Introduction
Information theory signals use entropy and mutual information to quantify how much uncertainty and shared information exist between time series. This lets you distinguish genuine predictive relationships from spurious correlations and reduce noise in your signal set.
Why does that matter to you as a trader or quant researcher? Because noisy features destroy model performance, inflate overfitting risk, and waste research time. How do you know when a signal is just noise, and when it carries real predictive power?
- Entropy measures unpredictability of a single variable, in bits if you use log base 2.
- Mutual information measures shared information between variables and captures nonlinear dependencies that correlation misses.
- Use discretization, k-nearest neighbor estimators, or Gaussian assumptions to estimate entropy and mutual information on return series.
- Apply information-theoretic feature selection methods, like mRMR, to maximize predictive relevance while minimizing redundancy across assets and timeframes.
- Condition on confounders with conditional mutual information to filter spurious relationships driven by volatility or market regimes.
Foundations: Entropy and Mutual Information
Entropy, H(X), quantifies uncertainty about a random variable X. For a discrete variable it is H(X) = -sum p(x) log2 p(x). A fair coin has H = 1 bit. For continuous variables you work with differential entropy but discretization makes interpretation easy for market data.
Mutual information, I(X;Y), measures the reduction in uncertainty about X given Y. Formally I(X;Y) = H(X) + H(Y) - H(X,Y). It is zero when X and Y are independent and positive otherwise. Importantly, mutual information captures nonlinear relations that linear correlation can miss.
Practical interpretation
If a candidate feature X has I(X;Y_future) = 0.2 bits for predicting next-day returns Y_future, that means observing X reduces the uncertainty about Y_future by 0.2 bits. That might be small in absolute terms, but relative to H(Y_future) it can be meaningful.
Mutual information is bounded above by min(H(X), H(Y)). So normalized measures, like I_norm = I(X;Y)/H(Y), let you compare features across different distributions and timeframes.
Practical Estimation in Market Data
Estimating entropy and mutual information from price series requires choices. You must choose discretization bins or a continuous estimator, handle nonstationarity, and control for sample bias. Bad choices produce biased estimates that mislead model selection.
Discretization and binning
One simple approach converts returns into a finite alphabet by quantiles. For example, split daily returns into five bins: extreme down, down, neutral, up, extreme up. Compute frequencies and then discrete entropy and mutual information in bits using log base 2.
Example numbers help. Suppose 10,000 daily observations for $AAPL returns are binned into five roughly equal quantile bins, so H(Y) approximates log2(5) = 2.32 bits if uniform. If a feature X yields I(X;Y) = 0.12 bits, that is about 5.2 percent of the maximum entropy and indicates low but measurable information.
Continuous estimators
If you prefer not to discretize, use k-nearest neighbor estimators for differential entropy and mutual information. These are robust to binning but require careful parameter selection for k and attention to edge effects. They also handle continuous-valued technical indicators like RSI or order flow measures.
Bias correction matters. With finite samples entropy is underestimated. Use permutation tests and shuffled surrogates to compute significance thresholds. For example, compute mutual information between X and shuffled Y many times to estimate a null distribution and report p-values or effect sizes.
Feature Selection and Noise Filtering Workflows
Information measures can be integrated into your feature engineering pipeline to remove noise early. The goal is to keep features that add unique information about target returns across the timeframe you care about.
Step-by-step mRMR style workflow
- Define target Y, for example next-day sign of return or excess return over $SPY on a 1-day horizon.
- Preprocess features, standardize scales, and discretize consistently across features and the target.
- Compute I(X_i;Y) for each candidate feature X_i to estimate relevance.
- Compute pairwise I(X_i;X_j) to estimate redundancy.
- Select features by maximizing relevance while minimizing redundancy using a scoring rule like score(X_i) = I(X_i;Y) - (1/(k)) sum_j I(X_i;X_j) for already chosen features.
- Validate selected features with cross-validated predictive performance and permutation tests for statistical significance.
This approach ensures you keep features that contribute nonredundant information about Y. You can extend it to time-aware selection by computing mutual information across lags and multiple horizons.
