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Fix mathematical equations for chapter 20-26
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‎_site/bayesian-learning-exercises/ex_8/index.html

+4-2
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@@ -168,7 +168,8 @@ <h3 class="masthead-title">
168168

169169
This exercise investigates properties of
170170
the Beta distribution defined in
171-
Equation (<a class="equationRef" title="" href="#">beta-equation</a><br />.
171+
Equation (<a class="equationRef" title="" href="#">beta-equation</a>).
172+
<br />
172173

173174
1. By integrating over the range $[0,1]$, show that the normalization
174175
constant for the distribution $[a,b]$ is given by
@@ -205,7 +206,8 @@ <h3 class="masthead-title">
205206

206207
This exercise investigates properties of
207208
the Beta distribution defined in
208-
Equation (<a class="equationRef" title="" href="#">beta-equation</a><br>.
209+
Equation (<a class="equationRef" title="" href="#">beta-equation</a>).
210+
<br>
209211

210212
1. By integrating over the range $[0,1]$, show that the normalization
211213
constant for the distribution $[a,b]$ is given by

‎_site/bayesian-learning-exercises/index.html

+2-1
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@@ -305,7 +305,8 @@ <h1 id="20-learning-probabilistic-models">20. Learning Probabilistic Models</h1>
305305

306306
This exercise investigates properties of
307307
the Beta distribution defined in
308-
Equation (<a class="equationRef" title="" href="#">beta-equation</a><br />.
308+
Equation (<a class="equationRef" title="" href="#">beta-equation</a>).
309+
<br />
309310

310311
1. By integrating over the range $[0,1]$, show that the normalization
311312
constant for the distribution $[a,b]$ is given by

‎_site/markdown/20-Learning-Probabilistic-Models/exercises/ex_8/question.md

+2-1
Original file line numberDiff line numberDiff line change
@@ -2,7 +2,8 @@
22

33
This exercise investigates properties of
44
the Beta distribution defined in
5-
Equation (<a class="equationRef" title="" href="#">beta-equation</a><br>.
5+
Equation (<a class="equationRef" title="" href="#">beta-equation</a>).
6+
<br>
67

78
1. By integrating over the range $[0,1]$, show that the normalization
89
constant for the distribution ${{\rm beta}}[a,b]$ is given by

‎_site/markdown/22-Natural-Language-Processing/exercises/ex_11/question.md

+2-2
Original file line numberDiff line numberDiff line change
@@ -27,8 +27,8 @@ each of the following models, propose a corresponding numeric measure.<br>
2727
end of the hour.<br>
2828

2929
5. The searcher will look through all the answers. Examining a document
30-
has cost \\$ A; finding a relevant document has value \\$ B; failing
31-
to find a relevant document has cost \\$ C for each relevant
30+
has cost \$ A; finding a relevant document has value \$ B; failing
31+
to find a relevant document has cost \$ C for each relevant
3232
document not found.<br>
3333

3434
6. The searcher wants to collect as many relevant documents as

‎_site/markdown/22-Natural-Language-Processing/exercises/ex_2/question.md

+3-3
Original file line numberDiff line numberDiff line change
@@ -1,10 +1,10 @@
11

22

3-
Write a program to do **segmentation** of
3+
Write a program to do <b>segmentation</b> of
44
words without spaces. Given a string, such as the URL
55
“thelongestlistofthelongeststuffatthelongestdomainnameatlonglast.com,”
6-
return a list of component words: \[“the,” “longest,” “list,”
7-
$\ldots$\]. This task is useful for parsing URLs, for spelling
6+
return a list of component words: [“the,” “longest,” “list,”
7+
$\ldots$]. This task is useful for parsing URLs, for spelling
88
correction when words runtogether, and for languages such as Chinese
99
that do not have spaces between words. It can be solved with a unigram
1010
or bigram word model and a dynamic programming algorithm similar to the

‎_site/markdown/23-Natural-Language-For-Communication/exercises/ex_10/question.md

