notes lec apr 14

index.rst

@@ -7,7 +7,7 @@ Welcome to cse142-notes's documentation!
 ========================================
 
 .. toctree::
-   :maxdepth: 2
+   :maxdepth: 4
    :caption: Contents:
 
    intro

prob.rst

@@ -1,6 +1,8 @@
 Probability Review
 ==================
 
+Useful notes: http://cs229.stanford.edu/section/cs229-prob.pdf
+
 Let's define some important things.
 
 - **Outcome Space**: :math:`\Omega` - contains all possible atomic outcomes
@@ -44,4 +46,124 @@ Let's define some important things.
 
         E(\text{# heads}) & = \sum_{r=1}^6 P(roll = r) E(\text{# heads} | roll = r) \\
         & = \frac{1}{6}(\frac{1+2+3+4+5+6}{2}) \\
-        & = \frac{21}{12} = 1.75
+        & = \frac{21}{12} = 1.75
+
+- Joint distributions factor
+    - if :math:`\Omega = (S*T*U)`, then :math:`P(S=s,T=t,U=u)` is :math:`P(S=s)P(T=t|S=s)P(U=u|S=s,T=t)`
+- Conditional distributions are also distributions
+    - :math:`P(A|B) = \frac{P(A, B)}{P(B)}`, so :math:`P(A|B, C)=\frac{P(A,B|C)}{P(B|C)}`
+
+Bayes Rule for Learning
+-----------------------
+
+- Assume joint distribution :math:`P(\mathbf{X=x}, Y=y)`
+- We want :math:`P(Y=y|\mathbf{X=x})` for each label :math:`y` on a new instance :math:`\mathbf{x}`
+- So, using Bayes' Rule, :math:`P(y|\mathbf{x}) = P(\mathbf{x}|y) \frac{P(y)}{P(\mathbf{x})}`
+- :math:`P(\mathbf{x})` doesn't matter here, so we care that :math:`P(y|\mathbf{x})` is proportional to :math:`P(\mathbf{x}|y) P(y)`
+- From the data, we can learn :math:`P(\mathbf{x}|y)` and :math:`P(y)`
+- Predict label :math:`y` with largest product
+
+So how do we learn :math:`P(\mathbf{x}|y)`?
+
+.. note::
+    Take for example a coin flip. You observe the sequence HTH; what is the probability that the next flip is H?
+
+    Mathematically, the answer is 2/3: taking the likelihood function :math:`L(\theta) = P(HTH|\theta)`
+    we get the probability equal to :math:`\theta^2 (1-\theta)`.
+
+    By finding the :math:`\theta` value at the zero derivative, we get 2/3.
+
+.. note::
+    But what if we have a prior belief :math:`P(\theta)` where :math:`\theta = P(H)`?
+
+    Now, the posterior on :math:`\theta` becomes :math:`P(\theta | HTH)`:
+
+    .. math::
+        P(\theta | HTH) = P(HTH | \theta) \frac{P(\theta)}{P(HTH)}
+
+    Or in this case:
+
+    .. math::
+        \frac{\theta^2 (1-\theta) P(\theta)}{normalization}
+
+    **Discrete Prior**
+
+    Taking :math:`P(\theta=0) = P(\theta=1/2) = P(\theta=1) = 1/3`, :math:`\theta^2 (1-\theta) P(\theta)` is
+    0, 1/24, and 0 for the 3 cases respectively. Thus, the posterior :math:`P(\theta = 1/2 | HTH) = 1`.
+
+    **Prior Density**
+
+    - :math:`P(\theta) = 1` for :math:`0 \leq \theta \leq 1`
+    - So :math:`\theta^2 (1-\theta) P(\theta)` is just :math:`\theta^2 (1-\theta)`
+    - and the posterior is :math:`\frac{\theta^2 (1-\theta)}{12}`
+    - If we plot this, the max is at :math:`\theta = 2/3`
+
+- Treat parameter :math:`\theta` as a random var with the prior distribution :math:`P(\theta)`, see training data :math:`Z`
+- :math:`posterior = \frac{prior * data likelihood}{constant}`
+- :math:`P(\theta | Z) = \frac{P(\theta) P(Z | \theta)}{P(Z)}`
+
+Bayes' Estimation
+-----------------
+
+Treat parameter :math:`\theta'` as a RV with the prior distribution :math:`P(\theta)`, use fixed data
+:math:`Z = (\mathbf{x}, y)` (RV :math:`S`)
+
+Maximum Likelihood
+^^^^^^^^^^^^^^^^^^
+
+.. math::
+    \theta_{ML} = \arg \max_{\theta'} P(S=Z|\theta = \theta')
+
+Maximum a Posteriori
+^^^^^^^^^^^^^^^^^^^^
+
+.. math::
+    \theta_{MAP} & = \arg \max_{\theta'} P(\theta = \theta' | S=Z) \\
+    & = \arg \max_{\theta'} P(S=Z | \theta = \theta')\frac{P(\theta = \theta')}{P(S=Z)}
+
+Predictive Distribution
+^^^^^^^^^^^^^^^^^^^^^^^
+aka Full Bayes
+
+.. math::
+    P(Y=y | S=Z) = \int P(Y=y | \theta=\theta') P(\theta=\theta' | S=Z) d\theta'
+
+Mean a'Post
+^^^^^^^^^^^
+
+.. math::
+    \theta_{mean} = E[\theta | S=Z] = \int \theta' P(\theta=\theta' | S=Z) d\theta'
+
+Use
+^^^
+
+- draw enough data so that :math:`P(Y=y | X=\mathbf{x})` estimated for every possible pair
+    - this takes a lot of data
+- another approach: class of models
+- think of each model :math:`m` as a way of generating the training set Z of :math:`(\mathbf{x}, y)` pairs
+
+Compound Experiment
+^^^^^^^^^^^^^^^^^^^
+
+- prior :math:`P(M=m)` on model space
+- models give :math:`P(X=x | M=m)` (where :math:`x` is a pair :math:`(\mathbf{x}, y)`)
+- The joint experiment (if data is iid given m) is:
+
+.. math::
+    P(\{(\mathbf{x_i}, y_i)\}, m) = P(m) \prod_i (P(\mathbf{x_i} | m) P(y_i | \mathbf{x_i}, m))
+
+Generative and Discriminative Models
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+- Generative model: :math:`P((\mathbf{x}, y) | m)`
+    - tells how to generate examples (both instance and label)
+    - learn :math:`P(\mathbf{x} | y, m)` and use Bayes' rule
+    - common assumptions:
+        - :math:`P(\mathbf{x} | y, m)` is Gaussian
+        - :math:`P(y | m)` is Bernoulli
+- Discriminative model: :math:`P(y | h, \mathbf{x})`
+    - tells how to create labels from instances
+    - often :math:`f(\mathbf{x}) = \arg \max_y f_y(\mathbf{x})`
+
+
+