A Model For The Probability Of Guilt In The Age Of #MeToo

The case of journalist-turned-government minister MJ Akbar in India got me thinking about the probability of guilt when a man is independently accused of sexual harassment by a number of different women. At last count, 16 different women have come forward as part of the #MeToo wave sweeping India, to accuse Akbar of various kinds of sexual misconduct.

At the time of writing, Akbar has chosen to brazen it out by filing a criminal defamation suit against the first woman who made her accusation public.

This case has since become a cause célèbre in India, with commentators taking various positions on it.

I personally believe that the weight of credibility is on the side of Akbar's accusers, but I was wondering whether there was a way to approach such a situation more objectively, even quantitatively. After all, each individual accusation could be dismissed as a case of "He said, she said", but put together, the accusations build up a more compelling argument.

How compelling?

Let's start with the simplest case. A man is accused of misbehaviour by a woman. He denies it. Neither of them has any evidence, or any witness, to back up their claim. Prima facie, both of them seem equally credible. What should an objective observer think?

With no further input and with no biases, an objective person would have to conclude that there is a 50% probability that the woman is telling the truth, and therefore a 50% probability that the man is guilty. The corollary is that there is a 50% probability that he is innocent.

Now, this obviously contradicts the legal principle that an accused person is 100% innocent until proven guilty, but this is not a court of law, where evidence is required beyond the shadow of a doubt before an accused can be convicted of a crime. This is a thought experiment. We are trying to compute what we can adopt as a working hypothesis, based on the balance of probabilities.

In graphical terms, this is what the picture looks like when a man is accused by just one woman:

Now let's make this more interesting. Let's say a second woman steps forward to make a similar accusation, based on an independent incident. And let's say he denies this accusation too.

Does our view of the man's guilt change? Do we still think it's just his word against his accusers', and therefore that he's no more guilty than we thought before? Or do we start to doubt him a little more now? If so, by how much?

My model is that each subsequent accusation halves the probability of his innocence. And so, this is what the picture should look like:

It's important to note that we are not treating the three claims as completely independent, and assigning a 33.33% probability to each. That's because the claims of the two women, while independent, nevertheless reinforce each other. The accusation of the second woman halves the probability of innocence that we granted him after the first accusation. So we assign a probability of only 25% to his innocence now, not 33.33%.

What happens when a third woman steps forward? It's easy to follow the logic now:

With three independent accusers, the man's probability of innocence reduces to 12.5%, notwithstanding his protestations.

Stated in mathematical terms, the probability of a man's guilt when independently accused by 'N' women is given by

P_g = 1 - (½)^N

When N = 1, P_g = 0.5, or 50%
When N = 2, P_g = 0.75, or 75%
When N = 3, P_g = 0.875, or 87.5%

Let's apply this model of guilt to the MJ Akbar case. The number of his accusers now stands at 16:

1. Priya Ramani
2. Ruth David
3. Majlie de Puy Kamp
4. Saba Naqvi
5. Ghazala Wahab
6. Shutapa Paul
7. Shuma Raha
8. Kanika Gahlaut
9. Suparna Sharma
10. Prerna Bindra
11. Harinder Baweja
12. Anju Bharti
13. Malini Bhupta
14. Kadambari Wade
15. Swati Gautam
16. Tushita Patel

What's the probability that Akbar is guilty?

P_g = 1 - (½)¹⁶ = 0.999984741210938, or 99.998%

In non-mathematical language, he's toast.

#TimesUp

Update 17/10/2018: MJ Akbar has resigned as Minister of State for External Affairs, after 20 women expressed their desire to testify in court against him. Three of them were from the original list (Tushita Patel, Kanika Gahlaut and Suparna Sharma), but the other 17 were all new, bringing the total number of his accusers to 33.

17. Meenal Baghel
18. Manisha Pande
19. Ramola Talwar Badam
20. Kaniza Gazari
21. Malavika Banerjee
22. A.T. Jayanthi
23. Hamida Parkar
24. Jonali Buragohain
25. Sanjari Chatterjee
26. Meenakshi Kumar
27. Sujata Dutta Sachdeva
28. Hoihnu Hauzel
29. Reshmi Chakraborty
30. Kushalrani Gulab
31. Aisha Khan
32. Kiran Manral
33. Christina Francis

Don't even bother calculating the probability of his guilt now.
(If you're curious, it's 99.99999999%.)

