Conclusion

Statistics gives ML its caution.

Optimization tells us how to fit a model. Statistics asks whether the fitted model should be trusted.

That question is not pessimistic. It is practical. A model score is only useful when we understand where it came from, how it might move with another sample, and what decisions it is strong enough to support.

What This Chapter Added

You now have a working vocabulary for:

• samples and datasets
• train, validation, and test splits
• estimators
• bias and variance
• likelihood and maximum likelihood
• Bayesian updating
• confidence intervals
• hypothesis tests
• cross-validation
• resampling

These ideas help you read experiments with less mystery and more discipline.

What To Carry Forward

Every number in ML has a source.

A validation score comes from data chosen during development. A test score should come after model choices are finished. A confidence interval says the point estimate is not exact. Cross-validation and resampling ask how much a number moves when the evidence is rearranged.

The habit is simple:

1. Ask what data produced the number.
2. Ask what choices were made after seeing related numbers.
3. Ask how stable the number is.
4. Ask whether the difference is large enough to matter.

This habit is not about distrusting every result. It is about knowing what kind of trust a result has earned.

What Comes Next

The next Mathematics chapter is Information Theory. It explains entropy, cross-entropy, KL divergence, negative log-likelihood, perplexity, and the link between prediction and compression.

Keep This Question Nearby

When you see a model score, ask:

What evidence produced this number, and how stable is it?