Editing Bayes' theorem (section)

== Use in genetics ==
In genetics, Bayes' rule can be used to estimate the probability that someone has a specific genotype. Many people seek to assess their chances of being affected by a genetic disease or their likelihood of being a carrier for a recessive gene of interest. A Bayesian analysis can be done based on family history or [[genetic testing]] to predict whether someone will develop a disease or pass one on to their children. Genetic testing and prediction is common among couples who plan to have children but are concerned that they may both be carriers for a disease, especially in communities with low genetic variance.<ref>{{cite journal|last1=Kraft|first1=Stephanie A|last2=Duenas|first2=Devan|last3=Wilfond|first3=Benjamin S|last4=Goddard|first4=Katrina AB|author-link4=Katrina A. B. Goddard|date=24 September 2018|title=The evolving landscape of expanded carrier screening: challenges and opportunities|journal=[[Genetics in Medicine]]|volume=21|issue=4|pages=790–797|doi=10.1038/s41436-018-0273-4|pmc=6752283|pmid=30245516}}</ref>

=== Using pedigree to calculate probabilities ===
{| class="wikitable"
!Hypothesis
!Hypothesis 1: Patient is a carrier
!Hypothesis 2: Patient is not a carrier
|-
!Prior Probability
|1/2
|1/2
|-
!Conditional Probability that all four offspring will be unaffected
|(1/2) &sdot; (1/2) &sdot; (1/2) &sdot; (1/2) = 1/16 
|About 1
|-
!Joint Probability
|(1/2) &sdot; (1/16) = 1/32 
|(1/2) &sdot; 1 = 1/2
|-
!Posterior Probability
|(1/32) / (1/32 + 1/2) = 1/17
|(1/2) / (1/32 + 1/2) = 16/17
|}
Example of a Bayesian analysis table for a female's risk for a disease based on the knowledge that the disease is present in her siblings but not in her parents or any of her four children. Based solely on the status of the subject's siblings and parents, she is equally likely to be a carrier as to be a non-carrier (this likelihood is denoted by the Prior Hypothesis). The probability that the subject's four sons would all be unaffected is 1/16 ({{frac|1|2}}&sdot;{{frac|1|2}}&sdot;{{frac|1|2}}&sdot;{{frac|1|2}}) if she is a carrier and about 1 if she is a non-carrier (this is the Conditional Probability). The Joint Probability reconciles these two predictions by multiplying them together. The last line (the Posterior Probability) is calculated by dividing the Joint Probability for each hypothesis by the sum of both joint probabilities.<ref name="Ogino et al 2004">{{cite journal |last1=Ogino |first1=Shuji |last2=Wilson |first2=Robert B |last3=Gold |first3=Bert |last4=Hawley |first4=Pamela |last5=Grody |first5=Wayne W |title=Bayesian analysis for cystic fibrosis risks in prenatal and carrier screening |journal=Genetics in Medicine |date=October 2004 |volume=6 |issue=5 |pages=439–449 |doi=10.1097/01.GIM.0000139511.83336.8F |pmid=15371910 |doi-access=free }}</ref>

=== Using genetic test results ===
Parental genetic testing can detect around 90% of known disease alleles in parents that can lead to carrier or affected status in their children. Cystic fibrosis is a heritable disease caused by an autosomal recessive mutation on the CFTR gene,<ref>"Types of CFTR Mutations". Cystic Fibrosis Foundation, www.cff.org/What-is-CF/Genetics/Types-of-CFTR-Mutations/.</ref> located on the q arm of chromosome 7.<ref>"CFTR Gene – Genetics Home Reference". U.S. National Library of Medicine, National Institutes of Health, ghr.nlm.nih.gov/gene/CFTR#location.</ref>

Here is a Bayesian analysis of a female patient with a family history of cystic fibrosis (CF) who has tested negative for CF, demonstrating how the method was used to determine her risk of having a child born with CF: because the patient is unaffected, she is either homozygous for the wild-type allele, or heterozygous. To establish prior probabilities, a Punnett square is used, based on the knowledge that neither parent was affected by the disease but both could have been carriers:
{| class="wikitable" style="text-align:center;"
! {{diagonal split header|<br /><br />Father|Mother}}
!W
Homozygous for the wild-<br />type allele (a non-carrier)
!M
Heterozygous<br />(a CF carrier)
|-
!W
Homozygous for the wild-<br />type allele (a non-carrier)
|WW
|MW
|-
!M
Heterozygous (a CF carrier)
|MW
|MM

(affected by cystic fibrosis)
|}
Given that the patient is unaffected, there are only three possibilities. Within these three, there are two scenarios in which the patient carries the mutant allele. Thus the prior probabilities are {{frac|2|3}} and {{frac|1|3}}.

Next, the patient undergoes genetic testing and tests negative for cystic fibrosis. This test has a 90% detection rate, so the conditional probabilities of a negative test are 1/10 and 1. Finally, the joint and posterior probabilities are calculated as before.
{| class="wikitable" style="text-align:center;"
!Hypothesis
!Hypothesis 1: Patient is a carrier
!Hypothesis 2: Patient is not a carrier
|-
!Prior Probability
|2/3
|1/3
|-
!Conditional Probability of a negative test
|1/10
|1
|-
!Joint Probability
|1/15
|1/3
|-
!Posterior Probability
|1/6
|5/6
|}
After carrying out the same analysis on the patient's male partner (with a negative test result), the chance that their child is affected is the product of the parents' respective posterior probabilities for being carriers times the chance that two carriers will produce an affected offspring ({{frac|1|4}}).

=== Genetic testing done in parallel with other risk factor identification ===
Bayesian analysis can be done using phenotypic information associated with a genetic condition. When combined with genetic testing, this analysis becomes much more complicated. Cystic fibrosis, for example, can be identified in a fetus with an ultrasound looking for an echogenic bowel, one that appears brighter than normal on a scan. This is not a foolproof test, as an echogenic bowel can be present in a perfectly healthy fetus. Parental genetic testing is very influential in this case, where a phenotypic facet can be overly influential in probability calculation. In the case of a fetus with an echogenic bowel, with a mother who has been tested and is known to be a CF carrier, the posterior probability that the fetus has the disease is very high (0.64). But once the father has tested negative for CF, the posterior probability drops significantly (to 0.16).<ref name="Ogino et al 2004"/>

Risk factor calculation is a powerful tool in genetic counseling and reproductive planning but cannot be treated as the only important factor. As above, incomplete testing can yield falsely high probability of carrier status, and testing can be financially inaccessible or unfeasible when a parent is not present.