Light And Dark Supernatural Teen Dramas - "Sabrina The Teenage Witch" And "Buffy The Vampire Slayer"

[Some spoilers ahead]

I've been enjoying watching Sabrina The Teenage Witch, and as I near the end of Season 3, I'm struck by some remarkable parallels with another serial I saw not too long ago - Buffy The Vampire Slayer.

Two shows with an intended audience of teenage girls have been strangely fascinating to a middle-aged man...

Both feature spunky teenage girls with powers out of the ordinary, and the serials show them dealing simultaneously with challenges from two worlds, the human and the supernatural. High school can be a fairly big deal by itself to a teenager, so the discovery of supernatural powers and the responsibilities that go with them is an additional burden. Yet the young ladies manage admirably.

Obviously, the two serials are very different in tone. Sabrina is pure lighthearted comedy. Buffy is darker, with a constant backdrop of evil, and many characters die, but it's not without some humour of its own.

The lead character

Sabrina Spellman (Melissa Joan Hart);

Buffy Summers (Sarah Michelle Gellar)

Both serials are almost entirely carried on the diminutive shoulders of their heroines. Sabrina is a cheerful, upbeat girl with a quick wit and a ready excuse for every inexplicable situation she finds herself in. Buffy belies her tiny size and packs a deadly wallop in a fight, not to mention a sharp tongue in everyday banter.

The high school

Westbridge High;

Sunnydale High

I guess American public schools all look alike, with their classrooms, corridors and student lockers where most of the action in the serials takes place. Both schools see their share of strange happenings, although Sunnydale High, being situated right on top of the Hellmouth, is in an arguably worse situation.

Family / adult supervision at home

Hilda and Zelda Spellman (Caroline Rhea and Beth Broderick);

Joyce Summers (Kristine Sutherland)

Sabrina's aunts provide a nice foil to each other, with the elder Zelda being (or trying to be) the more responsible one, and Hilda being almost as irresponsible as Sabrina herself. Buffy's home situation is far less humorous, with single mum Joyce doing her best to be supportive and strict at the same time.

The best (female) friend

Valerie Birkhead (Lindsay Sloane);

Willow Rosenberg (Alyson Hannigan)

Sabrina had a BFF in Season One called Jenny Kelley (Michelle Beaudoin), who disappeared mysteriously and was thereafter replaced by the angst-ridden Valerie Birkhead. Her closest parallel in the Buffy serial is Willow. Willow develops some magical abilities over the course of the series, but Valerie remains purely mortal. Indeed, she doesn't even know that Sabrina is a witch.

The mean girl

Libby Chessler (Jenna Leigh Green);

Cordelia Chase (Charisma Carpenter)

I guess the high school experience for girls isn't complete without at least one mean girl. In Sabrina's case, it's cheerleader Libby with her constant airs and putdowns ("Freak!"). Buffy has to endure the snob Cordelia ("I mean, they promised me they'd take me to St. Croix, and then they just decide to go to Tuscany. Art and buildings? I was totally beachless for a month and a half. No one has suffered like I have. Of course I think that that kind of adversity builds character. Well, then I thought, I already have a lot of character. Is it possible to have too much character?").

But both heroines know how to give it back.

Libby (as Sabrina approaches her table): Looking for the loser convention?
Sabrina: I am...and look...I've found it!

Cordelia: You're really campaigning for bitch of the year, aren't you?
Buffy: As defending champion, you nervous?

Buffy: Well, that works out great. You won't tell anyone that I'm the Slayer, and I won't tell anyone you're a moron.

The vindictive school authority

Vice-Principal Willard Kraft (Martin Mull);

Principal R Snyder (Armin Shimerman)

Sabrina is shown to be a straight-A student, and such kids are usually favourites of their teachers and principals. But for some strange reason, her vice-principal Willard Kraft is unreasonably suspicious of her and keeps looking for excuses to give her detention slips. Buffy's principal Snyder probably has more reason for his extreme lack of trust in her. After all, she had started a fire in her previous school. [Later on, it turns out that Snyder knows much more about Buffy than anyone suspected.]

