We often don’t want to just know if a particular event \(A\) has a certain probability, but also how other events (call them \(B\)) might depend on that outcome.
In other words:
We want the conditional probability of \(B\) given \(A\), denoted \(\mathbb{P}(B|A)\).
Conditional Probabilities
We can write conditional probabilities in terms of unconditional probabilities:
Extreme temperatures + many other mechanisms ⇒ power outages
Probability Distributions
The probability of possible values of an unknown quantity are often represented as a probability distribution.
Probability distributions associate a probability to every event under consideration (the event space) and have to follow certain rules (for example, total probability = 1).
Selecting a Distribution
The specification of distributions can strongly influence the analysis.
Probabilistic interactions between sea-levels and exposure
Selecting a Distribution
A distribution implicitly answers questions like:
What is the most probable event? How much more likely is it than the others?
Are larger or smaller events more, less, or equally probable?
How probable are extreme events?
Are different events correlated, or are they independent?
Key Features of Probability Distributions
Mean/Mode (what events are “typical”)
Skew (are larger or smaller events more or equally probable)
Variance (how spread out is the distribution around the mode)
Tail Probabilities (how probable are extreme events)
Common Distributions
Gaussian / Normal
Lognormal
Binomial
Uniform / Discrete Uniform
Probability Distribution Tails
The tails of distributions represent the probability of high-impact outcomes.
Key consideration: Small changes to these (low) probabilities can greatly influence risk.
Monte Carlo
Stochastic Simulation
Monte Carlo simulation: Propagating random samples through a model to estimate a value (usually an expectation or a quantile).
Goals of Monte Carlo
Monte Carlo is a broad method, which can be used to:
Obtain probability distributions of outputs;
Estimate deterministic quantities (Monte Carlo estimation).
Monte Carlo Estimation
Monte Carlo estimation involves framing the quantity of interest as a summary statistic (such as an expected value).
MC Example: Finding \(\pi\)
Finding \(\pi\) by sampling random values from the unit square and computing the fraction in the unit circle. This is an example of Monte Carlo integration.
\[\frac{\text{Area of Circle}}{\text{Area of Square}} = \frac{\pi}{4}\]
[ Info: Saved animation to /Users/vs498/Teaching/BEE4750/fall2023/slides/images/mc_pi.gif
MC Example: Dice
What is the probability of rolling 4 dice for a total of 19?
Can simulate dice rolls and find the frequency of 19s among the samples.
[ Info: Saved animation to /Users/vs498/Teaching/BEE4750/fall2023/slides/images/mc_dice.gif
Monte Carlo Estimation
This type of estimation can be repeated with any simulation model that has a stochastic component.
For example, consider our dissolved oxygen model. Suppose that we have a probability distribution for the inflow DO.
How could we compute the probability of DO falling below the regulatory standard somewhere downstream?
Monte Carlo and Uncertainty Propagation
This is an example of uncertainty propagation: draw samples from some distribution, and run them through one or more models to find the (conditional) probability of outcomes of interest (for good or bad).
For example (HW3): What is the probability that a levee will be overtopped given climate and extreme sea-level uncertainty?
Key Takeaways
Key Takeaways
Choice of probability distribution can have large impacts on uncertainty and risk estimates: try not to use distributions just because they’re convenient.
Monte Carlo: Estimate expected values of outcomes using simulation.
