Bayesian vs Frequentist, Study notes of Statistical Physics

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Bayesian vs Frequentist

Xia, Ziqing (Purple Mountain Observatory) Duan, Kaikai (Purple Montain Observatory) Centelles Chuliá, Salvador (Ific, valencia) Srivastava, Rahul (Ific, Valencia)

Taken from xkcd

The notion of probability - Frequentist

• If the number of trials approaches infinity , the relative frequency will converge exactly to the true probability.
• Repeatability of an experiment is the key concept. The number of trials where the event X occurred The total number of trials
• The Maximum Likelihood Estimator :

The notion of probability - Bayesian

Posterior Marginal likelihood Prior Likelihood function It is a direct consequence of the Bayes theorem :

• Bayes theorem relates the posterior probability (what we know about the parameter after seeing the data) to the likelihood (derived from a statistical model for the observed data) and the prior (what we knew about the parameter before we saw the data).
• A general rule to update our knowledge from the prior to the posterior.

Frequentist Approach to Simple Photon Counts

• The probability distribution of the measurement ：
• construct the likelihood function by computing the product of the probabilities for each data point:
• The best estimate ：

Bayesian Approach to Simple Photon Counts

• The posterior probability ：
• The model prior : a standard choice is to take a uniform prior.
• The Bayesian probability is maximized at precisely the same value as the frequentist result! In the case of a Gaussian likelihood and uniform prior, the posterior pdf and the profile likelihood are identical and thus the question of which to choose does not arise.

The Bayesian Billiard Game

• Alice and Bob can’t see the billiard table.
• Carol rolls a ball down the table, and marks where it lands. Once this mark is in place, Carol begins rolling new balls down the table.
• If the ball lands to the left of the mark, Alice gets a point; if it lands to the right of the mark, Bob gets a point.
• The first person to reach six points wins the game.
• Now say that Alice is leading with 5 points and Bob has 3 points. What can be said about the chances of Bob to win the game?

Frequentist (naive) Approach to The Billiard Game

• Five balls out of eight fell on Alice's side of the marker
• Maximum likelihood estimate of p that any given roll lands in Alice's favor.
• Assuming this maximum likelihood probability, we can compute the probability that Bob will win, which is given by:

Frequentist Probability of Bob Winning: 0.

Monte Carlo Approach to The Billiard Game

• Use a Monte Carlo simulation to determine the correct answer.

The correct Probability of Bob Winning: 0.09!

Bayesian win!!!

Open questions

• How to know when to use one approach or the other?
• What defines a good prior?
• Is there another approach ( the third one ) can resolve this case? References ： [1] Roberto Trotta: Bayesian Methods in Cosmology. arXiv: 1701. [2] Jake VanderPlas: Frequentism and Bayesianism: A Python-driven Primer. arXiv: 1411.

Thank you!