Bayesian Learning

Bayesian vs Frequentist

Bayesian vs Frequentist - the main difference is in interpreting "probability" and uncertainty

Frequentist Approach

In Frequentist, AKA classical stats, the probability is simply the long run frequency of things

It's the rate of event over rate of seen (among many frequent trials)

$P(Event) \over P(All)$

Characteristics:

• Parameters, $\theta$, are fixed and unknown
• Data is random, and uncertainty strictly stems from sampling variability
  • i.e. we choose data randomly in samples, so any uncertainty is simply because we're randomly choosing data points
• There are no prior beliefs; We only know what we know from what we see
• Hypothesis testing is done via p-values and confidence intervals
  • p-values are a way to interpret a hypothesis that was decided on before
    • Something similar to "we believe these two groups of patients are significantly different after one takes a drug and one doesn't"
    • For p-values you must assume the null hypothesis is true, and then the p-value tells you the probability of seeing that observed data $O$
    • The p-value is the probability of observing data as extreme as (or more extreme than) what you saw, given the null hypothesis
    • Therefore $P(O | \theta) = p$ is the likelihood of the observed data under the assumed parameters
      • So if the p-value is small, that means the probability of observing that data is small, and that we can reject the null hypothesis
      • If it's large we do not accept the null, but we fail to reject it

Example:

• Testing a new drug, and wanting to check it's effectiveness
• Null hypothesis: There is no effect (no statistical significance between observed and population)
  • Alternative hypothesis: There is statistical significance (doesn't specify direction or cause)
• Observed data: $O$
• P-Value: 0.035, with threshold 0.05
  • The threshold and significance values measure how close a hypothesis is to a dataset
  • To be clearest - it actually means "The threshold (significance level, e.g., 0.05) is the cutoff for how unlikely the observed data must be under the null hypothesis before we reject it."
    • This just means we choose some level of unlikeliness, and if the observed data is under that level of unlikeliness then we can reject null
    • This means the higher the significance, the more "wiggle room" we're giving
• Test: typically done with t-test, or some other one of the hundreds of different tests
  • Each test is valid only under certain conditions of population and observed datasets
  • Get p-value from these tests
• At this point the p-value is less than the threshold, and so you can reject the null

Bayesian Approach

In Bayesian statistics there's a degree of belief, and unknown in all of the probability calculations

This belief can be updated when new data comes in, and is constantly changing

Characteristics:

• All parameters, $\theta$ are treated as random variables with prior distributions
  • Prior distributions are our belief of knowledge of a parameter before observing any data
  • It's a distribution that allows us to express uncertainty about the parameter of interest
  • It's typically known as $P(\theta)$ where $\theta$ is the specific parameter of interest, for this we'll denote it as $\theta_{i}$
• These parameters, with their uncertain prior distributions, are updated as new data comes in
• Bayes Theorem allows us to update the parameters prior beliefs with new data, resulting in posterior distributions
• Inference at the end is based on the probability of hypothesis, not accepting or rejecting
  • Here we can say "there's a 95% chance the treatment effect is > 0"

Example:

• Testing a new drug, and wanting to check it's effectiveness
• Start with a prior that the drug has a small effect
  • How do you choose a prior? It's the subjcet of large debate
    • Duke Stats 732 Paper on Choosing Priors
    • Subjective Priors are priors based on subjective beliefs, typically we try to do this in lens of an "expert"
    • Default / Low-Informative / Uniform prior is an alternative that reflects a lack of strong and precisely quantified prior information
      • AKA Objective Prior, because we try to base this on non-subjective information of one person
  • Without diving too far into this, basically, you choose some prior
• After running trial(s) and observing data, you update your belief to get the posterior
  • You must run a large number of trials to update the posterior closer and closer to it's true value
  • If you only run one experiment, then it may not be that close to the true value
  • The expected difference from your outcome to the true value is known as ...?
• Finally, with this outcome, you can use your parameters and observed data to directly answer the effects question

Bayesian

Now we can discuss how Bayesian is used in ML and Analytics

The intuitive description is that we can update parameters that we created in / update parameters while running, our backpropogation algorithms

Ultimately, if we have an online system that runs $f(\theta, U_i) = \theta \cdot U_i$ of running these matrix parameters against a user input, we can update these parameters after seeing new data (feedback loop) of the user - this is the essence of Bayesian Updating

Bayes Theorem

$P(\theta \mid O) = { {P(O \mid \theta) \cdot P(\theta)} \over P(O)}$

• Parameters $\theta$
• Observed Data $O$
• $P(\theta)$: prior belief about parameters
• $P(O \mid \theta) \cdot P(\theta)$: likelihood (how likely the data is given parameters)
• $P(\theta \mid O)$: posterior (updated belief after seeing data)