1. Bayes theorem for events - Crash course Bayesian system identification

Exercise 1

Assuming that 1 in 100 reinforced concrete structures are affected by alkali-silica corrosion (alkali-silica reaction induced corrosion). At early stages the corrosion is not visible from the outside. Since in the long run it can lead to a considerable reduction of strength it is of utmost importance to detect the corrosion as soon as possible and take the necessary countermeasures.

Suppose that we have a newly developed non-destructive testing method which detects alkali-silica corrosion in 99% of the cases if there is corrosion (right decision). At the same time it signals detection in 5% of the cases when there is no corrosion (false positive). These numbers are determined based on extensive laboratory and field tests.

Question: What is the probability that a structure is affected by alkali-silica corrosion if the new method signals detection?

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Hint

We need to find:

$P(\text{corrosion}|\text{positive})$

Define the Known Probabilities¶

We will start by defining the probabilities given in the problem statement.

# Probability that a structure is affected by corrosion
p_corrosion = 0.01  # 1 in 100

# Probability of a positive test result given that there is corrosion
p_positive_given_corrosion = 0.99  # 99% detection rate

# Probability of a positive test result given that there is no corrosion
p_positive_given_no_corrosion = 0.05  # 5% false positive rate

Calculate the Total Probability of a Positive Test Result¶

Next, we need to calculate the total probability of a positive test result, which includes both true positives and false positives.

Hint

\begin{align*} P(\text{positive}) &= P(\text{corrosion}) P(\text{positive} | \text{corrosion}) \\ &\quad + [1 - P(\text{corrosion})] P(\text{positive} | \text{no corrosion}) \end{align*}

(1)

## Write your code here
p_positive = ...

Apply Bayes’ Theorem¶

Now, we can apply Bayes’ theorem to find the probability that a structure is affected by corrosion given a positive test result.

$P(\text{corrosion}|\text{positve}) = \frac{P(\text{positive}|\text{corrosion})P(\text{corrosion})}{P(\text{positive})}$

## Write your code here

p_corrosion_given_positive = ...

Complete Code! 📃💻

Here’s the complete code that you would run in your PC:


p_corrosion = 0.01

p_positive_given_corrosion = 0.99
p_positive_give_no_corrosion = 0.05

p_positive = p_positive_given_corrosion * p_corrosion + p_positive_give_no_corrosion * (1 - p_corrosion)

p_corrosion_given_positive = p_positive_given_corrosion * p_corrosion / p_positive
print(f"P(corrosion|positive) = {p_corrosion_given_positive: .3f}")

Crash course Bayesian system identification

Bayesian system identification

Exercises

2. Use of conjugate priors