hello, all. first post around here =^.^= I’ve been working my way through the core sequences, slowly but surely, and I ran into a question I couldn’t solve on my own. please note that this question is probably the stupidest in the universe.
what is the difference between the Bayesian and Frequentist points of view?
let me clarify: in Eli Yudkowsky’s explanation of Bayes’ theorem, he presented an iconic problem:
“1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?”
to my understanding of the Bayesian perspective, the answer would be 7.8% and would represent the degree of uncertainty that the subject has breast cancer
to my understanding of the Frequentist perspective, the answer would be 7.8% and would represent the frequency of subjects that both have cancer and tested positive.
a keen observer will understand where my confusion comes from- on my way through the core sequences, I have heard much from the Bayesian side, but nothing from the Frequentist side, making it seem artificially non-existent.
The bayesian/frequentist distinction can cover three different things that may occasionally be mixed up:
The core philosophical disagreement (the “proper” one) about whether probabilities an agent’s knowledge / uncertainty about the world, or whether they represent frequencies of some event. For example, a frequentist in this sense might say that it’s meaningless to talk about the probability that the millionth binary digit of pi is even or odd. I think frequentist epistemology is mostly discredited, but that it used to be dominant.
There are a bunch of hodge-podge statistical methods and tests (like p-values); and later on attempts to unify everything in terms of bayesian methods. People used to the “old” methods may not particularly call themselves “frequentists” or care that much about such labels; those pushing for the new (better) methods are the ones stressing the distinction (hunting down the sin of frequentism), sometimes to the annoyance of the rest.
Thinking in probabilities versus thinking in frequencies (80 women out of a hundred); the human brain works better when a problem is presented in terms of frequencies
The classical way of explaining the difference is through the example of a coin that you know is biased, but you don’t know whether heads or tails is favored and by how much. What is the probability that the next toss will be heads?
Supposedly, a frequentist would say that there is an objective answer, given by the bias of the coin which also equals the proportion of heads in a long run. You just don’t know what it is, the only thing you know is that it is not 1⁄2. A Bayesian would say by contrast that since you have no information to favor one side over the other, the probability (degree of belief) you have to assign at this point is 1⁄2.
This only explains the question of Frequentism vs Bayesianism as philosophical interpretations of “what probability is”. The practical issue of Frequentism vs Bayesianism as concrete statistical methods is often tangled with this one in discussions, but it is really a separate matter.
I had the same issue, and I’m personally not convinced there’s an actual “Bayesian vs frequentist” conflict as framed in the sequences. Both are useful ways of thinking in different scenarios.
To use Emile’s example, there’s a distinction between the probability that you think the millionth digit of pi is even or odd, and whether it really is even or odd. Even though you don’t know the millionth digit offhand, it can be computed and has a definite value, so it really doesn’t matter what you think it is. Saying 50:50, or more generally an equal probability distribution, is in my mind basically the same as saying “I don’t know” (i.e. “I have zero evidence for deciding one way or the other.”)
There’s also a difference between the parity of the millionth digit of pi, and, for example, the wind speed at an arbitrary place and future time. It’s impossible to calculate, so instead you can apply Bayesian methods and estimate a range of values based on prior knowledge, and any historical data you might have access to.
I don’t think Bayesians and Frequentists would answer that question differently; frequentists also use Bayes’ Theorem, they just don’t base all their philosophy on it.
to my understanding of the Bayesian perspective, the answer would be 7.8% and would represent the degree of uncertainty that the subject has breast cancer
The “degree of uncertainty” can be extracted from that figure but if becoming more certain decreases the figure then it requires a different word.
hello, all. first post around here =^.^= I’ve been working my way through the core sequences, slowly but surely, and I ran into a question I couldn’t solve on my own. please note that this question is probably the stupidest in the universe.
what is the difference between the Bayesian and Frequentist points of view?
let me clarify: in Eli Yudkowsky’s explanation of Bayes’ theorem, he presented an iconic problem:
to my understanding of the Bayesian perspective, the answer would be 7.8% and would represent the degree of uncertainty that the subject has breast cancer
to my understanding of the Frequentist perspective, the answer would be 7.8% and would represent the frequency of subjects that both have cancer and tested positive.
a keen observer will understand where my confusion comes from- on my way through the core sequences, I have heard much from the Bayesian side, but nothing from the Frequentist side, making it seem artificially non-existent.
The bayesian/frequentist distinction can cover three different things that may occasionally be mixed up:
The core philosophical disagreement (the “proper” one) about whether probabilities an agent’s knowledge / uncertainty about the world, or whether they represent frequencies of some event. For example, a frequentist in this sense might say that it’s meaningless to talk about the probability that the millionth binary digit of pi is even or odd. I think frequentist epistemology is mostly discredited, but that it used to be dominant.
There are a bunch of hodge-podge statistical methods and tests (like p-values); and later on attempts to unify everything in terms of bayesian methods. People used to the “old” methods may not particularly call themselves “frequentists” or care that much about such labels; those pushing for the new (better) methods are the ones stressing the distinction (hunting down the sin of frequentism), sometimes to the annoyance of the rest.
Thinking in probabilities versus thinking in frequencies (80 women out of a hundred); the human brain works better when a problem is presented in terms of frequencies
The classical way of explaining the difference is through the example of a coin that you know is biased, but you don’t know whether heads or tails is favored and by how much. What is the probability that the next toss will be heads?
Supposedly, a frequentist would say that there is an objective answer, given by the bias of the coin which also equals the proportion of heads in a long run. You just don’t know what it is, the only thing you know is that it is not 1⁄2. A Bayesian would say by contrast that since you have no information to favor one side over the other, the probability (degree of belief) you have to assign at this point is 1⁄2.
This only explains the question of Frequentism vs Bayesianism as philosophical interpretations of “what probability is”. The practical issue of Frequentism vs Bayesianism as concrete statistical methods is often tangled with this one in discussions, but it is really a separate matter.
I had the same issue, and I’m personally not convinced there’s an actual “Bayesian vs frequentist” conflict as framed in the sequences. Both are useful ways of thinking in different scenarios.
To use Emile’s example, there’s a distinction between the probability that you think the millionth digit of pi is even or odd, and whether it really is even or odd. Even though you don’t know the millionth digit offhand, it can be computed and has a definite value, so it really doesn’t matter what you think it is. Saying 50:50, or more generally an equal probability distribution, is in my mind basically the same as saying “I don’t know” (i.e. “I have zero evidence for deciding one way or the other.”)
There’s also a difference between the parity of the millionth digit of pi, and, for example, the wind speed at an arbitrary place and future time. It’s impossible to calculate, so instead you can apply Bayesian methods and estimate a range of values based on prior knowledge, and any historical data you might have access to.
I don’t think Bayesians and Frequentists would answer that question differently; frequentists also use Bayes’ Theorem, they just don’t base all their philosophy on it.
The “degree of uncertainty” can be extracted from that figure but if becoming more certain decreases the figure then it requires a different word.