Notes on Statistics

Statistics

Statistical Rethinking

• Probability, as has been mentioned before is just counting.
• It represents uncertainty numerically. Not so much that in x percent of the worlds this happens and 1-x percent the other outcome happens. It’s a matter of what could happen. I think it’s fundamentally unintuitive but it’s made seem like it should be.
• I ran into a weird one where I was counting outcomes on a tree, all paths are considered equally likely though so if that changes how does that fit into the ultimate outcomes count?
• Outcomes are weighted
• Conjecture what might be the case

Probability theory

• The sample space is the set of all possible worlds; they are mutually exclusive only one of them can ‘be the case’
• Random variables are a way to get subsets of the sample space

Notes

20/12/21 11:02:57

Probabilities are degrees of belief, even in a frequentist view, if only more moderately. When I had the issue of the garden of forking paths where all possible worlds (outcomes) in the sample spaces are not equally likely I tried to map it onto both the frequentist and subjectivist views. Eventually you end up defining hypothetical trials which is just an extrapolation from your belief. What I’m saying is that it might not make sense to try and intuitively understand it in the garden of forking paths.

In the garden of forking paths if we have two outcomes A and B. If the Pr(A) is 0.4 and that of B is 0.6 how do we visualise this? A frequentist approach is that ten trials might end up with <A, A, A, A, B, B, B, B, B, B> whereas the subjectivist says that its an internal representation of knowledge on the next trial. My issue was how can we determine joint probabilities with this. If another forking path is added then the Pr(A, A) is 0.40.4 and Pr(B) is 0.60.6

Can look at it as ropes. Where are finally set of paths are weighted ropes. The more weight to a rope the more likely it is to happen. If all the paths are equally likely all the ends of the ropes we’re holding weigh the same amount so we’re indifferent (weight being our indicator). If one path is weighted less than another we can factor that into the paths when counting them up. The weight is distributed the more forked paths we have (longer ropes) so that our belief of 1 is diluted by the amount of paths.

Why is a joint probability multiplication?

From reading the excellent ‘Artificial Intelligence a Modern Approach’

22/12/21 13:33:14

From the above why multiply with joint probability. Conditional probability is $P(a|b)=\frac{P(a\wedge b)}{P(b)}$

$P(a\wedge b)=P(a|b)\ast P(b)$

So in the sample space where b is the case and then further get the proportion of the sample space where a is the case given b. You are left with the proportion of the sample space where both and b are the case.

If a and b are independent then the P(a|b) is the P(a).

$P(a\wedge b)=P(a)\times (b)$

Because they are independent, we use each as a proportion of the whole. So the chances of a being withing the set refined by the chances of b is the same as the chances of it being in the whole set. When they are dependent the chance of it being in this refined by set is different that just being in the whole set (different how?)

03/02/22 11:50:59

Gaussian or normal distribution. The result of some sum Difference of heads or tails determines where you are in the field. For N tosses the amount of ways you could end up with an ‘outcome’ of zero (that is, when heads, say, is positive and tails negative in counting terms) Many tosses converge on half tails and half heads so to more tosses the more general outcomes will focus on zero

Arises out of some flunctuations. That tend to converge towards a mean. So for any set of observations they generally converge over a number of trials

statistics

Think this distinction between observation and variable is a good mental model for thinking about data frames. If the columns represent variables and the rows represent specific observations of these variables.

That is the columns are homogeneous (as they measure the same thing), but the rows need not be (as different variables are measuring different quantities). ¹

It’s good to get a sense of the functions that are available to act on data frames.

1. combine: create a new DataFrame populated with columns that are results of transformations applied to the source data frame columns.
2. select: create a new DataFrame that has the same number of rows as the source data frame populated with columns that are results of transformations applied to the source data frame columns.
3. select!: Perform select update in place on passed in data frame.
4. transform: the same as select but keeps the columns that were already present in the data frame (note though that these columns can be potentially modified by the transformations passed to transform).
5. transform!: the same as transform but updates the passed data frame in place.

Footnotes

1. https://jverzani.github.io/UsingJ/EDA/tabular-data-julia.html ↩