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update

The solution follows obtain the right answer now.

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The Question is here The problemFigure 13 (Bayesian Network)

And my answer is here

$P(a_0)=P(a_0|r_0) + P(a_0|r_1) = 0.4$

$P(d_0)=\sum_{r,c}P(d_0|r,c)\cdot P(r)\cdot P(c)=0.64$

\begin{align*} P(c_0|s_0,d_0,a_0) &= \frac{P(c_0,s_0,d_0,a_0)}{P(s_0,d_0,a_0)}\\ &=\frac{P(c_0,s_0,d_0,a_0,r_0)+P(c_0,s_0,d_0,a_0,r_1)}{P(s_0,d_0,a_0)}\\ &=\frac{P(c_0) P(s_0|a_0,d_0)[P(r_0)P(d_0|c_0,r_0) P(a_0|r_0)+ P(r_1) P(d_0|c_0,r_1) P(a_0|r_1)]}{P(s_0|a_0,d_0) \cdot \sum_{r,c}P(c)P(r)P(a_0|r)P(d_0|c,r)}\\ &=\frac{0.6\times 0.6 \times (0.5 \times 1 \times 0.2 + 0.5 \times 0.6 \times 0.6)}{0.6 \times 0.208}\\ &=0.807 \end{align*}

Where \begin{align*} \sum_{r,c}P(c)P(r)P(a_0|r)P(d_0|c,r) &=0.5\sum_{r,c}P(c)P(a_0|r)P(d_0|c,r)\\ &=0.5(0.6\times 1\times 0.2 + 0.6\times 0.6\times 0.6 + 0.4\times 0.7\times 0.2 + 0.4\times 0.1\times 0.6)\\ &=0.208 \end{align*}

I learned Bayesian Network point for the first time |-_-|, what's wrong with my answer?

Any advice will be appreciated!

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  • $\begingroup$ Hi. Jason. This is a mathjax-enabled site. Please use this to format your equations and write up your solution rather than a distant snap of a sheet. For help, check Instructions on how to use LaTeX on CrossValidated. $\endgroup$ Jan 5, 2023 at 10:34
  • $\begingroup$ Also, please add the self-study tag in your post. $\endgroup$ Jan 5, 2023 at 10:35
  • $\begingroup$ @User1865345 ok! I'll do that later. $\endgroup$
    – Jason Lee
    Jan 5, 2023 at 10:38
  • $\begingroup$ how did you obtain the expression in the denominator? I would say that $p(s,a,d)=p(a)p(d|a)d(s|d,a)$ but you have a different expression. And how did you get the numbers to plug in for $p(a)$ for example? $\endgroup$
    – Thomas
    Jan 5, 2023 at 12:12
  • $\begingroup$ @Thomas $p(s,a,d)=p(a,d)p(s|d,a)$, Here I should mistakenly think that $a$ and $d$ are independent, so I obtain my expression. But now I try to calculate $p(a,d)$ with the expression $p(a_0,d_0) = \sum_{r,c}p(c)p(r)p(a_0|r)p(d_0|c,r)$, I still can't get the right answer. I'm confused. $\endgroup$
    – Jason Lee
    Jan 6, 2023 at 7:18

2 Answers 2

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For the moment I write my brute force calculation as a reference. Maybe there are smarter ways using more properly the properties of Bayesian Networks, but at least I get the correct answer $(d)$. In case I find something smarter will update.

We first define:

$A=p(c_0,s_0,d_0,a_0)$, $B=p(c_1,s_0,d_0,a_0)$

And note that the probability that we want is:

$R=A/(A+B)$

Now:

$A=\sum_{r,f} p(c_0,r,f,d_0,a_0,s_0)=p(c_0)p(s_0|d_0,a_0)\sum_{r,f}p(r)p(d_0|c_0,r)p(a_0|r)p(f|c_0)$

Note that we have $p(r)=0.5$ and $p(f|c_0)=0.5$ for every $r,f$ so that this simplifies:

$A=p(c_0)p(s_0|d_0,a_0)*1/2*1/2*2*\sum_r p(d_0|c_0,r)p(a_0|r)$

Putting numbers inside:

$A=0.6*0.6*0.5*(0.2+0.6*0.6)$

$B$ is a bit more cumbersome. We arrive at:

$B=\sum_{r,f} p(c_1,r,f,d_0,a_0,s_0)=p(c_1)p(s_0|d_0,a_0)1/2 \sum_{r,f}p(d_0|c_1,r)p(a_0|r)p(f|c_1)$

and the four cases $(r_0,f_0),(r_1,f_1),(r_0,f_1),(r_1,f_0)$ must be considered seperately. Up to my calculations:

$B=0.4*0.6*0.5*(0.7*0.2*0.3+0.1*0.6*0.7+0.7*0.2*0.7+0.1*0.6*0.3)$


UPDATE: If we first sum over $f$ due to normaliazation we can simplify further before plugging in the numbers:

$B=p(c_1)p(s_0|d_0,a_0)1/2 \sum_{r}p(d_0|c_1,r)p(a_0|r)$

, leading to the simplified expression for B:

$B=0.4*0.6*0.5*(0.7*0.2+0.1*0.6)$

which is equivalent to the previous one.


Finally $R=A/(A+B) \sim 0.807$, which is answer $(d)$.

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  • $\begingroup$ get it!many thanks!! $\endgroup$
    – Jason Lee
    Jan 5, 2023 at 10:09
  • $\begingroup$ But I just wonder why should we take $f$ into consideration, my original thought is whether $f$ is considered or not, the result will be the same, because $f$ is independent of $a$,$d$ and $s$. $\endgroup$
    – Jason Lee
    Jan 5, 2023 at 10:44
  • $\begingroup$ I should reopen my book that I do not have here but I do not think right now that $f$ and $d$ are independent (for example). In the languange of $d-$separation the path $f->c->d$ is open I think so that $d$ and $f$ are not in general independent. They have a correlation due to a common cause (camera) $\endgroup$
    – Thomas
    Jan 5, 2023 at 10:54
  • $\begingroup$ I just noticed that in the calculation of B the sum over f goes away since we have sum_f p(f|c1) and this is 1 by normalizion. I will update the calculation with this trick. This suggests that the calculation may be done in a smarter way using some independence property but at the moment I do not see how... $\endgroup$
    – Thomas
    Jan 5, 2023 at 12:25
  • $\begingroup$ Indeed the same trick can be applied to A if we first some over f. Maybe you are right about the role of f I will think of a way to justify it without writing down the explicit summation... $\endgroup$
    – Thomas
    Jan 5, 2023 at 12:33
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I add an alternative approach, based on the comments of the OP, to evaluate $p(s_0,a_0,d_0)$, which is in the previous answer the denominator $A+B$.

As suggested by the comments of the OP: $p(s_0,a_0,d_0)=p(s_0|a_0,d_0)p(a_0,d_0)$ and since the Bayesian network provides $p(s_0|a_0,d_0)=0.6$ we just need the second factor, which is equal to:

$p(a_0,d_0)=\sum_{r,c} p(r)p(c)p(a_0|r)p(d_0|r,c)$

( variables $s$ and $f$ do not enter year because they have only outgoing edges from the set of variables $a,d,r,c$ )

Now this is equal to, since $p(r)=0.5$, and considering the four cases $(r,c)$:

$p(a_0,d_0)=0.5*(0.6*0.2*1.0+0.4*0.2*0.7+0.6*0.6*0.6+0.4*0.6*0.1)$

And therefore:

$p(a_0,d_0,s_0)=0.6*0.5*(0.6*0.2*1.0+0.4*0.2*0.7+0.6*0.6*0.6+0.4*0.6*0.1)$

One can numerically verify that this expression is equal to the denominator $A+B$ in the first answer. Of course one still needs to evaluate the numerator $A$ to get to the final result.

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  • $\begingroup$ That's right! This way I got the right answer! Much thanks to you again! $\endgroup$
    – Jason Lee
    Jan 12, 2023 at 2:54

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