probability that a randomly arriving trade is information-based under the structural assumptions.

7. PIN is not the probability that a named trader is an insider

PIN is an unconditional mixture probability derived from estimated arrival intensities.

It does not identify:

• which trade is informed;
• which trader is informed;
• whether information is illegal;
• whether a particular transaction should be investigated.

It is a structural market-microstructure statistic, not an enforcement classifier.

8. Why likelihood underflow is a major implementation problem

Daily buy/sell counts can be large. Terms such as:

e^−λλ^B/B!

can be extremely small.

The mixture adds several such tiny values together. Naively evaluating them in probability space can produce zeros or infinities even when the mathematical likelihood is well defined.

The correct approach is to work in log space.

9. Log-sum-exp stabilization

Suppose the three log mixture terms for one day are a,b,c.

Then:

log(e^a+e^b+e^c)

should be evaluated as:

m + log[e^a−m+e^b−m+e^c−m],

where:

m=max(a,b,c).

This log-sum-exp transformation avoids severe numerical underflow.

Factorial terms should use log-gamma:

log(k!) = log Γ(k+1).

10. Parameter constraints

The natural parameter space is:

• 0≤α≤1;
• 0≤δ≤1;
• μ≥0;
• ε_b≥0;
• ε_s≥0.

Optimization can enforce these through bounds or unconstrained transforms such as logits for probabilities and exponentials/softplus for intensities.

A generic unconstrained optimizer can wander into invalid negative intensities unless constraints are explicit.

11. Multiple local optima and flat regions

The mixture likelihood can be difficult to optimize.

For example, a sample with weak order imbalance may be fit by:

• small α and large μ;
• larger α and smaller μ;
• higher uninformed rates with modest information-event contribution.

Different parameter combinations can yield similar PIN or similar likelihood.

Diagnostic: use multiple starting points, inspect profile likelihoods, and report parameter/PIN stability rather than trusting one optimizer run.

12. Identification of α versus μ

PIN depends on the product αμ in the numerator.

The likelihood can sometimes identify α and μ separately from the shape/frequency of imbalanced days, but finite samples may leave them weakly separated.

Falsifier: profile α while reoptimizing μ and the other parameters. A broad flat likelihood ridge is evidence that the decomposition “rare strong events versus frequent weak events” is weakly identified.

13. The role of δ

δ controls the relative probability of bad-news versus good-news information events.

If buy/sell asymmetry is weak or classification is noisy, δ can be unstable.

In some applications, a symmetric assumption is imposed for simplicity, but doing so changes the structural interpretation.

14. Trade classification is an upstream dependency

The model requires daily counts of buyer-initiated and seller-initiated trades.

Historical datasets may not directly label aggressor direction.

If Lee–Ready or another classification rule mislabels trades, the observed B/S imbalance is distorted before PIN estimation even begins.

See Lee–Ready trade-signing algorithms.

15. Inputs and outputs

Inputs can include:

• daily buy counts B_t;
• daily sell counts S_t;
• trade-sign method;
• sample window;
• parameter bounds/transforms;
• optimizer;
• starting-value grid;
• convergence tolerance.

Outputs can include:

• α,δ,μ,ε_b,ε_s;
• PIN;
• maximized log likelihood;
• Hessian/standard errors;
• profile-likelihood plots;
• posterior state probabilities by day;
• model-implied buy/sell distributions;
• goodness-of-fit diagnostics;
• optimizer-start sensitivity.

16. Posterior probability that a day was an information day

Even though the baseline PIN is unconditional, Bayes’ rule can compute:

P(state | B_t,S_t,θ̂)

for no-news, good-news, and bad-news states.

A strongly one-sided day can therefore receive a high posterior probability of an information event.

But this posterior is still model-dependent and can confuse strategic order splitting or liquidity shocks with information.

17. Evidence polarity

Evidence for confidence includes:

• multiple starts converge to the same likelihood/PIN;
• model reproduces the empirical distribution of buy/sell imbalances;
• parameter estimates remain stable across adjacent windows;
• posterior information days align with independent information-event evidence without being mechanically defined by it;
• trade-sign uncertainty does not materially change PIN;
• bootstrap/profile intervals are reasonably informative;
• held-out daily count likelihood beats simpler Poisson benchmarks.

Evidence against confidence includes:

• optimizer dependence;
• boundary estimates α≈0/1 or μ≈0 without strong support;
• wide profile likelihoods;
• strong buy/sell serial dependence inconsistent with independent Poisson arrivals;
• large sensitivity to trade classification;
• order-size information ignored;
• model badly underpredicting extreme imbalance days.

18. Counterexample: order splitting looks like informed trading

A large uninformed institution splits a parent buy order into hundreds of child trades over one day.

