Say you have one continuous variable \(X\), and one discrete variable \(Y\), and you desire to express the probability of \(X\) conditioned upon \(Y\) using a gaussian model:

\begin{equation} p(x|y) = \begin{cases} \mathcal{N}(x \mid \mu_{1}, \sigma_{1}^{2}), y^{1} \\ \dots \\ \mathcal{N}(x \mid \mu_{1}, \sigma_{1}^{2}), y^{n} \\ \end{cases} \end{equation}

conditional plan is a POMDP representation technique. We can represent a conditional plan as a tree.

toy problem

crying baby POMDP problem:

• actions: feed, ignore
• reward: if hungry, negative reward
• state: two states: is the baby hungry or not
• observation: noisy crying (she maybe crying because she’s genuinely hungry or crying just for kicks)

formulate a conditional plan

we can create a conditional plan by generating a exponential tree based on the observations. This is a policy which tells you what you should do given the sequence of observations you get, with no knowledge of the underlying state.

There are many condition in the Great Depression caused

• by 1932, 1/4 had no work
• emigration exceeded immigration
• decrease in American birth
• increase of mental illness and suicide
• people create Hooverviles
• movies and radio became much more popular

proportional confidence intervals

We will measure a single stastistic from a large population, and call it the point estimate. This is usually denoted as \(\hat{p}\).

Given a proportion \(\hat{p}\) (“95% of sample), the range which would possibly contain it as part of its \(2\sigma\) range is the \(95\%\) confidence interval.

Therefore, given a \(\hat{p}\) the plausible interval for its confidence is:

\begin{equation} \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \end{equation}

where, \(n\) is the sample size, \(\hat{p}\) is the point estimate, and \(z*=1.96\) is the critical value, the z-score denoting \(95\%\) confidence (or any other desired confidence level).

Could a different choice of evaluation order change the terminating result of the program; note that this says nothing about whether or not particular evaluation order terminates.

constituents

requirements

A set of rewrite rules is confluent if for any expression \(E_0\), should \(E_0 \to^{* } E_1\) and \(E_0 \to^{* } E_2\), then there exists some \(E_3\) such that \(E_1 \to^{*} E_3\) and \(E_2 \to^{ *} E_3\)

Why this instead of saying we will end up at one state after all evaluations? Suppose a particular system never terminates, we have to specify a particular confluent state because otherwise the notion of all is bad.