From 485bffd0d63df8020a0f8d77f292fb6ca6da749c Mon Sep 17 00:00:00 2001 From: Wesley Barbosa Date: Wed, 5 Jun 2024 22:12:24 -0300 Subject: [PATCH] feat: Add Heredity --- README.md | 2 + .../bayesian-network/heredity/README.md | 212 ++++++++++++++++++ .../heredity/data/family0.csv | 4 + .../heredity/data/family1.csv | 7 + .../heredity/data/family2.csv | 6 + .../bayesian-network/heredity/heredity.py | 198 ++++++++++++++++ .../heredity/images/gene_network.png | Bin 0 -> 123782 bytes 7 files changed, 429 insertions(+) create mode 100644 uncertainty/bayesian-network/heredity/README.md create mode 100644 uncertainty/bayesian-network/heredity/data/family0.csv create mode 100644 uncertainty/bayesian-network/heredity/data/family1.csv create mode 100644 uncertainty/bayesian-network/heredity/data/family2.csv create mode 100644 uncertainty/bayesian-network/heredity/heredity.py create mode 100644 uncertainty/bayesian-network/heredity/images/gene_network.png diff --git a/README.md b/README.md index ce365bc..6a485ea 100644 --- a/README.md +++ b/README.md @@ -16,3 +16,5 @@ ## Uncertainty - **Markov Chain** - [PageRank](./uncertainty/markov-chain/pagerank/) + - **Bayesian Network** + - [Heredity](./uncertainty/bayesian-network/heredity/) diff --git a/uncertainty/bayesian-network/heredity/README.md b/uncertainty/bayesian-network/heredity/README.md new file mode 100644 index 0000000..1171ed5 --- /dev/null +++ b/uncertainty/bayesian-network/heredity/README.md @@ -0,0 +1,212 @@ +# Heredity + +Mutated versions of the GJB2 gene are one of the leading causes of hearing impairment in newborns. Each person carries two versions of the gene, so each person has the potential to possess either 0, 1, or 2 copies of the hearing impairment version GJB2. Unless a person undergoes genetic testing, though, it’s not so easy to know how many copies of the mutated GJB2 a person has. This is a "hidden state": information that has an effect that we can observe (hearing impairment), but that we don’t necessarily directly know. After all, some people might have 1 or 2 copies of mutated GJB2 but not exhibit hearing impairment, while others might have no copies of mutated GJB2 yet still exhibit hearing impairment. + +## Usage +```bash +python heredity.py data.csv +``` + +## Bayesian Network + +A [Bayesian Network](https://en.wikipedia.org/wiki/Bayesian_network) is a probabilistic graphical model that represents a set of random variables and their conditional dependencies via a directed acyclic graph (DAG). In the context of genetic traits like the mutated GJB2 gene, a Bayesian Network can help model and analyze the complex relationships between genes, traits, and inheritance patterns. + +![Gene Network](./images/gene_network.png) + +## Probabilities +### $P(G_{n})$: Probabilities of having $n$ copies of the gene + +$$ +\begin{aligned} +&P(G_{0}) = 96\\% = 0.96\\ +&P(G_{1}) = 3\\% = 0.03\\ +&P(G_{2}) = 1\\% = 0.01\\ +\end{aligned} +$$ + +- $P(G_{0})$: Probability of having no copies of the gene. +- $P(G_{1})$: Probability of having one copy of the gene. +- $P(G_{2})$: Probability of having two copies of the gene. + +### $P(T|G_{n})$: Probabilities of having the trait, given that you have $n$ copies of the gene + +$$ +\begin{aligned} +&P(T|G_{0}) = 1\\% = 0.01\\ +&P(\lnot T|G_{0}) = 1 - P(T|G_{0})\\ +&P(\lnot T|G_{0}) = 0.99\\ +&\\ +&P(T|G_{1}) = 56\\% = 0.56\\ +&P(\lnot T|G_{1}) = 1 - P(T|G_{1})\\ +&P(\lnot T|G_{1}) = 0.44\\ +&\\ +&P(T|G_{2}) = 65\\% = 0.65\\ +&P(\lnot T|G_{2}) = 1 - P(T|G_{2})\\ +&P(\lnot T|G_{2}) = 0.35\\ +\end{aligned} +$$ + +- $P(T|G_{0})$: Probability of having the trait, given that you have no copies of the gene. +- $P(\lnot T|G_{0})$: Probability of not having the trait, given that you have no copies of the gene. +- $P(T|G_{1})$: Probability of having the trait, given that you have one copy of the gene. +- $P(\lnot T|G_{1})$: Probability of not having the trait, given that you have one copy of the gene. +- $P(T|G_{2})$: Probability of having the trait, given that you have two copies of the gene. +- $P(\lnot T|G_{2})$: Probability of not having the trait, given that you have two copies of the gene. + +### $P(T)$: Probability of having the trait + +$$ +\begin{aligned} +&* P(T) = \sum_{i} P(G_{i})P(T|G_{i})\\ +&P(T) = P(G_{0})P(T|G_{0}) + P(G_{1})P(T|G_{1}) + P(G_{2})P(T|G_{2})\\ +&P(T) = 0.96 \times 0.01 + 0.03 \times 0.56 + 0.01 \times 0.65\\ +&P(T) = 0.0329\\ +\end{aligned} +$$ + +- $P(T)$: Probability of having the trait. + +### $P(G_{n}|T)$: Probabilities of having $n$ copies of the gene, given that you have the trait + +$$ +\begin{aligned} +&* P(G_{n}|T) = \frac{P(T|G_{n})P(G_{n})}{\sum_{i} P(G_{i})P(T|G_{i})} = \frac{P(T|G_{n})P(G_{n})}{P(T)}\\ +&\\ +&P(G_{0}|T) = \frac{P(T|G_{0})P(G_{0})}{P(T)} = \frac{0.01 \times 0.96}{0.0329}\\ +&P(G_{0}|T) = 0.2918\\ +&\\ +&P(G_{1}|T) = \frac{P(T|G_{1})P(G_{1})}{P(T)} = \frac{0.56 \times 0.03}{0.0329}\\ +&P(G_{1}|T) = 0.5106\\ +&\\ +&P(G_{2}|T) = \frac{P(T|G_{2})P(G_{2})}{P(T)} = \frac{0.65 \times 0.01}{0.0329}\\ +&P(G_{2}|T) = 0.1976\\ +\end{aligned} +$$ + +- $P(G_{0}|T)$: Probability of having no copies of the gene, given that you have the trait. +- $P(G_{1}|T)$: Probability of having one copy of the gene, given that you have the trait. +- $P(G_{2}|T)$: Probability of having two copies of the gene, given that you have the trait. + +### $P(M)$: Probability of a gene mutating + +$$ +\begin{aligned} +&P(M) = 1\\% = 0.01\\ +&P(\lnot M) = 1 - P(M)\\ +&P(\lnot M) = 0.99\\ +\end{aligned} +$$ + +- $P(M)$: Probability of a gene mutating. +- $P(\lnot M)$: Probability of a gene not mutating. + +### $P(M|G_{n})$: Probabilities of a gene mutating, given that you have $n$ copies of the gene + +$$ +\begin{aligned} +&P(M|G_{0}) = P(M)\\ +&P(M|G_{0}) = 0.01\\ +&\\ +&P(M|G_{1}) = \frac{1}{2} = 0.5\\ +&\\ +&P(M|G_{2}) = P(\lnot M)\\ +&P(M|G_{2}) = 0.99\\ +\end{aligned} +$$ + +- $P(M|G_{0})$: Probability of a gene mutating, given that you have no copies of the gene. +- $P(M|G_{1})$: Probability of a gene mutating, given that you have one copy of the gene. +- $P(M|G_{2})$: Probability of a gene mutating, given that you have two copies of the gene. + +### Example + +Consider the probability that: + +1. Lily (mother) has 0 copies of the gene and does not have the trait. + +$$ +\begin{aligned} +&P(\lnot T|G_{0}) = \frac{P(\lnot T \cap G_{0})}{P(G_{0})}\\ +&\\ +&P(\lnot T \cap G_{0}) = P(G_{0})P(\lnot T|G_{0})\\ +&P(\lnot T \cap G_{0}) = 0.96 \times 0.99\\ +&\boxed{P(\lnot T \cap G_{0}) = 0.9504}\\ +\end{aligned} +$$ + +2. James (father) has 2 copies of the gene and has the trait. + +$$ +\begin{aligned} +&P(T|G_{2}) = \frac{P(T \cap G_{2})}{P(G_{2})}\\ +&\\ +&P(T \cap G_{2}) = P(G_{2})P(T|G_{2})\\ +&P(T \cap G_{2}) = 0.01 \times 0.65\\ +&\boxed{P(T \cap G_{2}) = 0.0065}\\ +\end{aligned} +$$ + +3. Harry (son) has 1 copy of the gene and does not have the trait. + +There are two ways for this to happen. Either he gets the gene from his mother and not from his father, or he gets the gene from his father and not from his mother. + +His mother, Lily, has 0 copies of the gene, so the only way to get the gene from his mother is if it mutates with probability $P(M|G_{0})$. + +$$ +\begin{aligned} +&P(Mother) = P(M|G_{0})\\ +&P(Mother) = 0.01\\ +&\\ +&P(\lnot Mother) = 1 - P(Mother)\\ +&P(\lnot Mother) = 0.99\\ +\end{aligned} +$$ + +- $P(Mother)$: Probability of getting the gene from his mother. +- $P(\lnot Mother)$: Probability of not getting the gene from his mother. + +His father, James, has 2 copies of the gene, so he will get the gene from his father with probability $P(M|G_{2})$. + +$$ +\begin{aligned} +&P(Father) = P(M|G_{2})\\ +&P(Father) = 0.99\\ +&\\ +&P(\lnot Father) = 1 - P(Father)\\ +&P(\lnot Father) = 0.01\\ +\end{aligned} +$$ + +- $P(Father)$: Probability of getting the gene from his father. +- $P(\lnot Father)$: Probability of not getting the gene from his father. + +$$ +\begin{aligned} +&* P(G_{0}) = P(\lnot