I often hear of denialism dressed up as skepticism, such as “I’m a climate change skeptic“’ or of late “I’m a COVID19 skeptic“’, but no, that’s not skepticism, that’s denialism, there’s no better way to put it. I enjoy statistics, I try to live as far as possible, by evidence-based reasoning, but I’m human, I’m prone to pet theories and preconceived notions like anyone else. I found great solace in Bayes’ theorem in giving me science and philosophy to balance out my human failings with my desire for facts and reason. Today I want to share this with you.
Frequentist statistics looks at a bunch of observations and make an inference about them. The null hypotheses usually go with the status quo. We humans though have preconceived notions. I’m convinced that my dog is giving me the cold shoulder because I petted another one first, I have these feelings without knowing anything about the research into animal behavior. Are dogs really capable of such thought? If scientists provide me with evidence to the contrary, should I not change my mind, despite my strong internal feelings? Bayes’ theorem is more than a formula in a textbook, the very idea of rationality and changing one’s mind based on the evidence presented, is baked into it. The formula at its simplest derives very quickly from well-known laws of conditional probability.
That is the formula, where A and B are two events. P(A) and P(B) are the probabilities of occurrence of events A and B respectively. P(A|B) is the probability of event A occurring given event B occurred and P(B|A) is the probability of event B occurring given event A occurred. The beauty of Bayes’ theorem is that you can think of the formula this way.
The prior represents your belief that A will occur (or that A is true) before being presented the evidence, the event B. P(B|A) is the likelihood of the evidence occurring if A is true, and P(B) is the overall probability of the evidence occurring. P(A|B) is how you revise your estimate that A is true, given the new evidence. Not obvious? let’s talk of a little example.
Your friend has a coin, he claims it is a loaded coin, and using it will give you an edge if you call heads. He says this coin will turn up heads, 75 percent of the time. You don’t really believe your friend; he says all sorts of stuff. So, you have a preconceived notion that this coin is probably fair. You assign a 90% probability to the coin being fair. Fair enough!
Now your friend knows that you tend to get convinced by the evidence. He asks you to toss the coin 20 times and make up your own mind. You agree, toss it 20 times and now it turns up heads 14 out of 20 times. Now what? Do you believe your friend that the coin is loaded? Let’s keep the model simple and avoid integrating over a continuous range of possibilities by assuming that the coin is either totally fair or loaded to turn up heads 75 percent of the time.
Now let Fair be the event that the coin is fair and Fourteen is the event that you observe 14 heads in 20 tosses. You initially thought that P(Fair) = 0.9 and P(Loaded) = 1 – P(Fair) = 0.1 and but you know Bayes’ theorem, you know that
So, the probability that the coin is fair has to be revised after observing Fourteen. Our prior probability gets multiplied by a ratio. The numerator of the ratio is P(Fourteen|Fair), that is the probability you will observe fourteen heads out of twenty given a fair coin. Thankfully this is easy to compute given it follows the binomial distribution. (You can read about it here, but it isn’t necessary to understand it for the purpose of this discussion)
But what is P(Fourteen)? that is the total probability of getting 14 heads out of twenty tosses. Well, there are two ways this could have happened, either the coin was fair and we observed fourteen heads out of twenty tosses, or the coin was loaded and we observed 14 heads out of twenty tosses. In both cases it is possible to observe 14 heads! Just with different probabilities.
So,
This is the law of total probability. Again, from the binomial probability distribution, P(Fourteen|Loaded) = 16.9% and from the law of total probability,
P(Fourteen) = 0.037 * 0.9 + .169 * 0.1 = 0.05
Therefore, you need to adjust your probability that the coin is fair!
P(Fair|Fourteen) = 0.9*0.037/0.05 = 0.667
And voila, from being 90% sure, you are now only 66.7% sure that the coin is Fair! Just after a simple experiment, you have to change your preconceived notions.
