1. Bayesian learning finalized (with high probability)

2. Everything’s random... Basic Bayesian viewpoint: Treat (almost) everything as a random variable Data/independent var: X vector Class/dependent var: Y Parameters: Θ E.g., mean, variance, correlations, multinomial params, etc. Use Bayes’ Rule to assess probabilities of classes Allows us to say: “It is is very unlikely that the mean height is 2 light years”

3. Uncertainty over params Maximum likelihood treats parameters as (unknown) constants Job is just to pick the constants so as to maximize data likelihood Fullblown Bayesian modeling treats params as random variables PDF over parameter variables tells us how certain/uncertain we are about the location of that parameter Also allows us to express prior beliefs (probabilities) about params

4. Example: Coin flipping Have a “weighted” coin -- want to figure out  =Pr[heads] Maximum likelihood: Flip coin a bunch of times, measure #heads ; #tails Use estimator to return a single value for  This is called a point estimate

5. Example: Coin flipping Have a “weighted” coin -- want to figure out  =Pr[heads] Bayesian posterior estimation: Start w/ distribution over what  might be Flip coin a bunch of times, measure #heads ; #tails Update distribution, but never reduce to a single number Always keep around Pr[θ | data] : posterior estimate

6. Example: Coin flipping ? ? ? ? ? ? ? 0 flips total

7. Example: Coin flipping 1 flip total

8. Example: Coin flipping 5 flips total

9. Example: Coin flipping 10 flips total

10. Example: Coin flipping 20 flips total

11. Example: Coin flipping 50 flips total

12. Example: Coin flipping 100 flips total

13. How does it work? Think of parameters as just another kind of random variable Now your data distribution is This is the generative distribution A.k.a. observation distribution, sensor model, etc. What we want is some model of parameter as a function of the data Get there with Bayes’ rule:

14. What does that mean? Let’s look at the parts: Generative distribution Describes how data is generated by the underlying process Usually easy to write down (well, easier than the other parts, anyway) Same old PDF/PMF we’ve been working with Can be used to “generate” new samples of data that “look like” your training data

15. What does that mean? The parameter prior or a priori distribution: Allows you to say “this value of is more likely than that one is...” Allows you to express beliefs/assumptions/ preferences about the parameters of the system Also takes over when the data is sparse (small N ) In the limit of large data, prior should “wash out”, letting the data dominate the estimate of the parameter Can let be “uniform” (a.k.a., “uninformative”) to minimize its impact

16. What does that mean? The data prior: Expresses the probability of seeing data set X independent of any particular model Huh?

17. What does that mean? The data prior: Expresses the probability of seeing data set X independent of any particular model Can get it from the joint data/parameter model: In practice, often don’t need it explicitly (why?)

18. What does that mean? Finally, the posterior (or a posteriori) distribution: Lit., “from what comes after” (Latin) Essentially, “What we believe about the parameter after we look at the data” As compared to the “prior” or “a priori” (lit., “from what is before”) parameter distribution,

19. Example: coin flipping A (biased) coin lands heads-up w/ prob p and tails-up w/ prob 1-p Parameter of the system is p Goal is to find Pr[p | sequence of coin flips] (Technically, we want a PDF, f(p | flips) ) Q: what family of PDFs is appropriate?

20. Example: coin flipping We need a PDF that generates possible values of p p ∈ [0,1] Commonly used distribution is beta distribution: Normalization constant: “Beta function” Pr[heads]Pr[tails]

21. The Beta Distribution Image courtesey of Wikimedia commons

22. Generative distribution f(p|α,β) is the prior distribution for p Parameters α and β are hyperparameters Govern shape of f() Still need the generative distribution: Pr[h,t|p] h,t : number of heads, tails Use a binomial distribution:

23. Posterior Now, by Bayes’ rule:

24. Exercise Suppose you want to estimate the average air speed of an unladen (African) swallow Let’s say that airspeeds of individual swallows, x, are Gaussianly distributed with mean and variance 1 : Let’s say, also, that we think the mean is “around” 50 kph, but we’re not sure exactly what it is. But our uncertainty (variance) is 10 kph. Derive the posterior estimate of the mean airspeed.