Mother) \times P(\lnot Father)\\ +&* P(G_{1}) = P(Mother) \times P(\lnot Father) + P(Father) \times P(\lnot Mother)\\ +&* P(G_{2}) = P(Mother) \times P(Father)\\ +&\\ +&P(G_{1}) = P(Mother) \times P(\lnot Father) + P(Father) \times P(\lnot Mother)\\ +&P(G_{1}) = 0.01 \times 0.01 + 0.99 \times 0.99\\ +&\boxed{P(G_{1}) = 0.9802}\\ +&\\ +&P(\lnot T|G_{1}) = \frac{P(\lnot T \cap G_{1})}{P(G_{1})}\\ +&\\ +&P(\lnot T \cap G_{1}) = P(G_{1})P(\lnot T|G_{1})\\ +&P(\lnot T \cap G_{1}) = 0.9802 \times 0.44\\ +&\boxed{P(\lnot T \cap G_{1}) = 0.431288}\\ +\end{aligned} +$$ + +Calculate the joint probability. + +$$ +\begin{aligned} +&P(J) = P(\lnot T \cap G_{0}) + P(T \cap G_{2}) + P(\lnot T \cap G_{1})\\ +&P(J) = 0.9504 + 0.0065 + 0.431288\\ +&\boxed{P(J) = 0.0026643247488}\\ +\end{aligned} +$$ + +## References +- [CS50’s Introduction to Artificial Intelligence with Python](https://cs50.harvard.edu/ai/2024/) diff --git a/uncertainty/bayesian-network/heredity/data/family0.csv b/uncertainty/bayesian-network/heredity/data/family0.csv new file mode 100644 index 0000000..fa103d1 --- /dev/null +++ b/uncertainty/bayesian-network/heredity/data/family0.csv @@ -0,0 +1,4 @@ +name,mother,father,trait +Harry,Lily,James, +James,,,1 +Lily,,,0 diff --git a/uncertainty/bayesian-network/heredity/data/family1.csv b/uncertainty/bayesian-network/heredity/data/family1.csv new file mode 100644 index 0000000..786b300 --- /dev/null +++ b/uncertainty/bayesian-network/heredity/data/family1.csv @@ -0,0 +1,7 @@ +name,mother,father,trait +Arthur,,,0 +Charlie,Molly,Arthur,0 +Fred,Molly,Arthur,1 +Ginny,Molly,Arthur, +Molly,,,0 +Ron,Molly,Arthur, diff --git a/uncertainty/bayesian-network/heredity/data/family2.csv b/uncertainty/bayesian-network/heredity/data/family2.csv new file mode 100644 index 0000000..b609dc0 --- /dev/null +++ b/uncertainty/bayesian-network/heredity/data/family2.csv @@ -0,0 +1,6 @@ +name,mother,father,trait +Arthur,,,0 +Hermione,,,0 +Molly,,, +Ron,Molly,Arthur,0 +Rose,Ron,Hermione,1 diff --git a/uncertainty/bayesian-network/heredity/heredity.py b/uncertainty/bayesian-network/heredity/heredity.py new file mode 100644 index 0000000..042b2b1 --- /dev/null +++ b/uncertainty/bayesian-network/heredity/heredity.py @@ -0,0 +1,198 @@ +import sys +import csv +import itertools + +from typing import Dict, List, Set + +P = { + "G": { + 0: {True: 0.96, False: 0.04}, + 1: {True: 0.03, False: 0.97}, + 2: {True: 0.01, False: 0.99} + }, + "T|G": { + 0: {True: 0.01, False: 0.99}, + 2: {True: 0.65, False: 0.35}, + 1: {True: 0.56, False: 0.44} + }, + "M": { + True: 0.01, + False: 0.99 + }, + "M|G": { + 0: {True: 0.01, False: 0.99}, + 1: {True: 0.5, False: 0.5}, + 2: {True: 0.99, False: 0.01} + } +} + + +def main(): + if len(sys.argv) != 2: + sys.exit("Usage: python heredity.py data.csv") + people = load_data(sys.argv[1]) + probabilities = initialize_probabilities(people) + calculate_probabilities(people, probabilities) + print_probabilities(people, probabilities) + + +def load_data(filename: str) -> Dict[str, dict]: + """ + Load gene and trait data from a file into a dictionary. + File assumed to be a CSV containing fields name, mother, father, trait. + mother, father must both be blank, or both be valid names in the CSV. + trait should be 0 or 1 if trait is known, blank otherwise. + """ + data = dict() + with open(filename) as file: + reader = csv.DictReader(file) + for row in reader: + name = row["name"] + data[name] = { + "name": name, + "mother": row["mother"] or None, + "father": row["father"] or None, + "trait": (True if row["trait"] == "1" else + False if row["trait"] == "0" else None) + } + return data + + +def initialize_probabilities(people: dict) -> Dict[str, dict]: + return { + person: { + "gene": { + 0: 0, + 1: 0, + 2: 0 + }, + "trait": { + True: 0, + False: 0 + } + } + for person in people + } + + +def calculate_probabilities(people: dict, probabilities: dict) -> None: + """ + Calculate the probabilities for each person