The daily B/S imbalance can resemble a good-news informed day even if the parent trade was liquidity-motivated.

Falsifier: test same-sign serial dependence and parent-order proxies. Independent Poisson counts are a structural assumption, not a fact.

19. Counterexample: Hawkes clustering violates Poisson independence

Trade arrivals often cluster.

If each trade raises the short-run probability of more trades, a constant-rate Poisson mixture can attribute clustering to latent information states.

Falsifier: compare interarrival/dispersion statistics and a self-exciting benchmark.

See Hawkes-process algorithms.

20. Counterexample: buy/sell rates are asymmetric even without information

Market structure, rebalancing or directional liquidity demand can make ε_b≠ε_s.

Imposing ε_b=ε_s can force persistent liquidity asymmetry into the informed-trading component.

Falsifier: compare symmetric and asymmetric uninformed-rate specifications out of sample.

21. Counterexample: PIN is stable but α and μ are not

Two optimizer solutions may produce similar αμ and therefore similar PIN, while α and μ differ dramatically.

This means the summary metric is better identified than the latent event-frequency/intensity decomposition.

Falsifier: report parameter profiles, not PIN alone.

22. Counterexample: high-volume markets

Very large daily counts make naive likelihood arithmetic especially prone to underflow.

Falsifier: unit-test the log likelihood against arbitrary-precision/small-sample calculations and require finite values across the parameter search space.

23. Counterexample: long estimation windows hide time variation

The baseline model assumes stable parameters within the sample.

Information risk can change around earnings, crises or structural market changes.

Falsifier: estimate rolling/subperiod models and compare likelihood stability. A single annual PIN may average over materially different regimes.

24. PIN versus Glosten–Milgrom

Glosten–Milgrom algorithms own sequential Bayesian quote setting conditional on informed/noise trading.

PIN owns empirical likelihood estimation of information-event and order-arrival parameters from daily buy/sell counts.

25. PIN versus Kyle’s lambda

Kyle’s lambda measures price impact per unit signed order flow.

PIN estimates the proportion of order arrivals attributed to the informed component under a mixture model. The outputs are not interchangeable.

26. PIN versus VPIN

VPIN uses volume-bucket order imbalance and was designed as a more real-time toxicity metric.

It is not the same estimator as maximum-likelihood PIN, and the empirical interpretation of VPIN has been debated.

This page deliberately owns the classical structural PIN likelihood rather than broadening into all “informed trading” metrics.

27. Alternatives

Glosten–Milgrom: structural quote formation from asymmetric information.

Kyle models: strategic informed trading and price impact.

VPIN: volume-bucket imbalance toxicity proxy.

Hawkes/order-flow models: dynamic clustered arrival intensity.

State-space/dynamic PIN: allow information-risk parameters to vary over time.

Direct markout/adverse-selection models: infer information from post-trade price movement.

28. Weak links

• trade-sign classification;
• Poisson independence;
• constant parameters;
• daily aggregation;
• order size ignored;
• α/μ weak identification;
• likelihood underflow;
• optimizer local maxima;
• uninformed-rate asymmetry;
• strategic/order-splitting behavior.

29. What would falsify confidence?

Confidence should be withdrawn if the model cannot reproduce basic count/imbalance distributions; if results depend strongly on optimizer start or trade-sign method; if profile likelihoods are flat; if arrival clustering strongly violates Poisson assumptions; or if simpler/richer held-out models materially outperform PIN for the intended information-risk measurement job.

30. Verification and update triggers

Preserve the raw trades, sign-classification method, daily aggregation rule, optimizer, transformations, starts, likelihood code and parameter profiles.

Revalidate when:

• market structure changes;
• trade-sign labels change source;
• order splitting/clustering changes;
• sample volume changes materially;
• rolling parameters drift;
• optimization software changes;
• the use case requires intraday rather than daily information risk.

31. Primary and high-quality references

• David Easley, Nicholas Kiefer, Maureen O’Hara and Joseph Paperman, Liquidity, Information, and Infrequently Traded Stocks, Journal of Finance, 1996.
• RePEc record for the same paper: Liquidity, Information, and Infrequently Traded Stocks.
• PIN: Measuring Asymmetric Information in Financial Markets with R, R Journal, for stable likelihood computation and implementation details.
• For modern discussion of estimation limitations and alternative PIN specifications: Probability of Informed Trading During the COVID-19 Pandemic, which reviews several PIN estimation refinements.

Educational boundary: PIN is an estimated structural share of information-driven order arrivals under a specific Poisson-mixture model. It should not be interpreted as direct proof of illegal insider trading or as a model-free measure of market toxicity.