Statistical Rethinking

Interpretation of probability (intuitive understanding)

Frequentist (Objectivist)

Assign numbers to describe some objective state of the world. Most common version of this being freqentist approach. which claims that the probability of a random event denotes the relative frequency of occurrence of an experiment’s outcome when the experiment is repeated indefinitely. Another example is propensity probability.

Subjectivist

Assigns numbers as a degree of belief.

Frequentist

• Focuses on repeated sampling

Bayesian

• subjective probability read in term so subjective uncertainty
• Updating prior intuitions with information from our sample

How Bayes Theorem Works

• Pot with one cup of coffee in it. You can choose how that cup is distributed between cups on front of you. Its a distribution of your beliefs
• Belief about how some individual is distributed before you actually measure them
• Think of belief as a distribution, so you’re fairly sure its x but must allow for the other possibilities

Chapter 1

• I struggled with the example about neutral theory of evolution as a hypothesis
• It seems that he’s arguing against the falsifying of null hypotheses and for creating multiple non-null models of the natural phenomena. This is like creating explanations that can be falsified (Deutsch?)

• In order to challenge these process models with evidence, they have to be made into statistical models. This usually means deriving the expected frequency distribution of some quantity-a “statistic”- in the model

• Change your explanation to fit the process models and the statistic models they produce in accordance with the observed data

• Bayesian inference is no more than counting the number of ways things can happen, according to our assumptions. Things that can happen more ways are more plausible

• Frequentist approach struggles when there is no sampling invariance. The example used is Galileo looking at a blurred Saturn with its rings, no amount of re sampling will resolve the uncertainty present in the technology. The uncertainty is not a function of the repeated measurements
• With the Bayesian method this uncertainty in the information can be modelled.

• Bayesian golems treat “randomness” as a property of information, not of the world

• Nothing in the real world is actually random it just lacks information (hmmm?)

• We just use randomness to describe our uncertainty in the face of incomplete knowledge

Chapter 2

• Small world vs large world. The example of Behaim’s globe is used (it doesn’t have the Americas in it). While the small world model is internally consistent is doesn’t represent reality fully. It’s this interplay between your small world (model) and reality thats important.
• Follows on here from the Bayesian inference explanation above. Looking at the garden of forking paths in which alternative events are cultivated as we learn what did happen some of these alternatives are pruned. In the end what remains, is what is logically consistent with our knowledge
• Counting the possible paths then becomes a multiplication of the possible paths on each ring (in the example)
• Marble example (I think p here is just a way of numerically describing the no. of blue marbles)
  • A conjectured proportion of blue marbles p is usually called the parameter value. It’s just a way of indexing possible explanations of the data.
  • The relative number of ways that a value p can produce the data is usually called the likelihood. It is derived by enumerating all the possible data sequences that could have happened and then eliminating those sequences inconsistent with the data
  • The prior plausibility of any specific p is the prior probability
  • The new, updated plausibility of any specific p is the posterior probability

2.2 Building the model

• Creating a data story. A narrative of why we are getting the observations. Viewed as important as it makes you think of the variables you really need to consider, get a bit more exact about chain of events (creating something hard to vary?)
• walks through Bayesian updating, the amount of evidence we have is embodied in the plausibility (straight line at the start vs complex curve at the end). The final figure is normally shown but its important to know that it is just an iterative development from the first figure.
• Some tips given for evaluation, they seem rather abstract at the moment though with my current knowledge
• Now we look at mapping some of the concepts from the previous section to build up the model

Likelihood function (1) the number of ways each conjecture could produce an observation

• Because both outcomes (W and L) are equally likely, and independent we look at all the ways our sample size of 9 (n) could appear to us.
• The binomial distribution calculates the relative no. of ways to get six W’s with 9 tosses holding p at 0.5
• Looking at the parameters to the binomial function p, n and w they can each represent different conjectures once we can tell the likelihood and what has been observed
• In the sciences, the prior is considered part of the model, there is no reason not to interrogate it like other assumptions.
• Bayesian estimate is always a distribution over the parameters
• Posterior is proportional to the product of the prior and the likelihood
  • Count up all the ways you could see the data and multiply by the priors (look at table for marbles)
• Grid approximation builds up from the marble example. With just 3 possible values for water (0, 0.5 and 1) 0.5 wins out with in 3 tosses. If we bump up the possible values to 20 though we get a more accurate display of possible values (posteriors)