+6-7
Original file line numberDiff line numberDiff line change
@@ -3,9 +3,8 @@
33
In this exercise you will transform $\large \varepsilon_0$ into
44
Chomsky Normal Form (CNF). There are five steps: (a) Add a new start
55
symbol, (b) Eliminate $\epsilon$ rules, (c) Eliminate multiple words on
6-
right-hand sides, (d) Eliminate rules of the form<br>
7-
(${\it X}$<br>
8-
${{\;}}\rightarrow{{\;}}$${\it Y}$),<br>
6+
right-hand sides, (d) Eliminate rules of the form
7+
(${\it X} \rightarrow{{\;}}$${\it Y}$),
98
(e) Convert long right-hand sides into binary rules.<br>
109

1110
1. The start symbol, $S$, can occur only on the left-hand side in CNF.
@@ -20,21 +19,21 @@ ${{\;}}\rightarrow{{\;}}$${\it Y}$),<br>
2019

2120
3. A word can appear on the right-hand side in a rule only of the form
2221
(${\it X}$
23-
${{\;}}\rightarrow{{\;}}$*word*).
22+
${{\;}}\rightarrow{{\;}}$<i>word</i>).
2423
Replace each rule of the form (${\it X}$
25-
${{\;}}\rightarrow{{\;}}$…*word* …)
24+
${{\;}}\rightarrow{{\;}}$…<i>word</i> …)
2625
with (${\it X}$
2726
${{\;}}\rightarrow{{\;}}$…${\it W'}$ …)
2827
and (${\it W'}$
29-
${{\;}}\rightarrow{{\;}}$*word*),
28+
${{\;}}\rightarrow{{\;}}$<i>word</i>),
3029
using a new symbol ${\it W'}$.<br>
3130

3231
4. A rule (${\it X}$
3332
${{\;}}\rightarrow{{\;}}$${\it Y}$)
3433
is not allowed in CNF; it must be (${\it X}$
3534
${{\;}}\rightarrow{{\;}}$${\it Y}$
3635
${\it Z}$) or (${\it X}$
37-
${{\;}}\rightarrow{{\;}}$*word*).
36+
${{\;}}\rightarrow{{\;}}$<i>word</i>).
3837
Replace each rule of the form (${\it X}$
3938
${{\;}}\rightarrow{{\;}}$${\it Y}$)
4039
with a set of rules of the form (${\it X}$

‎_site/markdown/23-Natural-Language-For-Communication/exercises/ex_14/question.md

+1-1
Original file line numberDiff line numberDiff line change
@@ -5,6 +5,6 @@ context-free grammar cannot. Show an augmented context-free grammar for
55
the language $a^nb^nc^n$. The allowable values for augmentation
66
variables are 1 and $SUCCESSOR(n)$, where $n$ is a value. The rule for a sentence
77
in this language is<br>
8-
$$S(n) {{{{\;}}\rightarrow{{\;}}}}A(n) {{\;}}B(n) {{\;}}C(n) \ .$$
8+
$$S(n) \rightarrow A(n) B(n) C(n) \ .$$
99
Show the rule(s) for each of ${\it A}$,
1010
${\it B}$, and ${\it C}$.

‎_site/markdown/23-Natural-Language-For-Communication/exercises/ex_2/question.md

+2-2
Original file line numberDiff line numberDiff line change
@@ -7,7 +7,7 @@ such as $is$, $duck$, and so on). The HMM model includes a prior
77
${\textbf{P}}(N_0)$, a transition model
88
${\textbf{P}}(N_{t+1}|N_t)$, and a sensor model
99
${\textbf{P}}(W_t|N_t)$. Show that every HMM grammar can be
10-
written as a PCFG. \[Hint: start by thinking about how the HMM prior can
10+
written as a PCFG. [Hint: start by thinking about how the HMM prior can
1111
be represented by PCFG rules for the sentence symbol. You may find it
1212
helpful to illustrate for the particular HMM with values $A$, $B$ for
13-
$N$ and values $x$, $y$ for $W$.\]
13+
$N$ and values $x$, $y$ for $W$.]

