I didn’t watch the video, but I don’t see how that could be true. Occam’s razor is about complexity, while the conjunction fallacy is about logical strength.
Sure ‘P & Q’ is more complex than ‘P’, but ‘P’ is simpler than ‘(P or ~Q)’ despite it being stronger in the same way (P is equivalent to (P or ~Q) & (P or Q)).
(Another way to see this is that violating Occam’s razor does not make things fallacies).
″...which of course violates the conjunction rule of probability theory—also known as Occam’s razor—which says that a more complicated event cannot be more probable than an a strictly simpler event that includes the more complicated one.”
The conjunction fallacy applies only when you already have a probability law. (a specification of a probability space). It applies to events in a probability space. The conjunction rule proscribes assigning a subset event higher probability than the event containing it.
Occam’s razor is prescription for what probability laws should look like (e.g each program having a prior probability of (1/2) to the power of its code length in bits). i.e. what constitutes an outcome in the probability space, all outcomes having equal probability.
The conjunction fallacy really says nothing about prior probabilities. The conjunction rule is a theorem in probability. Occam’s razor is a working rule for assigning prior probabilities to hypotheses.
The conjunction fallacy really says nothing about prior probabilities. The conjunction rule is a theorem in probability. Occam’s razor is a working rule for assigning prior probabilities to hypotheses.
These are two distinct, legitimate uses of the term “Occam’s Razor”. The conjunction rule is the everyday sense; what you’re talking about is a deeper, philosophical sense.
Well I agree. In our everyday lives, Occam tells us to chose the simplest hypotheses. Conjunction rule tells us to keep any particular hypothesis we have as simple as possible.
Also, maybe we should apply conjunction rule first to our candidate hypotheses and then only Occam to chose the simplest among them. (of course, Occam by itself already picks out the simplest hypotheses, but i’m talking about a working procedure here)
The conjunction fallacy really says nothing about prior probabilities. The conjunction rule is a theorem in probability. Occam’s razor is a working rule for assigning prior probabilities to hypotheses.
Prior and posterior probabilities are not made of fundamentally different stuff, and posterior of one calculation can turn out the be prior in the next. Assuming fundamentally distinct sets of probabilities and new ways of popping probabilities into existence seems uncalled for.
You were also suggesting to first use conjunction rule to weed out hypotheses that are less likely, and then summoning Occam’s razor to do the exact same thing again. This too seems redundant.
Agreed, there is no fundamental distinction. You can certainly update existing probabilities which did not take into account Occam’s Razor, to take it into account. What makes Occam pertinent to priors in particular is that you can apply it to anything, which means it can always also be the first thing you apply to hypotheses. So think of Occam as ‘evidence’ that applies to all hypotheses. (note that the conjunction rule is not similarly ‘evidence’)
Yes it is ideally redundant, but i did emphasize I was suggesting it as a working rule. It seems to me less computationally expensive to remove extraneous elements from hypotheses than to calculate or at least rank their complexity.
The conjunction fallacy is a subset of Occam’s razor? Hmm. Yes—I had never thought of it like that before.
I didn’t watch the video, but I don’t see how that could be true. Occam’s razor is about complexity, while the conjunction fallacy is about logical strength.
Sure ‘P & Q’ is more complex than ‘P’, but ‘P’ is simpler than ‘(P or ~Q)’ despite it being stronger in the same way (P is equivalent to (P or ~Q) & (P or Q)).
(Another way to see this is that violating Occam’s razor does not make things fallacies).
The actual quote is:
″...which of course violates the conjunction rule of probability theory—also known as Occam’s razor—which says that a more complicated event cannot be more probable than an a strictly simpler event that includes the more complicated one.”
41 minutes in.
The conjunction fallacy applies only when you already have a probability law. (a specification of a probability space). It applies to events in a probability space. The conjunction rule proscribes assigning a subset event higher probability than the event containing it.
Occam’s razor is prescription for what probability laws should look like (e.g each program having a prior probability of (1/2) to the power of its code length in bits). i.e. what constitutes an outcome in the probability space, all outcomes having equal probability.
The conjunction fallacy really says nothing about prior probabilities. The conjunction rule is a theorem in probability. Occam’s razor is a working rule for assigning prior probabilities to hypotheses.
These are two distinct, legitimate uses of the term “Occam’s Razor”. The conjunction rule is the everyday sense; what you’re talking about is a deeper, philosophical sense.
Well I agree. In our everyday lives, Occam tells us to chose the simplest hypotheses. Conjunction rule tells us to keep any particular hypothesis we have as simple as possible.
Also, maybe we should apply conjunction rule first to our candidate hypotheses and then only Occam to chose the simplest among them. (of course, Occam by itself already picks out the simplest hypotheses, but i’m talking about a working procedure here)
Prior and posterior probabilities are not made of fundamentally different stuff, and posterior of one calculation can turn out the be prior in the next. Assuming fundamentally distinct sets of probabilities and new ways of popping probabilities into existence seems uncalled for.
You were also suggesting to first use conjunction rule to weed out hypotheses that are less likely, and then summoning Occam’s razor to do the exact same thing again. This too seems redundant.
Agreed, there is no fundamental distinction. You can certainly update existing probabilities which did not take into account Occam’s Razor, to take it into account. What makes Occam pertinent to priors in particular is that you can apply it to anything, which means it can always also be the first thing you apply to hypotheses. So think of Occam as ‘evidence’ that applies to all hypotheses. (note that the conjunction rule is not similarly ‘evidence’)
Yes it is ideally redundant, but i did emphasize I was suggesting it as a working rule. It seems to me less computationally expensive to remove extraneous elements from hypotheses than to calculate or at least rank their complexity.