COMS 4701 FINAL PRACTICE EXAM SPRING 2024 1.Consider a second-order hidden Markov model, in which Xt generally depends on both  Xt-1  and Xt-2.The initial distribution is    Pr(Xo,X₁),transition probabilities

COMS 4701

FINAL PRACTICE EXAM

SPRING 2024

1.Consider a second-order hidden Markov model, in which Xt generally depends on both  Xt-1  and Xt-2.The initial distribution is    Pr(Xo,X₁),transition probabilities are Pr(Xt|Xt-1,Xt-2)for t≥2, and observation probabilities are Pr(E|Xt)for   t≥1.

(a)Circle either true or false for each of the conditional independence statements below that are guaranteed to hold in the second-order HMM.

(b)  Give a minimal expression for Pr(X₁,…,X₅,e₁,…,e₅) using the HMM parameters. (Multiplica- tion of CPTs will be interpreted as multiplication of factors.)

(c)   Suppose we have αt=Pr(Xt-1,Xt|e1:t)and we want to compute αt+1 =Pr(Xt,Xt+1|e1:t+1). Give a minimal expression for αt+1 using at and the HMM parameters,normalizing if necessary.

2. Flying during the holidays can be a stressful time,since so many things can go wrong. Bad weather (W) or mechanical airplane problems (M) can delay your flight (D); mechanical problems can also affect the chances of your baggage (B) being lost. Suppose you have a probabilistic model of the relationships  between these Boolean events as follows:

(a)   Draw a representative Bayesian network of this model. Be sure to label your nodes and indicate directionalityon the edges.

(b)   Are weather (W) and whether your baggage (B) makes it back safely with you independent of each other?

(c)   Suppose  you  are  sitting  at  the   airport  and  you  tell  your   family  that  your   fight  was   indeed  delayed. Given  this  information,are  weather  and  baggage  arriving  safely  conditionally  independent  of  each other?

(d)   Write  an  analytical expression for Pr(W,B|D=+d), the  joint distribution of  weather  and baggage given that  your flight  is   delayed.Your expression should only include sums,products, and/or quotients of  terms fro  the model described  above.

(e)   Numerically compute  Pr(+w,+b,+d),the  joint  probability that bad weatheroccurred,your    bag- gage  got  lost,and  your  flight  was  delayed.

3.   A   recycling robot is trying to classify the objects that it sees as bottles(B=+b)or notbottles (B=-b).The robot considers three  binary  features:whether  the object is rounded(R=+r)or not (R=-r),whether  it  is  made of glass(G=+9)or plastic(G=-9),and whether it is small  (S=+8) or   large(S=-s).The robot  is given a labeled data set as follows:

(a)   Suppose we learn a  naive Bayes classifier from this data.Find the numerical parameters that would be learned usingα=1    smoothing. Please write your answers as reduced fractions.

(b)   Using   the   learned   model,how   does   the   robot   classify   the   feature   set (一r,-g,-s)?

(c)  Suppose  our  data   set  did  not  include  the  class  labels.If  we   were   to   learn   a  naive  Bayes  model using   expectation-maximization,are   we    guaranteed   to   recover   the    maximum-likelihood   parameters learned   from  the  labeled   data   set?Why   or  why  not?

(d)   Convert the  features to  numerical    values    by     treating    +as     +1     and-as-1.Consider    a     linear classifier    that     predicts B=-b    if     fw(x)≤0    and     B=+b     otherwise.What     is    the     classification accuracy   on   the   data   set   given  a  model   with   weight   vector   w=(1,1,0,1)?

(e)   Again   starting   from   w,compute   the   update   made   to   w   using   the   perceptron   learning   rule   after the  first  mistake  made  on  the  data  set.

(f)   A  sigmoid  activation  function  would  still  yield  the  same  predictions  and  same  classification  accu- racy  as  the  hard  threshold  function  described  above.Give  two  different  advantages  that  a  sigmoid function  has   over  the  hard  threshold.

(g)   Suppose we pass our data set through  the neural network below,where  x  is  R,y  is  G,and  z  is  S. Find  the  individual  outputs  of each forward pass

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