Bayesian Probability - two criminals, blood, and various things

In summary, the conversation discusses a problem involving calculating the odds ratio for a Bayesian probability scenario. The problem involves two criminals leaving blood traces at a crime scene and a suspect, Oliver, being tested and found to have type 'O' blood. The data suggests that there is a 60% chance that someone from the general population would leave 'O' type blood at the scene, and a 1% chance for 'AB' type blood. The conversation delves into the equation for calculating the odds ratio, with one person arguing that the prior information is missing and cannot be arbitrarily assigned. The expert clarifies that ignoring the prior information is a form of the prosecutor's fallacy.
  • #1
Astrofiend
37
0

Homework Statement



I am trying to work out what I believe to be a fairly simple 'odds ratio' problem using Bayesian probability for my physics class. Here is the statement of the problem:

"Two criminals have left traces of their own blood at the scene of a crime. A suspect,
Oliver, is tested and found to have type `O' blood. The blood groups of the two traces
are found to be of type `O' (a common type in the local population, having frequency
60%) and of type `AB' (a rare type, with frequency 1%). Do these data (type `O' and
`AB' blood were found at scene) give evidence in favour of the proposition that Oliver
was one of the two people present at the crime?
(b) What are the odds that Oliver was one of the two people at the scene of the
crime? Is the evidence in favor of this hypothesis?
(c) Suppose Oliver had blood type `AB'. What is the odds ratio now?"

I have tried to calculate part b as below:

Homework Equations



The odds ratio comparing two hypotheses is given by:

[tex]

O_1_2 = \frac{p(M_1 \mid D,I)}{p(M_2 \mid D,I)} = \frac{p(M_1 \mid I) p(D \mid M_1,I) }{p(M_2 \mid I) p(D \mid M_2,I} = \frac{p(D \mid M_1,I)}{p(D\mid M_2,I)}

[/tex]

Where I have applied this by making M1 the hypothesis that Oliver was present at the crime scene, M2 is the counter hypothesis, D is the data that 'o' type blood was present at the scene and I - the prior information, I'm not sure about.

The Attempt at a Solution



So, I equate [tex] p(D \mid M_1,I) [/tex] to 1, reasoning that, assuming o type blood is left by the criminals, if oliver is there he will leave o type blood.

I equate [tex]p(D \mid M_2,I)[/tex] to 0.6, reasoning that if oliver isn't there, then there is a 60% chance that someone from the general population will leave o type blood.

But I'm not sure what to do next. There is no information in the question that relates to the 'prior' I, which surely comes into this analysis as [tex]p(M_1 \mid I)[/tex] and [tex]p(M_2 \mid I)[/tex] in the equation. This would relate to the likelihood of Oliver being at the scene essentially by random chance as a member of the general population I think, but there is no such info in the question.

Am I completely off track? Am I trying to analyse with the wrong equation? Have I made errors of reasoning? Any help would be much appreciated!
 
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  • #2
You can't arbitrarily say [itex]p(M_1|I)=p(M_2|I)[/itex], which is exactly what you did in here:
Astrofiend said:
[tex]O_1_2 = \frac{p(M_1 \mid D,I)}{p(M_2 \mid D,I)} = \frac{p(M_1 \mid I) p(D \mid M_1,I) }{p(M_2 \mid I) p(D \mid M_2,I} = \frac{p(D \mid M_1,I)}{p(D\mid M_2,I)}[/tex]
Consider the case of minimal knowledge: Somebody alive somewhere in the world committed the crime, meaning there are 7 billion equiprobable suspects. In this case, [itex]P(M_1|I) = 1/(7e9)[/itex] and [itex]P(M_2|I)=1-P(M_1|I)\approx 1[/itex]. You wiped out a factor of 1.4×10-10. Ignoring this factor is a form of the prosecutor's fallacy.
 
  • #3
Thanks for that - yep, I had come around to that conclusion. What about the other conditional probability assignments that I made? Are they valid?
 

What is Bayesian Probability?

Bayesian Probability is a statistical approach to calculating the likelihood of an event occurring based on prior knowledge or information.

How does Bayesian Probability apply to two criminals?

In the context of two criminals, Bayesian Probability can be used to determine the probability of each criminal committing a crime based on prior information such as their criminal record, motive, and alibi.

How can Bayesian Probability be used in blood analysis?

Bayesian Probability can be used in blood analysis to determine the probability of a particular blood sample belonging to a specific individual based on prior information such as DNA profiles and blood type.

What other scenarios can Bayesian Probability be applied to?

Bayesian Probability can be applied to a wide range of scenarios, such as stock market predictions, medical diagnosis, and weather forecasting.

What are the advantages of using Bayesian Probability?

One of the main advantages of Bayesian Probability is its ability to update probabilities as new information is obtained, making it a more accurate and flexible approach compared to traditional statistical methods.

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