Summary

15/12/21 15:22:11

Models operate on a small, internally consistent world. It’s important to recognise this when describing the large world with it and making predictions about the large world

The garden of forking paths sets up the abstract model for probability as just counting. I’m still kind of struggling with this, how each branched path has an implicit or explicit probability assigned and how that contributes to the general likelihood of some observation or conjecture in the garden of forking paths.

Another area I’m trying to wrap my head around is the basic logic for a distribution. For something like the globe toss where you don’t know the parameter (in this case, the proportion of the Earths surface water covers) you can generate a distribution on what it could be based on observations. In the example if the globe is tossed 9 times with 6 W’s and 3 L’s you can generate a binomial distribution with the binomial function where x goes from 0 → 1. This function requires the amount of times the observation comes up in n trials. This becomes our likelihood function, based on how many W’s we’ve seen in our trials the output of this functions assigns different plausibilities to different values of p.

• The function says, based on a probability how likely is it that we end up with the supplied W’s and n?
• I’m still trying to wrap my head around the counting aspect of the binomial distribution. But when probability is merely considered as just uncertainty, it does not say if this trial was performed x amount of times the probability assigned would appear in p% of the trials (although this might happen) its just a numerical value on uncertainty

Chapter 3

• Whenever the condition of interest is very rare, having a test that fins all the true cases is still no guarantee that a positive result carries much information at all.

• randomness is always a property of information, never of the real world

• Why sampling of model output, should apply to every model in the book ?

  • It is often easier and more intuitive to work with smaples from the posterior, than to work with probabilities and integrals directly

  • Where getting integrals is computationally expensive?
• Interesting note box here. Why statistics can’t save bad science. Suppose the probability of a positive finding and a false positive rate thats very low. If the probability of the prior, that is the probability of any hypotheses you posit in general being true is low, the best you could probably do is 0.5 (in terms of the posterior that the finding indicates the hypothesis is true). The lesson here being that no amount of accurate instrumentation can account for bad hypothesis (or explanations)

• These posterior intervals report two parameter values that contain between them a specified amount of posterior probability, a probability mass

  • parameter value being something on the spectrum of the parameter value (in this case 0 → 1 the proportion of water)

• 95% is a small world number.. true in the model’s logical world

  • On interpreting confidence intervals
• Loss function. If you were to make a bet on what the correct parameter value is. Where to cost is proportional to your distance from the ‘correct’ answer

• Given a realized observation, the likelihood function says how plausible the observation is. And given only the parameters, the likelihood defines a distribution of possible observations that we can sample from, to simulate observation

• Posterior predictive distribution
  • The top graph is the posterior distribution, for each parameter we are running is through a binomial distribution of 9 tosses as if the correct proportion p is that chosen parameter (so set p to 0.1 and run a simulation with p as 0.1 and see what posterior distribution you get as a result from 9 tosses). The initial posteriors are multiplied then by each sampling distribution and the final predictive distribution is shown
  • Passing the uncertainty of all parameters down is important so that the model does not appear more confident than it in the prediction
• Highlights different ways to analysis the model through alternative ways of a predictive distribution looking to see if the globe could be bias by the amount of times its switched, as an example

Summary

15/12/21 15:13:26

Sampling is introduced as a way to get the general gist of a posterior distribution without having to run a large integral computation (to get some area under the graph). Will need to see more examples of this before I fully understand it’s benefits

Simulations done with your model with Posterior predictive distributions. Run a sample of your model with each value of your sample of the posterior distribution. Multiply each of the mini distribution values by its initial posterior value. Because the posterior is incorporated all the way along it’s operates as a posterior predictor

Chapter 4

• Linear Regression is the geocentric model of applied statistics. A family of simple statistical golems that attempt to learn about the mean and variance of some measurement, using an additive combination of other measurements (compared to a fourier series)

• Useful in a lot of cases, and foundational. Not universal though (obviously)

• Any process that adds together random values from the same distribution converges to a normal.

