Lecture 14: Conditional Random Variables
Feb 12th, 2021
Lecture Materials
Slides Ppt --> --> Slides PDF Concept Check Recording
Learning Goals
Learn what it means to talk about conditional probabilities with random variables. Be able to calculate conditional expectations. Be able to use the law of total expectation.
Concept Check
https://www.gradescope.com/courses/226051/assignments/1018506
Questions & Answers
Q: Would you mind defining obfuscate again?
A1: here, obfuscate means obscure X_i half of the time with a random coin flip. Y_i is still mathematically defined in terms of X_i, but we introduce an unpredictable fair coin flip so you’re not fully revealing the private data that might be associated with X_i.
A2: However, the statistics of X_i (expectation, for instance) can still be derived from Y_i, and if you’re only interested in the statistics, you can still get them without disclosing the private data 100% of the time.
Q: So we could also use the fact that one table sums to one accross columns and the other does across rows?
A1: absolutely.. that’s what I did :)
A2: And that’s just what Chris did too. :)
Q: do you need the parentheses in #8 or is it just there for fun?
A1: Just for fun... not needed.
Q: Is there a difference between E[X|Y] and E[X|Y=y]?
A1: nope, just two different notations.
Q: How did we do the first step?
A1: E[X|Y] is a function of Y, so the average value of that function, where each value of that function is weighted by the probability that Y = y.
Q: so we could find the expectation of S by summing over diffrent values for D_2?
A1: You certainly can. But just to clarify, do you mean computing E[S] from E[S|D_2]? That’s how I’m reading your question.
Q: Can you explain again why E[X,Y] doesn't make sense?
A1: You can only compute the expectation of a single random variable, like X, or Y, or Z = X + Y. X, Y isn’t a single random variable, because it’s not clear how X and Y are being combined.
Q: sorry I missed E[X, Y]. Is this asking E[X and Y], so a number?
A1: No, that particular one doesn’t make sense.
A2: You can only compute the expectation of a single random variable, like X, or Y, or Z = X + Y. X, Y isn’t a single random variable, because it’s not clear how X and Y are being combined.
Q: I mean we could find the expectation of S (overall) by inputing all valid numbers for D_2 (1,2,3, …) into E[E[S|D_S]]
A1: Absolutley. E[S| D_2] (which is what I think you mean) is a function of D_2, so you can now compute the weighted average of that function for all valid D_2 outcomes.
Q: how is the position of the second best engineer figured into k/i+1
A1: Technicaly, all engineers better than any in the first k are ones that stop the process. We basically want to maximize the likelihood that the best engineer isn’t among the first k engineers, and that instead is the first engineer of all better-than-first-k engineers to be interviewed once you can actually hire someone.
COMMENTS
A conditional random field may be viewed as an undirected graphical model, or Markov random field [3], globally conditioned on X, the random variable representing observation sequences. Formally, we define G = (V,E) to be an undirected graph such that there is a node v ∈ V corresponding to each of the
Sep 9, 2022 · I have a question about conditional random assignment. The simplified dataset looks as below: COMPANY BOARDROLE INSIDER A Acting Director Yes B CEO Yes C Independent Director No D Chairman Unknown E Chairman Unknown F Member Unknown G Independent Director Outsider H Member Unknown I Member Unknown J Member Unknown
Conditional Random Fields Jacob Andreas / MIT 6.864 / Spring 2020. Admin. ... assignment of tags to observations? argmax Q p(O,Q) best length-t tag seq. ending in j
discrete case in this chapter. We denote an assignment to X by x, and we denote an assignment to a set A ⊂ X by x A, and similarly for Y. We use the notation 1 {x=x0} to denote an indicator function of x which takes the value 1 when x = x 0 and 0 otherwise. A graphical model is a family of probability distributions that factorize according
Jun 19, 2023 · Conditional Random Field (CRF) is the statistical and computational framework that allow for modeling linear chains and more complex sequentially dependent events [1].
Conditional Independence and Random Variables 7. Variance, Bernoulli, and Binomial RVs 8. Poisson, Geometric, and Negative Binomial RVs 9. Continuous 10. Gaussian 11. Probabilistic Models 12. Independent Random Variables 13. Joint Statistics 14. Conditional Random Variables 15. Continuous Probabilistic Models 16. Continuous Inference 17.
Sep 8, 2019 · Conditional Random Field is a special case of Markov Random field wherein the graph satisfies the property : “When we condition the graph on X globally i.e. when the values of random variables in X is fixed or given, all the random variables in set Y follow the Markov property p(Yᵤ/X,Yᵥ, u≠v) = p(Yᵤ/X,Yₓ, Yᵤ~Yₓ), where Yᵤ~Y ...
3. Conditional Random Fields In what follows,-is a random variable over data se-quences to be labeled, and. is a random variable over corresponding label sequences. All components.0/ of. are assumed to range over a finite label alphabet 1. For ex-ample,-might range over natural language sentences and. range over part-of-speechtaggings of those ...
A solution to this problem is to model the conditional distribution p(yjx) directly, which is all that is needed for classi cation. This is a conditional random eld (CRF). CRFs are essentially a way of combin-ing the advantages of classi cation and graphical modeling, combining the ability to compactly model multivariate data with the ability to
sum of two independent random variables, each having a continuous distri-bution. Example <8.10> Suppose X has a continuous distribution with den-sity f and Y has a continuous distribution with density g. If X and Y are independent then the random variable Z = X+ Y has a continuous distribution with density <8.9> h(z) = Z 1 1 g(z x)f(x)dx for ...