$29.99
VE 492 Homework10
Question 1: Rejection SamplingWe will work with a Bayes' net of the following structure.
In this question, we will perform rejection sampling to estimate P(C=1|B=1,E=1). Performone
round of rejection sampling, using the random samples given in the table below. Variables are
sampled in the order A,B,C,D,E. In the boxes below, choose the value (0 or 1) that each variable
gets assigned to. Note that the sampling attempt should stop as soon as you discover that the
sample will be rejected. In that case mark the assignment of that variable. When generating random samples, use as many values as needed from the table below, which we
generated independently and uniformly at random from [0,1). Use numbers from left to right. To sample a binary variable W with probability P(W=0)=p and P(W=1)=1-p using a value a fromthe table, choose W=0 if a<p and W=1 if a>=p
Enter either a 0 or 1 for each variable that you assign a value to. Upon rejecting a sample, enter its
assigned value, and leave the fields for the remaining variables blank (use “b” for blank). For
example, if C gets rejected, fill in “b” for D and E. A____B____C____D____E____ Which variable will get rejected? If no variables will get rejected, leave the field belowblank. ____ Sample answer:
01bbbB
Question 2: Estimating Probabilities fromSamples
Below are a set of samples obtained by running rejection sampling for the Bayes' net fromthe
previous question. Use them to estimate P(C=1|B=1,E=1) . The estimation cannot be made
whenever all samples were rejected. In this case, input -1.
Sample answer:
0.2
Question 3: Likelihood WeightingWe will work with a Bayes' net of the following structure.
In this question, we will perform likelihood weighting to estimate P(C=1|B=1,E=1). Generate a
sample and its weight, using the random samples given in the table below. Variables are sampledin the order . In the table below, select the assignments to the variables you sampled. When generating random samples, use as many values as needed from the table below, which we
generated independently and uniformly at random from [0,1). Use numbers from left to right. To sample a binary variable W with probability P(W=0)=p and P(W=1)=1-p using a value a fromthe table, choose W=0 if a<p and W=1 if a>=p
Enter either a 0 or 1 for each variable assigned by a pass of likelihood weighting with the
generated samples above. A____B____C____D____E____ What is the weight for the sample you obtained above? (Keep 2 decimal places) ____ Sample answer:
011010.16
Question 4: Estimating Probabilities fromWeighted Samples
Below are a set of weighted samples obtained by running likelihood weighting for the Bayes' net
from the previous question. Use them to estimate P(C=1|B=1,E=1). Input -1 in the box belowif
the estimation cannot be made.
____ (Keep 2 decimal places)
Sample answer:
0.25
Question 5: HMMs, Part I
Consider the HMM shown below. The prior probability 0) , dynamics model t1
) , and sensor model ) are as
follows:
We perform a first dynamics update, and fill in the resulting belief distribution 1). We incorporate the evidence 1 . We fill in the evidence-weighted distribution 1 1)1), and the (normalized) belief distribution 1). Note: Please write your answer for each table in one row, that is, there will be 3 rows for this
question. Besides, please use values rounded to 3 decimal places. Sample Answer:
0.160,0.170
0.200,0.211
0.222,0.180
1) You get to perform the second dynamics update. Write your answer to fill in the resultingbelief distribution ). )
0
1
2) Now incorporate the evidence ‸ . Write your answer to fill in the evidence-weighteddistribution ‸)), and the (normalized) belief distribution ). ‸))
0
1
)
0
1
Question 6: HMMs, Part II
Consider the same HMM. The prior probability 0) , dynamics model t1
) , and sensor model ) are as
follows:
In this question we'll assume the sensor is broken and we get no more evidence readings. We are
forced to rely on dynamics updates only going forward. In the limit as , our belief about should converge to a stationary distribution ) defined as follows:
Recall that the stationary distribution satisfies the equation
for all values in the domain of . In the case of this problem, we can write these relations as a set of linear equations of the formIn the spaces below, fill in the coefficients of the linear system. The system you have written has
many solutions (consider (0,0), for example), but to get a probability distribution we want the
solution that sums to one. Write your answer to fill in the table below. (Hint: to check your answer, you can also write some code and run till convergence.)
Note: Please write your answer for each table in one row, that is, there will be 2 rows for this
question. Besides, please use values rounded to 3 decimal places. Sample Answer:
0.160,0.170,0.160,0.170
0.200,0.211
coefficient value
a
b
c
d
)
0
1