As an aside, does this line bother anyone else?
> programming from a computation/understanding-first, mathematics-second point of view
Mathematics is understanding. What's implicit? The suggestion that mathematics isn't understanding, it's just funny symbols that no one understands.
> The latter path is much more useful, as it denies the necessity of mathematical intervention at each step, that is, we remove often-intractable mathematical analysis as a prerequisite to Bayesian inference.
In fact, I would say that by removing maths you're removing at least some amount of understanding. You're left with a workflow of "if x use model a else use model b." That will work until it doesn't, i.e. until you have to face a problem that isn't predicted by the workflow. Not saying that's a bad approach (it's certainly the most pragmatical) but it shows that the phrase "understanding first maths second" is misguided at best and disengenious at worst.
I think a more appropriate phrase for what the author describes in the prologue would be "practicalities first, theory second".
The whole idea of notation/language is:
1.to have a common reference for each definition that is explicit
2. forces you to define and handle proofs from a proper angle of detail (think of it like trying to write the specification of a data structure and an algorithm in prose instead of Haskell, C, some assembly dialect. This is actually interesting because what people call "math language" comprises a wide variety of different styles, like programming languages do.
2. concise -- again and again using notation and prior well-defined notions saves a lot of time and space; you don't tangle on the properties of registers for everything you do.
I would say that mathematics has a variety of languages that were, and are introduced per area, to go along. If you have a programming languages background you are familiar with the setting (and joke) that each new work defines a new (somewhat) programming language.
In general, there is a reason computer science is regarded to be that close to math.
However, I can also understand what the author is getting at when they say "computation first", "mathematics second."
Here they are using the layperson's notion of "mathematics" as a bunch of a Greek symbols detached from concrete meaning.
Of course, the layperson's notion is wrong, but nonetheless if that's how people think I can see why the author would try to communicate in their terms.
Real mathematics can of course be described with Greek letters, English sentences, or computer code - it's all the same mathematics!
The more experience you have in mathematical thinking the more you understand why using a "bunch of Greek letters" is actually easier than English sentences or computer code, but the layperson with limited mathematical experience doesn't grasp that, so they want to learn in code or plain English.
For more in-depth treatment, see Probabilistic Models of Cognition
Instead of 3 doors, imagine there are 1e9 doors. Then you choose one door (at random), and the host reveals all but one other door.
I think then it's clear why you would switch doors.
What complicates things is that these people are right, too. The generalisation of the problem does not fix the number of doors opened by the host. It could be that the host opens just one door. Or 5e8 doors, splitting the set in the middle. Or any other number in between. In fact, the number of doors opened by the host could be decided by dice before each round starts.
So now the person you talk to goes, "I'm still not convinced, but let's say for the sake of the argument that switching is good in the situation you mention. Now tell me why it is better to switch in all other cases."
You're ending up in formal proof country a lot faster than you'd want for an intuitive explanation.
(Odd observation: if k is the number of doors opened by the host, I find any case where n=2k+1 a lot harder to explain than when k=1 and n is some large number. Maybe that can give some insight into why it's a tough problem.)
Assume you pick one door. It's the right door with a chance (1/3) and the wrong door with a chance (2/3).
Now if you could pick both other doors (2/3) you would choose those. And this is exactly what you're doing when you switch after the reveal.
The other two doors always have one wrong one, which can be revealed. So switching after the reveal is the same as having the chance to pick two doors before the reveal.
I have even noticed one otherwise intelligent guy pronounce it Bython.
Mathematics brings with it some necessary baggage that by itself not only does not help understanding but may interfere with it when one is trying to make sense of things. A mathematical model usually includes a lot of "scaffolding" that does not correspond to anything in the real world, and it is often all too easy to get confused about which parts of the model do and which do not. This problem of interpretation is an especially difficult one in cases when the reality is outside of the domain of one's immediate experience, quantum mechanics being one example.
Give me a specific example.
Yes and no. Even as a mathematician I find it much easier understand the actual mathematics if I first have an intuitive 'feel' for the thing. Even if my initial mental model is simple and incomplete I find it much easier to go from there to the formal mathematical understanding than trying. Conversely I can also find myself in situations where I can understand all the math on the page, not really grasp what it is trying to tell me until I've had a chance to 'play' around with it in some from.
Now, understanding implies being able to do calculations. And you said that you could understand without maths, which means you can do calculations without maths. So try that and let me know how it goes.
There is a free book for it at:
Daphne Koller's 3 coursara classes on Probabilistic Graphical Models are also really good.
There's a full version of another good book ("Think Bayes: Bayesian Statistics in Python") available as a PDF from the publisher here:
(related Amazon reviews)
Having said that, I much prefer "Doing Bayesian Data Analysis" by Kruschke. It is extremely clear. It introduces concepts with intuition first, then math, then code, which I find to be an extremely useful order.
I personally find exceptional clarity once I see the code for a certain technique in Machine Learning. Often times, the theory skips certain implementation details which always leaves a void for me. After having read quite a bit on Bayesian Learning sometime ago, it is easy to connect this guide back to theory. That immediate click of ideas is rewarding!
I'm inclined towards Machine Learning and hence the bias. Not sure if this would cover the statistics parts but I think at least the fundamentals are the same.
Support great work like this and buy the book!
although if I knew more about Bayesian methods I would be in a better place to recommend it.