Chp 03 Review

[aloctavodia]

Consider improving the quality of the images

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Consider using only the first person of the plural

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Consider hiding the code used to render the images

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Potential confusion

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You used the same representation (a red circle with a black border) to represent both event A and a red ball. Consider using something that looks more like 3D balls than flat disks or use squares instead.

It may also be beneficial to show at least the first time the content of the sets inside the Venn diagram and then switch to the most abstract representation of just circles.

Probability definition vs interpretation

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At the beginning of section "3.3 Probability", it may be a good idea to add a disclaimer saying something like

While Probability has a well-defined mathematical definition. We don't want to go into much detail. And instead in this book, we will adopt a more practical definition. In particular, we adopt the Bayesian interpretation, stating that a probability is a measure of our belief that a particular event will occur, and we express it with a number ranging from 0 to 1

Don't be that mathematical guy :-)

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Instead of saying "By definition, the probability of the entire sample space S is unity, or P{S}=1" explain it.

use \text in math mode

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Change :

P(A) = \frac{success \ cases} {total \ cases}

Into :

P(A) = \frac{\text{success cases}}{\text{total cases}}

Add more text

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At the end of section 3.3 there are a bunch of examples of probability calculations. Consider adding more text between the mathematical expression. The union and intersection operators are used but not defined, define them.

Also there is a missing curly bracket in A \cap B = {(G,W),(B,W)

show the truncation of the sample space

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Use a figure to show that "since we know B occurred we can truncate the sample space to the B event". not conditioned/truncated --> conditioned/truncated

Add more text II

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Consider adding more text with details of the calculations after the sentence "P(A|B) is interpreted as the probability of picking a red and a green ball knowing that I already picked one red ball."

Unify notation

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In section 3.3 you use $\cap$ and in section 3.4 you use "and"

conditional probability definition

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When saying

"To calculate the joint probability of these two events, we use the formula..."

considering add that this is the same formula for the conditional probability introduced in the previous section, but re-arranged

Add more text II

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The derivation of the Bayes formula needs a few more steps and explanations.

Bayes theorem vs Bayesian interpretation of probability

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Consider deleting this sentence.

"This theorem does not only give us a practical way to calculate conditional probabilities but also is the fundamental building block of the Bayesian interpretation of probability."

Bayes' theorem is agnostic of the interpretations of probability. It is possible to use Bayes's theorem and adopt a frequentist interpretation of probability.

Maybe a caricature

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"In a Bayesian way of thinking one could assign (and actually it is done extensively) a probability to events such as the election of some politician, while in the frequentist view this would make no sense, since we can’t make large repetitions of the election to know the frequency underlying that event."

You can not make large repetitions in real life, but you can still use the analogy as a mental scaffold. And that's what frequentists actually do in practice.

I suggest avoiding comparison between freq and Bayesian interpretations and just focusing on explaining Bayes' theorem in practice. Don't take this the wrong way, I have also spent time in the past doing these comparisons, but I think now that is better to just avoid as much as possible any mention of frequentist ideas. In particular when the mentions are very brief.

avoid using the word "belief"

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I know a lot of people use it. But I think it can be detrimental. Better reword to don't need to use belief or similar words or use words as knowledge.

Platonic worlds

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The sentence

"What is nice about the Bayesian framework is that we always account for the uncertainty of the world..."

It is potentially confusing, as the Bayesian framework can only account for uncertainty given a model or set of models, and models can be completely detached from "the real word".

Maybe change it to something like:

What is nice about the Bayesian framework is that provides a useful and theoretically grounded method to account for the uncertainty. We start with some probability and end with another probability, uncertainty is always present.

Redundant plot

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In section "3.7.1 Discrete Case", there are two plots, not clear what the second one is adding to the first one. Also, the second one has "x" as a label instead of "Spam e-mails"

Consider using the metric system

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The plot in "3.7.2 Continuous cases" uses inches instead of centimeters.

Consider use a more thecnical definiton

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As most people is familiar with the concept of the mean and standard deviation use them to define the parameters of the Gaussian, or instead write something like

Roughly speaking, μ is associated with the center of the distribution, and σ with how wide it is. Formally μ is the mean σ the standard deviation.

Show it

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After saying

Every probability density that is defined by a mathematical function, has a set of parameters that defines the distribution’s shape and behavior, and changing them will influence the distribution in different ways, depending on the one we are working with.

add a or more plots showing 2 or 3 different sets of parameters for the Gaussian, so readers can check the previous statement by themselves. Also, use this to generate intuition on why we can use probability distribution to represent uncertainty

(Bayesian) Bandits

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The connection of this example with Bayesian stats/theorem may be not very clear for newcomers. Spend some time making the connection between Bayes' theorem the prior, the likelihood, the beta, the binomial, and the conjugated priors. And only then move to build the code to actually compute the results.

bandit_plot

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Add a point or line or something with colors matching the distributions to show where the true values are

Summary

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"we went over a few examples and saw why Bayesians use them to represent probability."

I guess it should be

"we went over a few examples and saw why Bayesians use them to represent uncertainty."

Besides that, this not may be totally clear from the content of the chapter, I already made a comment about this earlier.