Q19

[sptripathi]

Need help in verifying if below understanding is correct ?

The Bayesian:
P(h=f | D) = P(D | h=f) * P(h=f) / P(D)

For this Q, we are given:
P(h=f) is uniform in [0,1]
D: one-person-with-heart-attack
Pick f = c (constant)

To simplify, I assume that h and f are a discrete random-variables with 10 possible values from (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0)
and each is equally likely with P=1/10. Essentially simplifying here to make P(h=f) a pmf which is actually a pdf.


Now:

P (D | h=f)
= Pr( one-person-with-heart-attack | h=f )
= Probability of one-person-with-heart-attack, given (h=f)
= c

( because if h=f were given, then the Prob of one picked person getting heart-attack is c, as defined by f )

Plug in above to get:
P(h=f | D) = c * P(h=f) / P(D)

Does above sound correct ?
Also P(D) =1 in this case ?

Thanks.

[Dorian]

I find this exercise simple but very useful. If one thinks of the series of following measurements (1s and 0s for heart attack or not) one can clearly form an idea how this transforms step-by-step from a uniform distribution to a Bernoulli one.

Does this mean that this example represents one of those cases where the initial prior is irrelevant and we can safely use it for learning? Also, is this some form of reinforcement learning?

thanks,
Dorian.

[yaser]

Originally Posted by Dorian

Does this mean that this example represents one of those cases where the initial prior is irrelevant and we can safely use it for learning? Also, is this some form of reinforcement learning?

In this case, with sufficient number of examples, the prior indeed fades away. Noisy examples blur the line between supervised and reinforcement learning somewhat as the information provided by the output is less definitive than in the noiseless case.

[nkatz]

I am very confused by this problem. Perhaps this questions will help:
Is P(D|h=f) a function of D or h or both? It looks to me like it's a function of D, but we need to convert it to a function of h to get the posterior... Is this correct?

[yaser]

Originally Posted by nkatz

Is P(D|h=f) a function of D or h or both?

Let us first clarify the notions. The data set has one data point in it which is either (heart attack) or (no heart attack). Being a function of means being a function of that value (), so indeed is a function of , and it so happens in this problem that the value is fixed at . The probability is also a function of (which happens to be the same as according to what we are conditioning on).

Since is fixed, this leaves as a function of only.

[Elroch]

There is an analogy that may be enlightening, which I thought of because of the presentation of the first part of this course.

Suppose you have a large number of urns each containing a large number of black and white balls in varying proportions. You are told how many urns there are with each proportion.

Then you go up to one of the urns and take out a ball which you find is black. The question is how likely it is that specific urn has each particular fraction of black balls.