The Kingfisher Manor Mystery

A Bayesian Investigation

When Should We Close the Case? Exploring the Kingfisher Manor Mystery

A version of this Bayesian mystery appears in Twisted Logic: Puzzles, Paradoxes, and Big Questions, by Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.

DISCOVERING THE BODY

Set on the brooding Moors of Southwest England stands Kingfisher Manor, a stately and imposing presence with a storied past. Its hallways and long corridors tell tales of intrigue and menace, and its ancient walls hold secrets that have never been exposed. This is a tale of one fateful summer’s afternoon, on which the manor’s eerie silence was shattered by the discovery of Lord Montgomery-Newton, a renowned archaeologist known for his documentation of the secrets of the Egyptian Pharaohs. He lay lifeless on the floor, grasping to his chest an ancient Egyptian amulet, believed by some to hold mystical and medical powers.

News was quick to spread of the tragic fate of the noted academic and within the hour Detective Inspector Anna Marchbank was at the scene, ready and eager to unravel the mystery that lay before her.

UNVEILING THE SUSPECTS

As the chaos resulting from the initial shock died down, DI Marchbank looked around the room, scanning every detail. The study was home to elegant Edwardian furniture and bookshelves filled with rare volumes and manuscripts. It was already clear that Lord Montgomery-Newton’s death was no accident—the discarded revolver was ample testimony to that.

Turning her attention to the suspects, Marchbank learned that five individuals had been present at the manor on the day of the murder. Each of these exuded some degree of suspicion, their personal secrets intertwining with those of the ancient abode.

Mr. Hadleigh: A loyal servant of the household for over 30 years, Mr. Hadleigh was known for his profound devotion and loyalty towards Montgomery-Newton. However, there were rumours of a grudge stemming from a well-guarded incident in the distant past.

Captain Blackwood: A career soldier with a charming side, Captain Blackwood had recently returned from a secret military operation overseas. Whispers of his involvement in clandestine affairs had long circulated.

Dr. Winterbottom: A well-respected doctor with a fascination for the history of medicine, Dr. Winterbottom’s special interest in the ancient dark arts made her an intriguing suspect.

Miss Sinclair: The beautiful and enigmatic Miss Sinclair was a frequent guest at Kingfisher Manor, although her precise connection to the Montgomery-Newton family was veiled in mystery. The motivation for her ubiquitous presence had raised suspicions in the eyes of some.

Professor Adamant: A distant relative of Mr. Hadleigh, Professor Adamant, an expert in medieval theology, had arrived at the manor just days before the murder. His debonair bearing and Edwardian dress sense marked him out from the crowd.

With the suspects identified, Detective Inspector Marchbank knew that putting together the pieces of this jigsaw would require a sharp mind and a keen focus. She had enough information to know that there was only one person involved in the murder of the noble Lord and started by assigning an equal probability to each of the suspects, assigning them each a 20% chance of being the guilty party.

THE DANCE OF CLUES

The investigation commenced, and DI Marchbank began the process of putting together the fragments of evidence.

TWO HOURS INTO THE INVESTIGATION: ELIMINATING MR. HADLEIGH

The first breakthrough came when an airtight alibi emerged for Mr. Hadleigh. He had been attending a high-profile charity event in the neighbouring village at the time of the murder. The detective swiftly eliminated him from the list of suspects, narrowing down the field to Captain Blackwood, Dr. Winterbottom, Miss Sinclair, and Professor Adamant.

With Mr. Hadleigh’s elimination, the probability of guilt for the remaining four suspects increased to 25% each. Marchbank knew that each subsequent clue would alter these probabilities, moving her closer to the elusive killer.

FOUR HOURS IN: QUESTIONING CAPTAIN BLACKWOOD’S ALIBI

As the investigation progressed, doubts began to emerge regarding Captain Blackwood’s alibi. Witness testimonies conflicted, creating a fog of uncertainty around his whereabouts on the afternoon of the murder. Marchbank sensed a crack in his armour and increased her evaluation of his probability of guilt to 40%.

The detective understood the delicate balance of probabilities, acknowledging the importance of assigning weight to each suspect based on the available evidence. She embraced the Bayesian approach, allowing it to guide her through the labyrinthine twists of the investigation.

SIX HOURS AND 45 MINUTES IN: EXONERATING DR. WINTERBOTTOM

The Detective Inspector was made aware of a crucial piece of evidence that placed Dr. Winterbottom far from the scene of the crime. Reliable witnesses confirmed her presence at a medical conference during the time of the murder, eliminating her as a suspect.

The investigation was gaining momentum, but the truth still eluded them. Marchbank looked for the thread that could potentially unravel the entire tapestry of deception.

