I want to use a few words of terminology. This again is a Bayes network of which the hidden variable C causes the still stochastic test outcomes T1 and T2. And what’s really important is that we assume not just that T1 and T2 are identically distributed. It is the same 0.9 for test one … Read more
We have tried the same trick as before, where we use exact same prior of first plus gives the following factors, 0.9, 0.2. But our minus gives us the probability 0.1 for a negative test result treatment of cancer. And a 0.8 for the inverse of a negative result of not getting cancer. You multiply … Read more
Calculate for me the probability of cancer, given that have received one positive and one negative test result. Please write your number into this box.
So the correct answer is 0.1698 approximately. And to compute this, I used the trick I’ve shown you before. Let me write down the running count for cancer, and for not cancer, as I integrate the various multiplications in Bayes rule. My prior for cancer was 0.01, and for non-cancer was 0.99. Then I get … Read more
The reason why I gave you all this is because I wanted to apply it now to a slightly more complicated problem, which is the two test cancer example. In this example, we, again, might have our unobservable cancer c, but now we’re running two tests, test one and test two. As before, the prior … Read more
So we just encountered our very first Bayes network and did a number of interesting calculations. Let’s now talk about Bayes Rule and look into more complex Bayes networks. I want to look at Bayes Rule again and make an observation that is being non-trivial. Here is Bayes Rule, and in practice what we find … Read more
And the answer is yes. F is independent of A. Well defined by our rules of the separation is that F is dependent on D and A is dependent on D. But, if you don’t know D, you can’t govern any dependency between A and F at all. Now, if you do know D, then … Read more
This leads me to the general study of Conditional independence in Bayes Networks that works often called D-Separation or Reachability. D-Separation is best studied by so called Active Triplet’s and Inactive Triplet’s. We have Active Triplet’s render variables dependent and Inactive Triplet’s render them independent. Any chain of three variables like this makes the initial … Read more
And the answer to this one is really interesting. A is clearly not independent of E because through C we can see an influence of A to E. Given B that doesn’t change. A still influences C despite the fact we know B. However if we know C, the influence is cut off. There is … Read more
In this specific example, the rule applies very, very simple. Any two variables are independent, if they’re not linked by just unknown variables. So for example, if we know B, and everything downstream of B, becomes independent of anything upstream of B. E is now independent of C, conditioned B, however, knowledge of B does … Read more
So C is not independent of A. In fact, A influences C by virtue of B. But if you know B, then A becomes independent of C, which means the only determinant to C is B. If you know B for sure, then knowledge of A won’t really tell you anything about C. C is … Read more
The next concept I’d like to teach you is called D-Separation. And let me start the discussion of this concept by a quiz. We have here a Bayes network and then when we ask to conditional independence question. Is C independent of A, please tell me yes or no. Is C independent of A given … Read more
So it takes 47 numerical probabilities to specify the joint compared to 65,000 if we didn’t have the structure. I think this example really illustrates the advantage of compact Bayes network representations. Over unstructured joint representations.
The answer is 3. It takes one parameter to specify P(A) from which we derive P(-A). It takes two parameters to specify P(B | A) and P(-A). >From which we can derive P(-B | A) and P(-B | -A). So it’s a total of 3 parameters for this Bayes’ Network. [BLANK_AUDIO]
To answer this question, let us add up these numbers. [INAUDIBLE] is 1, 1, 1, this is one incoming arc, so it’s 2, two incoming arc makes 4, one incoming arc is 2, 2 equals 4. Four incoming arcs make 16, we add all these numbers we get 47.
And here’s our car network which we discussed at the very beginning of this unit. How many parameters do we need to specify this network? Remember, there are 16 total variables and the naive join over those 16 will be 2 to the 16- 1, which is 65,535. Please write your answer into this box … Read more
And the answer is 19, so one here, one here, one here, two here, two here. Two arrows pointing to G which makes for four and three arrows pointing to D, two to the three is eight. So you get 1, 2, 3, 8, 2, 2, 4, if you add those up it’s 19
Here’s another quiz. How many parameters do we need to specify the joint distribution for this phase map over here where A B and C point into D. D points into E. F and G and C also point into G? Please write your answer into this box.
And the answer is 13. 1 over here, 2 over here, and 4 over here. Simplified this speaking, any variable that has K inputs requires 2 to the K search variables. So in total we have 1, 9, 13.
So we are now ready to define Bayes networks in a more general way. Bayes networks define probability distributions over graphs of random variables. Here’s an example of a graph of five variables and this Bayes network defines the distribution over those five random variables. Instead of enumerating all possibilities of combinations of these five … Read more