‎_site/markdown/23-Natural-Language-For-Communication/exercises/ex_7/question.md

+13-12
Original file line numberDiff line numberDiff line change
@@ -12,18 +12,19 @@ $Article \rightarrow \textbf{the} \quad Noun \rightarrow \textbf{supermarket}$<b
1212
Which of the following three grammars, combined with the lexicon,
1313
generates the given sentence? Show the corresponding parse tree(s).<br>
1414

15-
| $\quad\quad\quad\quad (A):\quad\quad\quad\quad$ | $\quad\quad\quad\quad(B):\quad\quad\quad\quad$ | $\quad\quad\quad\quad(C):\quad\quad\quad\quad$ |
16-
| --- | --- | --- |
17-
| $S\rightarrow NP\space VP$ | $S\rightarrow NP\space VP$ | $S\rightarrow NP\space VP$ |
18-
| $NP\rightarrow Pronoun$ | $NP\rightarrow Pronoun$ | $NP\rightarrow Pronoun$ |
19-
| $NP\rightarrow Article\space Noun $ | $NP\rightarrow Noun$ | $NP\rightarrow Article\space NP$ |
20-
| $VP\rightarrow VP\space PP$ | $NP\rightarrow Article\space NP$ | $VP\rightarrow Verb\space Adv$ |
21-
| $VP\rightarrow VP\space Adv\space Adv$ | $VP\rightarrow Verb\space Vmod$ | $Adv\rightarrow Adv\space Adv$ |
22-
| $VP\rightarrow Verb$ | $Vmod\rightarrow Adv\space Vmod$ | $Adv\rightarrow PP$ |
23-
| $PP\rightarrow Prep\space NP$ | $Vmod\rightarrow Adv$ | $PP\rightarrow Prep\space NP$ |
24-
| $NP\rightarrow Noun$ | $Adv\rightarrow PP$ | $NP\rightarrow Noun$ |
25-
| $\quad$ | $PP\rightarrow Prep\space NP$ | $\quad$ |<br>
26-
15+
$$
16+
\quad\quad\quad\quad (A):\quad\quad\quad\quad \quad\quad\quad\quad(B):\quad\quad\quad\quad \quad\quad\quad\quad(C):\\
17+
\quad\quad\quad\quad S \rightarrow NP \space VP \quad\quad\quad\quad \quad\quad\quad\quad S\rightarrow NP\space VP \quad\quad\quad\quad S\rightarrow NP\space VP\\
18+
\quad\quad\quad\quad NP\rightarrow Pronoun \quad\quad\quad\quad NP\rightarrow Pronoun \quad\quad\quad\quad NP\rightarrow Pronoun\\
19+
\quad\quad\quad\quad NP\rightarrow Article\space Noun \quad\quad\quad\quad NP\rightarrow Noun \quad\quad\quad\quad NP\rightarrow Article\space NP\\
20+
\quad\quad\quad\quad VP\rightarrow VP\space PP \quad\quad\quad\quad NP\rightarrow Article\space NP \quad\quad\quad\quad VP\rightarrow Verb\space Adv\\
21+
\quad\quad\quad\quad VP\rightarrow VP\space Adv\space Adv \quad\quad\quad\quad VP\rightarrow Verb\space Vmod \quad\quad\quad\quad Adv\rightarrow Adv\space Adv\\
22+
\quad\quad\quad\quad VP\rightarrow Verb \quad\quad\quad\quad Vmod\rightarrow Adv\space Vmod \quad\quad\quad\quad Adv\rightarrow PP\\
23+
\quad\quad\quad\quad PP\rightarrow Prep\space NP \quad\quad\quad\quad Vmod\rightarrow Adv \quad\quad\quad\quad PP\rightarrow Prep\space NP\\
24+
\quad\quad\quad\quad NP\rightarrow Noun \quad\quad\quad\quad Adv\rightarrow PP \quad\quad\quad\quad NP\rightarrow Noun\\
25+
\quad\quad\quad\quad\quad \quad\quad\quad\quad PP\rightarrow Prep\space NP \quad\quad\quad\quad \quad\quad\quad\quad
26+
27+
$$
2728

2829
For each of the preceding three grammars, write down three sentences of
2930
English and three sentences of non-English generated by the grammar.

‎_site/markdown/25-Robotics/exercises/ex_1/question.md

+9-14
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@@ -7,27 +7,22 @@ expected value—because of the way particle filtering works. In this
77
question, you are asked to quantify this bias.<br>
88

99
To simplify, consider a world with four possible robot locations:
10-
$X=\{x_{{\rm 1}},x_{{\rm 2}},x_{{\rm 3}},x_{{\rm 4}}\}$. Initially, we
10+
$X=\{x_1,x_2,x_3,x_4\}$. Initially, we
1111
draw $N\geq {{\rm 1}}$ samples uniformly from among those locations. As
1212
usual, it is perfectly acceptable if more than one sample is generated
1313
for any of the locations $X$. Let $Z$ be a Boolean sensor variable
1414
characterized by the following conditional probabilities:<br>
1515

