IID Samples In supervised learning, we usually assume that data points are sampled independently and from the same distribution IID assumption: data are independent and identically distributed ⇒ joint PDF can be written as product of individual (marginal) PDFs:

The max likelihood recipe Start with IID data Assume model for individual data point, f(X; Θ ) Construct joint likelihood function (PDF): Find the params Θ that maximize L (If you’re lucky): Differentiate L w.r.t. Θ, set =0 and solve Repeat for each class

Exercise Find the maximum likelihood estimator of  for the univariate Gaussian: Find the maximum likelihood estimator of  for the degenerate gamma distribution: Hint: consider the log of the likelihood fns in both cases

Solutions PDF for one data point: Joint likelihood of N data points:

Solutions Log-likelihood:

Solutions Log-likelihood: Differentiate w.r.t.  :

Solutions Log-likelihood: Differentiate w.r.t.  :

Solutions Log-likelihood: Differentiate w.r.t.  :

Solutions Log-likelihood: Differentiate w.r.t.  :

Solutions What about for the gamma PDF?

Putting the parts together [X,Y][X,Y] complete training data

Putting the parts together Assumed distribution family (hyp. space) w/ parameters Θ Parameters for class a: Specific PDF for class a

Putting the parts together

Gaussian Distributions

5 minutes of math... Recall your friend the Gaussian PDF: I asserted that the d-dimensional form is: Let’s look at the parts...

5 minutes of math...

Ok, but what do the parts mean? Mean vector, : mean of data along each dimension

5 minutes of math... Covariance matrix Like variance, but describes spread of data

5 minutes of math... Note: covariances on the diagonal of are same as standard variances on that dimension of data But what about skewed data?

5 minutes of math... Off-diagonal covariances ( ) describe the pairwise variance How much x i changes as x j changes (on avg)

5 minutes of math... Calculating from data: In practice: you want to measure the covariance between every pair of random variables (dimensions): Or, in linear algebra:

Bayesian Wrap-Up (probably)

5 minutes of math... Marginal probabilities If you have a joint PDF:... and want to know about the probability of just one RV (regardless of what happens to the others) Marginal PDF of or :

5 minutes of math... Conditional probabilities Suppose you have a joint PDF, f(H,W) Now you get to see one of the values, e.g., H=“183cm” What’s your probability estimate of W, given this new knowledge?

5 minutes of math... Conditional probabilities Suppose you have a joint PDF, f(H,W) Now you get to see one of the values, e.g., H=“183cm” What’s your probability estimate of A, given this new knowledge?

5 minutes of math... From cond prob. rule, it’s 2 steps to Bayes’ rule: (Often helps algebraically to think of “given that” operator, “|”, as a division operation)

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”

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

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  Bayesian (MAP): 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

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

Example: Coin flipping 1 flip total

Example: Coin flipping 5 flips total

Example: Coin flipping 10 flips total

Example: Coin flipping 20 flips total

Example: Coin flipping 50 flips total

Example: Coin flipping 100 flips total

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:

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

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

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

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?)

What does that mean? Finally, the posterior (or a posteriori) distribution: Lit., “from what comes after” or “after the fact” (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” or “before the fact”) parameter distribution,

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.