THE BAYESIAN BALANCE

As Detective Inspector Marchbank meticulously evaluated the evidence, she was acutely aware that each new piece of evidence needed to be filtered through the prism of prior probabilities.

Analysing the case through this Bayesian lens, she considered the individual probabilities assigned to each suspect. Captain Blackwood had been assigned a probability of guilt of 40%, leaving a 60% chance of it being one of the other remaining suspects. As such, she was able to assign an equal 20% probability of it being Miss Sinclair, Professor Adamant, or Dr. Winterbottom. Now that Winterbottom had been eliminated, her 20% share of the probability needed to be distributed to the other suspects. Critically, the Bayesian approach dictated that Captain Blackwood’s probability of guilt should be adjusted twice as much as the probabilities for the other two suspects in this process, since his prevailing assigned chance of being the culprit (40%) was twice that of each of the others (Miss Sinclair and Professor Adamant) before the Doctor was eliminated.

Marchbank was quick to raise Captain Blackwood’s probability of guilt, therefore, by 10%, to 50%, reflecting the weight of the evidence against him. Simultaneously, she increased the probabilities assigned to Miss Sinclair and Professor Adamant from 20% to 25% each.

SUMMARISING THE INVESTIGATION

To summarise where we have got to, Marchbank identified five suspects, each with potential motives and secrets. Initially, she assigned each a 20% probability of guilt, and then used a Bayesian approach to adjust these probabilities as new evidence emerges.

THE INVESTIGATION: A SERIES OF BREAKTHROUGHS

Eliminating Mr. Hadleigh: An airtight alibi for Mr. Hadleigh removes him from suspicion, increasing the remaining suspects’ probabilities to 25% each.

Doubting Captain Blackwood’s Alibi: Conflicting testimonies about Captain Blackwood raise his probability of guilt to 40%.

Exonerating Dr. Winterbottom: Evidence places Dr. Winterbottom away from the scene, eliminating her as a suspect.

REASSESSING PROBABILITIES

With each clue, Marchbank recalculates the probabilities:

Captain Blackwood’s Increased Chances: After eliminating Dr. Winterbottom, Blackwood’s probability of guilt rises to 50%, reflecting the growing suspicion against him.

Remaining Suspects: Miss Sinclair and Professor Adamant’s probabilities increase to 25% each.

THE TRIAL AND TWISTS OF FATE

The revised probabilities paved the way for the trial of Captain Blackwood, his fate hanging in the balance.

Trapped within the walls of a courtroom, Blackwood was at the mercy of a prosecution barrister who skilfully guided the jury to the trap door of the Prosecutor’s Fallacy. Like so many juries before them, they confused the likelihood that someone is guilty in light of the evidence with the likelihood of observing the evidence if they were guilty. The likelihood that Montgomery-Newton was killed in the study if the Captain was guilty of his murder was naturally rather high, and this led to his conviction. Unfortunately for Captain Blackwood, the relevant probability (that he was guilty of murder given that the great man was killed in the study) was somewhat smaller but bypassed in the deliberations.

It’s certainly true that the evidence was consistent with the Captain’s guilt. Yet it was equally consistent with the guilt of the other suspects. But they were not in the dock! Unfortunately for the Captain, he was. The verdict of guilty was not long in coming.

Trial and Error: The Prosecutor’s Fallacy

The jury had fallen prey to the Prosecutor’s Fallacy, confusing the likelihood of Captain Blackwood’s guilt given the evidence with the likelihood of the evidence if he were guilty. This led to his wrongful conviction.

The Twist

The true killer, Miss Sinclair, evaded justice entirely. Concealed within the pages of an ancient manuscript, she had hidden a letter, a damning piece of evidence linking her to a nefarious smuggling operation. Lord Montgomery-Newton’s discovery of the letter sealed his fate. In a desperate act to protect her secrets, Miss Sinclair had resorted to murder.

Captain Blackwood, a victim of circumstance and statistical misinterpretation, would serve a life sentence for a crime he did not commit. Meanwhile, Miss Sinclair eluded the clutches of justice, disappearing into the shadows as a tax exile in a distant land.

CONCLUSION: THE COMPLEXITY OF TRUTH AND PROBABILITY

Kingfisher Manor, once a place of mystery and intrigue, stands now as a solemn witness to the twists and turns of fate. The echoes of the past whisper through its halls, reminding us of the delicate balance between evidence and probability. More broadly, the Kingfisher Manor mystery underscores the intricacies of evidence interpretation and probability assessment. It highlights the challenges in drawing conclusions from circumstantial evidence and emphasises the critical role of Bayesian reasoning in great investigative endeavours.