Temporal and cross-asset selection
Information relationships change across timeframes. A feature may have high mutual information with hourly returns but none with daily returns. Compute I(X_t;Y_{t+h}) across h and pick features conditional on your trading horizon.
For cross-asset signals, compute conditional mutual information I(X;Y|Z) to control for market-wide drivers Z like $SPY returns or realized volatility. This helps reveal features that truly add asset-specific information rather than simply reflecting a common factor.
Real-World Examples and Walkthroughs
Here are concrete examples that show how entropy and mutual information reveal signal quality and guide selection. These examples use intuitive numbers so you can reproduce the calculations in your codebase.
Example 1: Momentum indicator across assets
Take 5-day momentum for $AAPL and $MSFT, discretize returns into 5 quantiles, and compute I(momentum; next-day return). Suppose you find I = 0.18 bits for $AAPL and I = 0.03 bits for $MSFT on the same sample. That tells you the momentum predictor is much more informative for $AAPL in this period.
Next compute redundancy. If you include both momentum and an RSI feature, and find I(momentum;RSI) = 0.12 bits, then the marginal benefit of adding RSI after momentum is limited. Use the mRMR score to rank and choose the best set.
Example 2: Conditional mutual information to remove volatility confound
Suppose you test an order flow imbalance feature X and find I(X;Y) = 0.15 bits for next-hour returns Y. But when you compute I(X;Y|V) conditioning on realized volatility V, you get I_cond = 0.02 bits. That implies the apparent predictability is largely explained by volatility, not by unique signal content in X.
At that point you either transform features to remove volatility dependence or drop the feature for your chosen horizon. Conditioning guards against spurious inclusion of features that simply mirror a confounder.
Example 3: Quantifying signal decay across lags
Compute I(X_t;Y_{t+h}) across h = 1, 5, 20 days. You might see I = 0.20 bits at h = 1, 0.05 bits at h = 5, and ~0 at h = 20. That curve tells you the signal decays quickly and should be used only for short-horizon strategies.
Use this information to set feature lifetimes in your pipeline and to choose the correct retraining cadence for your models.
Common Mistakes to Avoid
- Interpreting small MI as zero, without normalization, you might dismiss useful features. Always compare I to H(Y) or to a shuffled null distribution to assess significance.
- Relying solely on linear correlation. Correlation can be near zero while mutual information is significant. Use both to get a full picture.
- Using inconsistent binning across features. Inconsistent discretization makes MI comparisons meaningless. Standardize bin edges or use rank-based quantiles.
- Ignoring sample bias. Small samples underestimate entropy. Use bias correction, bootstrapping, or permutation tests to validate estimates.
- Failing to condition on common drivers. Market-wide factors like volatility or index moves can create spurious MI. Use conditional mutual information to control for these confounders.
FAQ
Q: How sensitive are mutual information estimates to discretization choices?
A: They can be quite sensitive. Coarse bins understate MI, while overly fine bins increase variance and bias. Use quantile-based binning or k-nearest neighbor estimators and validate stability across choices.
Q: Can mutual information replace correlation in my feature selection?
A: It should complement correlation, not replace it. MI detects nonlinear relations so it can reveal features correlation misses. Use both metrics to form a richer selection criterion.
Q: How much mutual information is practically useful for trading performance?
A: Small MI values can still matter if they are persistent and the signal scales with capital. Rather than focusing on absolute bits, consider effect size relative to H(Y), robustness over time, and economic costs like transaction fees.
Q: Is mutual information robust in high-dimensional feature sets?
A: Pairwise MI can guide selection, but it does not capture all multivariate dependencies. Use approaches like mRMR to reduce redundancy and consider multivariate information measures or joint modeling for final validation.
Bottom Line
Entropy and mutual information give you principled, model-agnostic tools to quantify unpredictability and shared information in financial time series. When you apply them thoughtfully, they help separate informative features from noise and control redundancy across assets and timeframes.
Next steps for you are to implement robust estimators with bias correction, integrate MI into a systematic feature selection pipeline like mRMR, and use conditional mutual information to control for common market drivers. At the end of the day, these tools won't eliminate uncertainty, but they will make your feature engineering measurably more disciplined and defensible.
