I remember watching the Hollywood mystery thriller Knives Out, leaning towards the screen, as if the case were mine to crack. As detective Blanc’s team questions each person at the Thrombey Mansion, I, too, crossed off names in my head, only to reinstate them after a twist or two. Back then, it never struck me that this old-fashioned whodunit was making me do math in my head. While it might seem like a stretch, I strongly feel that Benoit Blanc’s investigative style closely mirrors Bayesian Inference. But those who remember the interrogations in the movie will quickly realize that Benoit Blanc wasn’t even actively interrogating. He was seated beside a piano, letting his team (Lieutenant Elliot and Trooper Wagner) ask questions. Then why do I say that Blanc’s investigative style had anything to do with Bayesian Inference? Blanc himself mentioned this in the movie, and I quote:
“I observe the facts without biases of the head or heart.” (Benoit Blanc, Knives Out [1])
This is the very essence of Bayesian Inference, where your conclusions are not driven by intuition but by evidence. Let’s solve this murder mystery together using Bayesian Inference.
Here’s a quick note before we begin. Throughout the movie, contradictions are presented in two forms. There are contradictions presented in the form of flashbacks, which are shown only to the audience and are mostly unknown to Blanc. Then, there are contradictions revealed by verbal inconsistencies that Blanc witnesses during the investigation. Therefore, we will focus only on the verbal inconsistencies noted by Blanc.
Also, a note on the probability weight assignments and updates. These are not calculated using the Bayesian formula, as likelihood values are difficult to assign to behavioral evidence such as behaving evasively or lying. Instead, we use informed estimates as a teaching tool and not as mathematical proof. So, hope you enjoy this journey.
Setting the Stage — Establishing the Initial Beliefs
Detective Blanc was hired anonymously by a family member to investigate the possibility of Harlan Thrombey being murdered. When his team begins the interrogation, Blanc quietly observes the potential suspects from behind. When the interrogation steers off course, he redirects the team to realign by tapping a piano key.
He observes that each interaction is muddled with lies and contradictions. What he does right is not tossing aside a narrative as being baseless while holding on to another based on gut feeling. He understands that misleading accounts may contain fragments of truth. He carefully assesses each interaction, assigns weights to each observation, and then combines them to arrive at a conclusion. He starts from uncertainty but slowly builds towards the most probable truth, keeping his personal biases aside.
Blanc begins by listing the probable causes of death. In the Bayesian world, this is called a Prior Model. A prior model is the set of assumptions we hold before we have any evidence. In this case, the prior model is the initial hypotheses about Thrombey’s death before the investigation commences.
Assessing the Completeness of Initial Beliefs
Let’s assess the initial beliefs to see if we’ve overlooked any other possibility. Have we overlooked the possibility that this was an attempt to frame someone? If so, should that be included as the sixth hypothesis?
This is where the most important rule (MECE Principle) for formulating a hypothesis in Bayesian Inference comes into play. Each hypothesis formulated as part of Bayesian Inference should be Mutually Exclusive and Collectively Exhaustive (MECE).
Let’s revisit the sixth potential hypothesis, ‘Trying to Frame Someone’. While the chosen hypothesis should answer what might have caused the death, this potential hypothesis talks more about the motive behind the death, provided it is proven that it was a murder. So, it breaks the mutual exclusivity rule of the MECE principle and hence cannot be a direct hypothesis.
Assigning Probabilities (Prior Probabilities)
Let’s stick with the hypotheses we had formulated earlier, as they consider all possible causes of death (collectively exhaustive). The next logical step is to assign probabilities to our initial beliefs. This means we start with an educated guess about how likely each hypothesis is to have caused Harlan Thrombey’s death. Since we assign probabilities before we have any direct evidence or data, we call this the prior probability. The below visual shows us assigning equal weightages to all hypothesis. Let’s assume that these are our prior probabilities for a moment.
A question that naturally comes to our mind is whether each hypothesis carries the same probability of occurring. No, not always. It is a common misconception in Bayesian inference that we must assign equal probability to all hypotheses. In the absence of prior evidence, we assume that Detective Blanc assigns equal probability to each hypothesis. But that’s not always the case.
We may also assume non-uniform (unequal) probabilities if we have prior knowledge suggesting that a hypothesis is more probable than the others. General crime statistics may also be useful for estimating prior probabilities. For instance, according to FBI homicide data [2], it is said that in most homicides, murder victims know their murderer. Homicides by an outsider often require a motive involving burglary or some kind of revenge. Therefore, H4 receives greater weight, as family members have greater access to the victim. Moreover, in Harlan Thrombey’s case, the hypothesis that a family member caused his death carries more weight as his family members could be motivated by the inheritance of his wealth and estate. The ideal prior probabilities in our scenario would be an unequal distribution.
Updating Probabilities based on Evidence
Let’s try to recall the scene where Marta is being interrogated. Marta has a pathological condition that causes her to vomit whenever she lies. But since Marta initially thinks that she caused Thrombey’s death by accidentally switching drugs, she tackles the situation by giving incomplete answers and half-truths.