  • The more terms the more likely values get cancelled out, so large deviations from the center by other large (or lots of small) deviations the other way.
  • Interacting deviations for growth (multiplications) as long as they are sufficiently small converge to a Gaussian distribution
  • The flip of a coin and go right or left is a good example. The further away you get from the midline the less likely you are to go further away (as the coin is fair). Any distribution of people doing this is more likely to end up around the midline.

• Justifications for Gaussian

  • ontological: Measurement errors, variations in growth, and the velocities of molecules all tend towards Gaussian distributions. They do this because they all add together fluctuations. The Gaussian distribution is a member of the exponential family of distributions.
  • Epistemological: It represents a particular state of ignorance. When all we know about a series of measurements is there mean and variance, this distribution arises. If we produced any other distribution it assumes we have some other particular knowledge about measurements we are taking. Or something that the Golem should know about.

• Probability distributions with only discrete outcomes, like the binomial are usually called probability mass functions. Continuous ones like the Gaussian are called probability density functions. Probability densities can be greater than 1

• p.77 the language for describing models (can I translate this to the chpt 2 example?)

  • A set of outcome variables that we hope to predict or understand. It’s an outcome in both sense whether measured or predicted.
  • For each of these outcome variables we define a likelihood distribution that defines the plausibility of individual observations
  • We have predictor variables that we hopefully use to understand or predict the outcome
  • Fundamentally, these models define the ways values of some variables can arise, given values of other variables

• tilda used for stochastic. It marks a relationship between a parameter or variable onto a distribution. For instance, where we don’t know the true proportion p of water.

06/02/23 15:29:24

@daily

• I’m interested to see if there is a heuristic to use, to notice patterns in the world for applying Bayesian Inference.
• ”When we don’t know what caused the data, potential causes that may produce the data in more possible ways are more plausible”.
• The garden of forking paths starts out with trying to determine the composition of a bag of marbles based on some draw of the bag of marbles.
• We count the paths to the draw we got in the sample space of the bag having all its possible compositions of marbles.
• The conjecture, is what we think the composition of the bag is.
• How to relate counts to multiplications. It works here because we are starting from square on each path fork.
  • Statement on this at the end of chapter 2.
• https://www.youtube.com/watch?v=R1vcdhPBlXA&list=PLDcUM9US4XdPz-KxHM4XHt7uUVGWWVSus&index=2
• Bayesian data analysis
  • For each possible explanation of the sample.
  • Count all the ways the sample could happen.
  • The explanations with more ways to produce the sample will be more ‘plausible’.
• Many possible stories (datasets) that can arise depending upon what happens in the process of sampling.
• All the things that could have happened, relative plausibility (no. of ways) of the things that actually happened.
• Builds up this forking terminology, to basic probability.
• If we think of it in the sense of Absence of Evidence, evidence of absence. It’s not clear how common alien life is. How likely our life is as a prior times the number of planets. Here, the bag of marbles is the state of the world, or what is the case. We might conjecture that they exist, but for each sample we get, the plausibility isn’t increasing dramatically (that is, the sample could exist in many possible worlds than our own).
• In @david-kipping lecture, he talks about how we can’t judge whether life is common, because we exist. There are many possible worlds in which that is the case but life is uncommon.
• The notion of evidence being linked to probability is engaging, that we can quantify, in some sense belief. Or at least acknowledge unknowns.
• Also, it’s interesting when you try and make statements about abiogenesis. There’s a repeatability issue. We might see earth like planets but how do we know they had a similar collision with a Mars sized planet to create the moon? This has to influence our belief because in the long time scale, there are a lot of factors that come into play. If total replication is needed for establishing life, there’s just a vast array of other possible ways it could have gone, although, if the amount of earth like planets was massive this could be tempered slightly.