In plain English
When the standard assumptions do not hold, you need methods that either weaken the assumptions or simulate the answer instead of solving for it.
The advanced view
Causal inference (difference-in-differences, instrumental variables, regression discontinuity, matching) trades assumptions for identification. Bayesian methods put a distribution on parameters and update with evidence, which suits small samples and prior knowledge. Monte Carlo replaces intractable integrals with sampling, and the bootstrap replaces analytic standard errors with resampling.
Bayesian statistics treats parameters as random variables with priors. Combine prior and likelihood to get the posterior — a full distribution of plausible values, not a point estimate. With small samples informative priors help; with large samples the prior washes out and Bayesian and frequentist answers converge. Bayesian methods underpin probabilistic programming, A/B testing and many risk models.
Deep learning stacks layers: feedforward networks for tabular data (often beaten by gradient boosting), RNNs and LSTMs for sequences, transformers for attention-based modelling. In finance it shows up in trading signals, NLP on earnings calls and filings, and non-linear modelling where accuracy beats interpretability. NLP extracts information from text — sentiment on calls and news, topic modelling, named-entity recognition on filings. High-dimensional analysis handles p ≥ n, where standard regression breaks: regularise, reduce dimensions, or select variables.
Essential vocabulary
- Posterior distribution
- Prior × likelihood, normalised. The complete Bayesian answer.
- Overparameterisation
- More parameters than observations. Requires regularisation or dimension reduction.
- Embedding
- Dense vector representation of text or categories that preserves similarity.
- Walk-forward validation
- Expanding or rolling window testing that respects temporal ordering.
Why it works
Difference-in-differences works because the control group carries the counterfactual trend: subtracting the control's change removes anything that would have happened to both groups anyway. Its validity rests entirely on the parallel-trends assumption, which is why any credible answer shows the pre-period lines moving together.
Common pitfalls
- ×Claiming causality from a diff-in-diff without pre-trend evidence.
- ×Using a weak instrument — it biases toward OLS while looking rigorous.
- ×Running Monte Carlo with independent draws when the drivers are correlated.
How it is used — measure a pricing pilot
Step 1 of 4
- 1Test region revenue: 100 before, 118 after. Control: 100 before, 108 after.
Deeper
Deeper: causal inference beyond controls
Adding controls does not create causality. The credible designs are: randomised experiment; difference-in-differences (needs parallel pre-trends); instrumental variables (needs an instrument that affects treatment but not the outcome directly); regression discontinuity (needs a sharp cutoff rule); and synthetic control (build a weighted comparison unit).
When you cannot get any of them, state the assumption your causal claim rests on and how you would falsify it. That sentence is what separates an analyst from a dashboard.
Must know cold
- ✓Randomisation solves selection; nothing else does it as cleanly.
- ✓Difference-in-differences requires parallel trends before treatment.
- ✓Monte Carlo turns a point estimate into a distribution of outcomes.
- ✓Bootstrap gives confidence intervals when the formula is unknown.
Exercises
Try each one on paper before revealing the worked solution.
Exercise 1
Stores that got a refit grew 8%; the others grew 3%. The refits went to the best locations. How would you estimate the true effect?
Exercise 2
A DCF gives 1,355. How do you turn it into a range with Monte Carlo, and what do you report?