The portal to the supernatural realm

The linen closet;

the Hellmouth

For Sabrina and her family, stepping across to the "Other Realm" is easy. They just walk into the linen closet at home and close the door behind them. A crash of thunder and a flash of lightning from under the door tell us they've crossed over. Occasionally, unwelcome visitors from the Other Realm enter the house through the same doorway. Buffy's Hellmouth is a much nastier business, with vampires and worse crossing over to our human world with alarming regularity. The school is situated right over the Hellmouth, and this is taken as a matter-of-fact explanation for why many strange things keep happening.

The mentor

Salem Saberhagen (voiced by Nick Bakay);

Rupert Giles (Anthony Head)

This parallel may be a bit of a stretch. Salem is a "witch familiar", cursed to take the form of a cat for a hundred years, all for the crime of plotting to take over the world, an ambition he has never quite given up. Nevertheless, he comes to Sabrina's aid from time to time with advice based on his years of witching knowledge. Giles is Buffy's "Watcher" and is responsible for her training. While Giles takes his job extremely seriously, Salem would probably run from any such responsibility, sobbing in his trademark way if forced to take it on.

The (male) classmate and/or boyfriend

Harvey Kinkle (Nate Richert);

Xander Harris (Nicholas Brendon), Angel (David Boreanaz), and Spike (James Marsters)

There is no clean parallel between the Sabrina and Buffy serials in this department. Harvey is Sabrina's classmate as well as her boyfriend. Xander initially wants to be Buffy's boyfriend but ends up being just her classmate and friend. At various times, the vampires Angel and Spike end up being Buffy's boyfriends. The big difference is that Harvey remains innocent of the realisation that Sabrina is a witch, whereas all of Buffy's male companions know that she's a slayer.

I wonder how many of these parallels were deliberate. After all, both these serials were made and aired at around the same time. Sabrina ran from 1996 to 2000, while Buffy ran from 1997 to 2001.

The Comics That Made Me What I Am (A Nerd In The Very Best Sense)

A few days ago, I received a pack of DVDs from an online seller that made me delirious with joy. The DVDs contained scanned comic books from the 60s and 70s (the so-called Silver Age), and among these were many that I had owned when I was a child.

Going through this set gave me many nostalgic moments, and I realised that my very way of thinking has been profoundly altered by their influence.

My favourite comics were not of the light, ha-ha funny kind. Many of my friends enjoyed reading Archie, Richie Rich, Donald Duck, Sad Sack, and the like. My favourites were Science Fiction and Superhero comics.

Being middle-class, my parents couldn't afford to indulge my every request, but they did occasionally buy me the comics I asked for, and these gave me so much pleasure that I read them over and over, memorising the dialogues even when they made little sense at the time.

In those days, Indian schoolteachers were strongly discouraging of comics. The received wisdom was that "comics spoil your English". Our teachers wanted us to read classics in book form instead. But I maintain that I learnt a lot more from comics. I not only built up an impressive vocabulary at a young age, but also learnt a number of scientific concepts long before they made their formal appearance in my classroom. That's why I would encourage all parents to give their children Science Fiction comics from a very early age, and indulge them in their fantasies. My father was quite indulgent, and often laughed along as I made up my own scenarios and dialogues using scientific and pseudo-scientific terminology.

Here are some examples of what I read:

Doctor Solar - Man of the Atom

One of my first SF comics, if not the very first. This was published in 1967, and I got my hands on it sometime in 1969, I believe. I was six years old then.

In retrospect, the Doctor Solar comics were a bit amateurish, because the hero's powers were not well-defined, and he could apparently do anything. Still, this comic familiarised me with a number of scientific terms that I would learn the meanings of only much later.

For a 6 year old, my vocabulary began to expand in strange ways.

"Outer Space" became a common word in my vocabulary, and this was not the only comic to use it

"Atomic radar vision"? Oh, well.

Germs, viruses, contaminated

An isotope of Uranium

The speed of light

Vaporized lead, orbits, spores

Atomic artillery, laser beams, matter, energy

Short-circuit

Molecules, absolute zero

High-frequency sound, airless void

"No sound can pass through an airless void"

Expansion can explode vessels like steam overloads a boiler

Cosmic storm

Looking back on this five decades later, I'm in awe of how much this one issue taught me. I was mesmerised by the story, the superhero and his powers, and all the scientific terms that were casually thrown about. I think this was the comic that laid the foundation for my later career choices in favour of engineering and computer technology, and my lifelong interest in science in general.

Magnus Robot Fighter

I had two comics of Magnus Robot Fighter, and both were mind-blowing. Looking back now, I realise that it was Russ Manning's visionary artwork that brought a future world to life in such breathtaking and convincing detail.