Well what if you hadn’t observed 14 heads? What if you had observed 13 or 15? Well, with a few short lines of Python, you can graph how the evidence changes your Posterior probability, despite the Prior probability (preconceived notion) of the coin being fair.
from scipy.stats import binom
import numpy as np
p_fair = 0.9
p_loaded = 0.1
outcomes = list(range(0,21))
# Get the probability of 0 - 20 heads in a trial given
# the coin is fair
p_outcomes_given_fair = binom.pmf(outcomes,20,0.5)
# Get the probability of 0 - 20 heads in a trial given
# the coin is loaded #to give heads 75% of the time.
p_outcomes_given_loaded = binom.pmf(outcomes,20,0.75)
p_outcomes = p_outcomes_given_fair * p_fair + p_outcomes_given_loaded*p_loaded
p_fair_given_outcomes = p_fair * (p_outcomes_given_fair) / p_outcomes
#Draws the plots
ax = sns.lineplot(x = outcomes, y = p_fair_given_outcomes)ax.
set_ylabel('Probability of coin being Fair')
ax.set_xlabel('Number of Heads in twenty tosses')
ax.set_title('Loaded at 75% heads or Fair?')
And there you see how evidence impacts your preconceived notions. This is what I love about Bayes’ theorem.
- You are allowed to have preconceived notions; skepticism is in fact good!
- You change your mind when presented with evidence.
- How much the evidence changes your mind, depends on the likelihood of the evidence occurring given what you think is true.
- Not changing one’s mind despite the evidence isn’t skepticism, don’t call it that, it is denialism.
So, if you believe X is true and Y happens which is very unlikely to happen if X is true but quite likely to happen if X is false, you reset your mental probability of X being true! Plus, it is iterative, you can keep changing your mind as new evidence is presented.
Let’s take the same approach to climate change or COVID19, I understand it is hard to believe outright that the climate is changing due to our actions or that there is a killer pandemic out there, our brains want to believe differently, that things are okay, but let us change those beliefs when presented with evidence. If you’d hang on here for a little while more, we’ll apply this theorem to climate science, and think a little. We know that 97% of climate scientists agree that anthropogenic climate change is happening.
Let NinetySeven be the event that 97% of climate scientists agree on anthropogenic climate change, and ClimateChange be the event that anthropogenic climate change is occurring. Then using our previous formulae,
Of course, we can only wonder about the probabilities here, but even if you are initially skeptical about climate change, assigning it a 20% probability of happening.
What do you think the probability is that ninety-seven percent of the world’s scientists would agree that it is happening, given it is actually happening. Let’s say you doubt the scientists’ ability to accurately measure climate change and hence assign only an 80% probability, that they would agree it is happening, given it is happening.
More interestingly, what do you think the probability is that ninety-seven percent of the world’s scientists agree on climate change happening, when in fact it is not! That seems absurd! But even if you think that would happen 10% of the time. (I find that unlikely, but I’ll be generous). You now have to revise your prior belief to
P(ClimateChange|NinetySeven) = 0.2 * (0.8)/(0.8*0.2 + 0.1*0.8)
= 0.67!
So now you at least have to be on the fence about climate change, and then you can read up more and revise your belief as you encounter more evidence.
In real life, we’ll only have guesstimates for a lot of these probabilities, but think about this, every time you hear a conspiracy theory they try to claim that there is a large probability that the evidence is manipulated, that massive amounts of evidence exists despite the hypothesis being false, and this is a result of some large scale coordinated effort. Think about the probability that this could be true, for eg: That 97% percent of the world’s scientists were coerced into claiming a falsehood is true, if there isn’t a good explanation (with evidence) for how this could be, then the skepticism is just denialism.
I’ll leave this discussion here, there can be more said about what evidence is real or good. Should I believe everything I see, but that is another topic.
Do check out this article for a beautiful Bayesian argument for anthropogenic climate change.
— author, Gowri Thampi