07/02/23 10:27:43
@statistical-rethinking @probability @book
It’s useful to draw on examples I read about, say a Sean Carroll article, McElreath’s marbles, Kipping on aliens
Looking at taking the notion of counts and using the new tool of multiplication and normalising over sum of products.
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Because one of the conjectures must be the case, the sum of conjectures must equal 1 (somehow). That is, one of the conjecture paths must be the case. So you take the plausibility for one conjecture and put it over the sum of all possible conjecture paths.
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It allows us to put a number of plausibility on a conjecture (which is a guess at the state of the world). Once again, we’re assuming one of our conjectures is the case.
Chapter 2
2.2
- A conjecture is usually called a parameter.
I kind of like this example of the water on the globe. It’s kind of what you’re trying to always do with Bayesian Analysis (I think) which is postulate a state of the world.
Basic design loop:
- Data story: Narrating how the data might arise. Can be descriptive and causal. Almost as if the data is a character, and you’re trying to find motivations of why it exists or the reason it is the way it is.
”Bayesian data analysis usually means producing a story for how the data came to be”.
- A Bayesian model begins with a set of plausibilities for each conjecture (priors).
Evaluate
The inference is perfect, provided the model is perfect (which it’s not always, map and the territory).
Components of Model
- The no. of ways each conjecture could produce the observation.
- The accumulated no. of ways each conjecture could produce the entire data.
Unobserved variables (in our case, the proportion of water) are usually called parameters. We can then have observed variables, like what we draw from the bag of marbles, or what comes up under our finger tossing the globe (W or L).
Assign plausibility of p with the data (observables). We were able to defined the ‘state of the world’ through one variable p in the marble case (that is, the proportion that were blue).
The story as McElreath puts it is that we have to events, W and L. Nothing else can happen. We are given a string of 9 events (in this examples). Out of all the possible worlds where 9 events occur, with our parameter p defining what is the case, what is the plausibility of the string of 9 events we have.
A binomial distribution is counting the paths for you. For a given proportion of water to land, it’s saying how many So we have some variable p, that constrains our sample space. On determining a new path, p is ever present. We have W, L which we might consider the data. The observables. Given that
- Rethinking datum/paramater? Data is normally considered ‘known’ and ‘parameters’ unknown
2.4
Or story is we want to know the plausibility of p given some observable W out of N tosses.
The binomial distribution gives us a set of plausibilities for P(W,L|p). We just want this for every p.
which is the same as equating these leaves us with
The initial goal was to determine which conjecture, out of a set of conjectures was the most plausible given some data. In the marble example, we had 4 possible conjectures. Moving on to the globe example, the conjectures are all the possible states of the world (literally), this state is defined by the proportion of water to land.
Plausibility for a given conjecture is proportional to the plausibility of the data given the state of the world is our conjecture times the plausibility of that world being the case (prior).
This prior can also be thought of as the prior number of paths (for some previous data say). So it’s just counting paths.
Grid approximation?
2M3.
The conjecture here is that the tossed globe is Earth given we’ve seen land. Both are equally likely to be tossed, so the priors of it being either is uniform. We know that if it’s Mars the relative plausibility is 100%. We know that if it’s Earth the relative plausibility is 30%.
The Pr(L) is 1 + 0.3.
I didn’t enumerate Pr(L) properly, this is the probability across all spaces that land exists. Land if it were earth plus land if it were mars. This cancels the Pr(E) on top.