16+
1617
$$\begin{aligned}
17-
P(z\mid x_{{\rm 1}}) &=& {{\rm {0.8}}} \qquad\qquad P(\lnot z\mid x_{{\rm 1}})\;\;=\;\;{{\rm {0.2}}} \\
18-
P(z\mid x_{{\rm 2}}) &=& {{\rm {0.4}}} \qquad\qquad P(\lnot z\mid x_{{\rm 2}})\;\;=\;\;{{\rm {0.6}}} \\
19-
P(z\mid x_{{\rm 3}}) &=& {{\rm {0.1}}} \qquad\qquad P(\lnot z\mid x_{{\rm 3}})\;\;=\;\;{{\rm {0.9}}} \\
20-
P(z\mid x_{{\rm 4}}) &=& {{\rm {0.1}}} \qquad\qquad P(\lnot z\mid x_{{\rm 4}})\;\;=\;\;{{\rm {0.9}}}\ .\end{aligned}$$<br>
18+
P(z | x_1) = 0.8 \qquad\qquad P(z | x_1) = 0.2 \\
19+
P(z | x_2) = 0.4 \qquad\qquad P(z | x_2) = 0.6 \\
20+
P(z | x_3) = 0.1 \qquad\qquad P(z | x_3) = 0.9 \\
21+
P(z | x_4) = 0.1 \qquad\qquad P(z | x_4) = 0.9
22+
\end{aligned}$$
2123

22-
\begin{table}[]
23-
\begin{tabular}{ll}
24-
P(z\textbackslash{}mid x\_\{\{\textbackslash{}rm 1\}\}) \&=\& \{\{\textbackslash{}rm \{0.8\}\}\} & 1 \\
25-
1 & 1 \\
26-
1 & 1 \\
27-
1 & 1
28-
\end{tabular}
29-
\end{table}
3024

25+
<br>
3126

3227
MCL uses these probabilities to generate particle weights, which are
3328
subsequently normalized and used in the resampling process. For
@@ -38,7 +33,7 @@ probability distribution over $X$.<br>
3833

3934
1. What is the resulting probability distribution over $X$ for this new
4035
sample? Answer this question separately for
41-
$N={{\rm 1}},\ldots,{{\rm {10}}}$, and for $N=\infty$.<br>
36+
$N=1,\ldots,10$, and for $N=\infty$.<br>
4237

4338
2. The difference between two probability distributions $P$ and $Q$ can
4439
be measured by the KL divergence, which is defined as

‎_site/markdown/26-Philosophical-Foundations/exercises/ex_12/question.md

+1-1
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@@ -1,6 +1,6 @@
11

22

33
Some critics object that AI is impossible, while others object that it
4-
is *too* possible and that ultraintelligent machines pose a
4+
is <i>too</i> possible and that ultraintelligent machines pose a
55
threat. Which of these objections do you think is more likely? Would it
66
be a contradiction for someone to hold both positions?

‎_site/nlp-communicating-exercises/ex_11/index.html

+4-4
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@@ -193,8 +193,8 @@ <h3 class="masthead-title">
193193
end of the hour.<br />
194194

195195
5. The searcher will look through all the answers. Examining a document
196-
has cost \\$ A; finding a relevant document has value \\$ B; failing
197-
to find a relevant document has cost \\$ C for each relevant
196+
has cost \$ A; finding a relevant document has value \$ B; failing
197+
to find a relevant document has cost \$ C for each relevant
198198
document not found.<br />
199199

200200
6. The searcher wants to collect as many relevant documents as
@@ -248,8 +248,8 @@ <h3 class="masthead-title">
248248
end of the hour.<br>
249249

250250
5. The searcher will look through all the answers. Examining a document
251-
has cost \\$ A; finding a relevant document has value \\$ B; failing
252-
to find a relevant document has cost \\$ C for each relevant
251+
has cost \$ A; finding a relevant document has value \$ B; failing
252+
to find a relevant document has cost \$ C for each relevant
253253
document not found.<br>
254254

255255
6. The searcher wants to collect as many relevant documents as

‎_site/nlp-communicating-exercises/ex_2/index.html

+6-6
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@@ -166,11 +166,11 @@ <h3 class="masthead-title">
166166

