1. You’re right. The previous lecture is Tselil’s blog post; I’ll add a link to it shortly. Thanks for pointing that out.

2. Almost: the probability is defined to be proportional (not equal to) the right hand side. So in the example, the probability distribution is uniform over every satisfying assignment. It’s probably worth mentioning that this scenario is very typical of inference tasks: we can easily write down the pdf of a model, but only up to normalizing constant, and the normalizing constant often gives us useful information about the entire distribution. Moreover, calculating the normalizing constant can be computationally hard (in the example for 3SAT, it would be #P complete), and is often a computational problem of interest in many fields (e.g. sampling methods in statistics, models of physical systems, bayesian inference, and computational problems).

]]>Some comments:

1. There’s no link to the previous lecture though you refer to it.

2. The first displayed equation is unclear to me. You say you want to “decompose” the probabilistic model. What does a decomposition mean? I understood that it means that the probability P(x) of getting x, is defined to be as the right hand side of the equation. But then the example seems to say that for every satisfying assignment x, it’s probability is 1. Which doesn’t make sense. Where did I get it wrong?

]]>once again, I didn’t try to take a side in this debate, but just to point out that the debate exists, and the fact of its existence shows a shift in the value-system of physics. (Moreover, arguably the majority of working physicists implicitly take the SUAC side of it, which says that such questions are a topic for blog posts or pub discussion over drinks, but they are not part of the main working agenda.)

]]>I was referring to the following observations about physics, which as far as I know are true:

1) Wavefunctions are the standard model for states of systems in quantum mechanics as taught in undergraduate courses and used in physics research.

2) There is no universal agreement among physicists if wavefunctions “really exist” or not. Wavefunction realism is one interpretation of quantum mechanics but there are other interpretations as well.

3) The issue of interpretation is largely considered a matter for philosophy and not for physics. The reason that wavefunctions are widely used and successful is because they are a clean mathematical model to predict observations. You don’t have to believe they exist to do that, and to a large extent physicists don’t care.

There are of course important reasons (corresponding to both “quantum weirdness” and actual experiments with atoms) why physicists don’t normallly talk about the interpretation of atoms and debate whether they really exist.

But what I find interesting is that by essentially deciding that the interpretation question is irrelevant, physics implicitly changed its goal. (Or perhaps physicists realized that their goal all along was to find mathematical models to predict observations, rather than to find mathematical models that have a direct correspondence with reality.)

]]>I mean is there some operational sense in which atoms differ from wavefunctions in terms of our observational practice? In the case of atoms their supposed realness is merely deduced from their usefulness in making observational predictions about the world and wavefunctions and other aspects of modern physics are no different in that regard. Operationally, it seems the only difference is that the atomic model doesn’t seem highly weird (i.e. divorced from our daily experience of the world) while the quantum model does. But if this is really all there is to the distinction why dress up the well known fact about qm weirdness in these terms.

On the other hand if you really meant to express some deep, non-operational, metaphysical claim about the nature of reality then you really ought to specify the metaphysical principles that render atoms obviously ‘real’ but raise doubts about wavefunctions as well as specifying what sense of real you even mean. I mean such claims depend critically on one’s metaphysical assumptions. For instance, if one is an idealist (ultimately the only thing that’s ‘real’ is conscious experience and the external world is merely a certain kind of regularity in those experiences) then there is absolutely no difference at all between explaining how gasses and liquids work in terms of atoms and explaining how subatomic particles work in terms of wavefunctions. In contrast, others might even take issue with the coherence of such a distinction between real aspects of the world and mere mathematical models.

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