having a gene and/or trait. + """ + names = set(people) + for have_trait in powerset(names): + # Check if current set of people violates known information + if evidence_violated(people, have_trait): + continue + # Loop over all sets of people who might have the gene + for one_gene in powerset(names): + for two_genes in powerset(names - one_gene): + # Update probabilities with new joint probability + p = joint_probability(people, one_gene, two_genes, have_trait) + update(probabilities, one_gene, two_genes, have_trait, p) + # Ensure probabilities sum to 1 + normalize(probabilities) + + +def powerset(s: Set[str]) -> List[Set[str]]: + """ + Return a list of all possible subsets of set s. + """ + s = list(s) + return [ + set(s) for s in itertools.chain.from_iterable( + itertools.combinations(s, r) for r in range(len(s) + 1) + ) + ] + + +def evidence_violated(people: dict, trait: set) -> bool: + """ + Check if the current set of people with the trait violates known information. + """ + return any(people[person]["trait"] != (person in trait) + for person in people + if people[person]["trait"] is not None) + + +def joint_probability(people: dict, one_gene: set, two_genes: set, trait: set) -> float: + """ + Compute and return a joint probability. + + The probability returned should be the probability that + * everyone in set `one_gene` has one copy of the gene, and + * everyone in set `two_genes` has two copies of the gene, and + * everyone not in `one_gene` or `two_gene` does not have the gene, and + * everyone in set `have_trait` has the trait, and + * everyone not in set` have_trait` does not have the trait. + """ + probability = 1 + + for person in people: + num_genes = 1 if person in one_gene else 2 if person in two_genes else 0 + have_trait = True if person in trait else False + + mother = people[person]["mother"] + father = people[person]["father"] + + if not mother and not father: + probability *= P["G"][num_genes][True] * P["T|G"][num_genes][have_trait] + else: + parents = dict() + for parent in [mother, father]: + n_genes = 2 if parent in two_genes else 1 if parent in one_gene else 0 + parents[parent] = P["M|G"][n_genes] + + if num_genes == 0: + probability *= parents[mother][False] * parents[father][False] + elif num_genes == 1: + probability *= ( + parents[mother][True] * parents[father][False] + + parents[father][True] * parents[mother][False] + ) + else: + probability *= parents[mother][True] * parents[mother][True] + + probability *= P["T|G"][num_genes][have_trait] + + return probability + + +def update(probabilities: dict, one_gene: set, two_genes: set, trait: set, p: float): + """ + Add to `probabilities` a new joint probability `p`. + Each person should have their "gene" and "trait" distributions updated. + Which value for each distribution is updated depends on whether + the person is in `have_gene` and `have_trait`, respectively. + """ + for person in probabilities: + num_genes = 1 if person in one_gene else 2 if person in two_genes else 0 + probabilities[person]["gene"][num_genes] += p + probabilities[person]["trait"][person in trait] += p + + +def normalize(probabilities: dict) -> None: + """ + Update `probabilities` such that each probability distribution + is normalized (i.e., sums to 1, with relative proportions the same). + """ + for person in probabilities.copy(): + for field in probabilities[person]: + total = sum(probabilities[person][field].values()) + for value in probabilities[person][field]: + normalized = probabilities[person][field][value] / total + probabilities[person][field][value] = normalized + + +def print_probabilities(people: dict, probabilities: dict) -> None: + for person in people: + print(person) + for field in probabilities[person]: + print(f" {field.capitalize()}:") + for value in probabilities[person][field]: + p = probabilities[person][field][value] + print(f" {value}: {p:.4f}") + + +if __name__ == "__main__": + 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