The twist here is that Detective Blanc is already aware of her condition. Do Marta’s half-baked responses raise suspicion and consequently shift weights? One possibility is that Martha had a motive to kill Mr. Harlan (supporting the outsider theory – H5). Another possibility is that Marta, being the nurse, may have committed a fatal mistake that cost Mr. Thrombey’s life (H2). The Bayesian Likelihood function is helpful in such ambiguous situations. The Bayesian Likelihood Function measures how well each hypothesis explains the observed evidence. Martha’s demeanor is insufficient to distinguish between H2 and H5. So, the probabilities will shift only slightly, not dramatically. Probabilities for H2 and H5 would increase slightly, and those for H1 and H3 would decrease.
An important point to note about probabilities. The moment we get some form of evidence (minor or major) and start updating our weights, we call it posterior probability. Based on the above, we re-assign the probabilities as shown.
From the visual, it is clear that the weights have shifted slightly towards H2 but there is no considerable shift yet.
Simple yet Direct Contradictions — Bayesian Gold
There was a striking contradiction around who was immediately next to Harlan Thrombey during his birthday party. Harlan’s daughter Linda mentioned that she was next to Harlan, along with her husband and son. However, Walt mentioned that he and his family were next to Harlan. While this contradiction may not point to any one individual, it raises suspicion about their collective credibility. This raises weights around H4.
Below are the updated probabilities.
Walt’s Deflection towards Ransom
Lieutenant Elliot asks Walt why Harlan took him aside for a talk and why Walt seemed chastened later on. Walt hesitated for a minute and then deflected the argument to Ransom. He mentioned that Harlan had an argument with Ransom. This suggests that Walt is actively hiding his conversation with Harlan. Let’s reassign the probabilities based on these pieces of evidence.
Mom-Daughter Contradictions
When Blanc’s team asks why Joni came in early, she says she wanted to meet with Harlan about an issue with wiring the school fees for her daughter. But Joni’s daughter, Meg, says that her grandfather, Harlan, never missed wiring money for her school fees. This contradiction greatly increases the probability of H4.
The Will Reading Scene — Refining Your Hypothesis
So far, the weights have been the highest for H4, supporting the theory around murder by a family member. But when we see that all assets have been awarded to the nurse and caretaker, Marta, the entire suspicion shifts to her. The weights almost triple for H5 after this dramatic change in events. The family suspects her of manipulating Harlan to change his will in her name. Below are the updated probabilities.
This is where an important concept called ‘Hypothesis Refinement’ comes into play. Bayesian Inference doesn’t restrict you to sticking with the initial set of hypotheses. Instead, it lets you refine a hypothesis and branch it out when you have more evidence. In this case, H5 (Murder by an outsider) was a broader umbrella term. Now, we can branch into a more granular sub-hypothesis. Our updated hypothesis space and corresponding weights are shown below.
All of a sudden, the family who adored Marta sees her as a prime suspect. However, Blanc still isn’t convinced that Marta had a motive, as the toxicology report shows that Harlan didn’t die due to a morphine overdose. Unlike the family members, Blanc is not reacting on intuition but on evidence. As he follows the trail of evidence, it points him in a different direction, towards Ransom.
The Climax — The Ultimate Probability Shifter
During the investigation, almost every family member (including staff) spoke of a fallout between Ransom Drysdale and his grandfather, Harlan, causing Ransom to storm out of the party earlier than expected. In addition, Ransom not being present the day after Harlan’s death served as more evidence. However, the motive remained unclear until Ransom arrived on the day the will was being read. Jacob, another grandson of Harlan mentioned that he overheard Ransom saying ‘The Will’ and ‘I’m warning you’ to his grandfather before storming out. When confronted by his family, Ransom admitted that he already knew that he was cut out of the will. Detective Blanc, who was observing all this, realized that this may be Ransom’s motive to kill Harlan. Based on this evidence, we update our hypotheses. Since H4 (Murder by a family member) is a broader umbrella term, we branch into a more granular sub-hypothesis. Our updated hypothesis space and corresponding weights are shown below.
Notice how the likelihood of Marta being the killer drops drastically based on new evidence that the toxicology report didn’t show a morphine overdose, and the fact that Ransom was angry that he was not included in the will. The posterior shifts as and when solid evidence arrives. This is what makes Bayesian so intuitive. Being based on Conditional Probability, it asks the most honest question ‘Given everything I know so far, what is the most probable answer?’.
In the above diagram, notice how Marta’s probabilities plummet once in a while, while Ransom’s probabilities skyrocket towards the end based on new evidence.
Conclusion — Failure to converge to H3?
As we have seen, Knives Out serves as a great example to illustrate reasoning under uncertainty, which is essentially the underlying premise of Bayesian Inference. Initially, the likelihood of murder by a family member rose as there were contradictions in every conversation. But as new evidence about Marta emerged, suspicion shifted towards her. However, upon Ransom’s arrival and subsequent revelations about his quarrel with Harlan, the probabilities converged onto him. The reality is that Harlan had actually committed suicide to protect Marta, as they both believed that she had given him a lethal dose of morphine. So, is Bayesian Inference failing, as it didn’t converge to H3 (Death by Suicide)? Sometimes, truth can be layered, as in this case, where Ransom switched the drugs on purpose and took away the antidote with the sole intention of causing Harlan’s death. Therefore, while Ransom didn’t physically murder Harlan, he did plan his death. The Bayesian Reasoning approach went deeper than the direct cause of Harlan’s death, which was suicide. When handled with a neutral mind, Bayesian Inference can effectively guide you to the layers buried beneath the surface-level truth.
References
[1] The Official Transcript of Knives Out by Director Rian Johnson