Bunda the Great

This was a story about a robot that grows too big for its own boots, even demanding to be treated as a god until Magnus brings it down.

As I said before, the artwork in this series was simply spectacular.

The drawings were so detailed and consistent that one could form a mental picture of the future society called "North Am".

Magnus defeats the mighty Bunda that weighs four tons. It crashes to its death because it does not contain anti-gravity units!

There is an element of religion here, and perhaps my first exposure to the idea that religion is just quackery.

The birth of a new religion...

...and its debunking by science.

Robot Ghost

Another beautiful Magnus story was of a group of invincible robots surrounded by a force field, robots built by a scientist to exact revenge on society after his death. The robot Nadmot filled me with delicious chills. It became my favourite villain for a while. The only way to defeat it was by trickery.

Note the beauty of Russ Manning's artwork, and the imagery it conjures up.

Guard-robs and riot-robs

The guard-robs prove ineffective against the evil robots' force-field

I had memorised this sentence so well that older kids in my school bus would ask 7-year-old me to recite it for their amusement: "Even Magnus's steel-smashing strength is unable to penetrate the mysterious barrier that veils the evil robots"

Ultimately, Magnus fools the evil robots with an illusion...

...and sends them off on a one-way trip into deep space. Once again, look at Russ Manning's depiction of the world of the future.

Space Family Robinson - Lost in Space

Much later, I learnt that this was a futuristic take on the classic adventure story "Swiss Family Robinson". The issue I had was called "Attack of the Plant Creatures".

Once again, the artwork in this comic conjured up a very romantic image, this time of space travel in general. 

How lovely to be able to live in a home like this!

Ooh, laser pistols! And they can burn holes through a ship's casing unless you set them for close range!

The notion of vacuum has never been explained more graphically

How nice it would be to live in a space station! Sigh.

Voyage to the Bottom of the Sea

Closer home, the adventures of a nuclear submarine called the Seaview captured my imagination. I had two comics in this series. The first was about an intelligent dolphin called Star that organised the creatures of the sea against humans.

The Emperor of the Oceans

The artwork here was fabulous, too, and it gave me chills.

It was scary to see whales hijacking a submarine. The reddish light in the submarine's cabin gave it an atmosphere of crisis.

The Life and Death of the Seaview

This issue was even more dramatic. It introduced something called the Moho fluid, believed to be responsible for all life on earth. But a subterranean explosion opens up an underwater reservoir of the Moho, and the Seaview is forced at one point to fill its ballast tanks with the fluid. That turns the Seaview itself into a living creature.

The artwork in this issue was nothing short of magnificent. I was by turns fascinated and terrified by the underwater scenes it depicted.

I wouldn't want to be caught in something like this!

Not to mention being inside something like this!

This was one of the most thrilling scenes for me - how the Seaview was taken back

And the thrilling race back to the surface, because there's a time bomb that will seal them in along with the Moho fluid if they don't make it out in time!

Admiral Nelson of the Seaview was one of my early heroes, and it gave me a thrill later on in history class when I learnt about the real Admiral Horatio Nelson

And the mandatory philosophy lesson right at the end.

I simply loved submarines, thanks to the adventures of the Seaview

Superman

The School for Superman Assassins

One of my aunts bought this issue for me, and I remember this with gratitude.

Superhero comics were a slightly different experience from Science Fiction comics, but there was still a great element of science fiction in them.

In those early issues, it was necessary to remind readers that Superman's secret identity was Clark Kent. 

Since this was my very first Superman comic, I certainly needed to be told that.

This was my first introduction to the notion of matter and anti-matter...

...and to the notion of time travel. I would read HG Wells's classic 'The Time Machine' only much later.

"Friction causes heat" - I learnt this from a Superman comic long before I heard about it in any science class

In-between some humour, a number of concepts were taught

Again, long before I learnt Newton's Third Law in a classroom, Superman had already taught me that "the physics principle behind rocketry is reaction"

This is just a sample of what my eclectic childhood was like. I grew up on a university campus, with academics for parents. I would never trade my childhood for anyone else's. And a large part of the magic of that childhood was thanks to the wonderful Science Fiction and Superhero comics I grew up with.

Once again, I would exhort parents to give their children the gift of this magic. Five decades later, I'm still tingling with excitement to re-read these beloved stories!