167167
<div id="hiddden">
168168

169-
Write a program to do **segmentation** of
169+
Write a program to do <b>segmentation</b> of
170170
words without spaces. Given a string, such as the URL
171171
“thelongestlistofthelongeststuffatthelongestdomainnameatlonglast.com,”
172-
return a list of component words: \[“the,” “longest,” “list,”
173-
$\ldots$\]. This task is useful for parsing URLs, for spelling
172+
return a list of component words: [“the,” “longest,” “list,”
173+
$\ldots$]. This task is useful for parsing URLs, for spelling
174174
correction when words runtogether, and for languages such as Chinese
175175
that do not have spaces between words. It can be solved with a unigram
176176
or bigram word model and a dynamic programming algorithm similar to the
@@ -195,11 +195,11 @@ <h3 class="masthead-title">
195195
<div class="card-body">
196196
<p class="card-text">
197197

198-
Write a program to do **segmentation** of
198+
Write a program to do <b>segmentation</b> of
199199
words without spaces. Given a string, such as the URL
200200
“thelongestlistofthelongeststuffatthelongestdomainnameatlonglast.com,”
201-
return a list of component words: \[“the,” “longest,” “list,”
202-
$\ldots$\]. This task is useful for parsing URLs, for spelling
201+
return a list of component words: [“the,” “longest,” “list,”
202+
$\ldots$]. This task is useful for parsing URLs, for spelling
203203
correction when words runtogether, and for languages such as Chinese
204204
that do not have spaces between words. It can be solved with a unigram
205205
or bigram word model and a dynamic programming algorithm similar to the

‎_site/nlp-communicating-exercises/index.html

+5-5
Original file line numberDiff line numberDiff line change
@@ -185,11 +185,11 @@ <h1 id="22-natural-language-processing">22. Natural Language Processing</h1>
185185
<div class="card-body">
186186
<p class="card-text">
187187

188-
Write a program to do **segmentation** of
188+
Write a program to do <b>segmentation</b> of
189189
words without spaces. Given a string, such as the URL
190190
“thelongestlistofthelongeststuffatthelongestdomainnameatlonglast.com,”
191-
return a list of component words: \[“the,” “longest,” “list,”
192-
$\ldots$\]. This task is useful for parsing URLs, for spelling
191+
return a list of component words: [“the,” “longest,” “list,”
192+
$\ldots$]. This task is useful for parsing URLs, for spelling
193193
correction when words runtogether, and for languages such as Chinese
194194
that do not have spaces between words. It can be solved with a unigram
195195
or bigram word model and a dynamic programming algorithm similar to the
@@ -385,8 +385,8 @@ <h1 id="22-natural-language-processing">22. Natural Language Processing</h1>
385385
end of the hour.<br />
386386

387387
5. The searcher will look through all the answers. Examining a document
388-
has cost \\$ A; finding a relevant document has value \\$ B; failing
389-
to find a relevant document has cost \\$ C for each relevant
388+
has cost \$ A; finding a relevant document has value \$ B; failing
389+
to find a relevant document has cost \$ C for each relevant
390390
document not found.<br />
391391

392392
6. The searcher wants to collect as many relevant documents as

‎_site/nlp-english-exercises/ex_10/index.html

+12-14
Original file line numberDiff line numberDiff line change
@@ -169,9 +169,8 @@ <h3 class="masthead-title">
169169
In this exercise you will transform $\large \varepsilon_0$ into
170170
Chomsky Normal Form (CNF). There are five steps: (a) Add a new start
171171
symbol, (b) Eliminate $\epsilon$ rules, (c) Eliminate multiple words on
172-
right-hand sides, (d) Eliminate rules of the form<br />
173-
(${\it X}$<br />
174-
$\rightarrow$${\it Y}$),<br />
172+
right-hand sides, (d) Eliminate rules of the form
173+
(${\it X} \rightarrow$${\it Y}$),
175174
(e) Convert long right-hand sides into binary rules.<br />
176175

177176
1. The start symbol, $S$, can occur only on the left-hand side in CNF.
@@ -186,21 +185,21 @@ <h3 class="masthead-title">
186185

187186
3. A word can appear on the right-hand side in a rule only of the form
188187
(${\it X}$
189-
$\rightarrow$*word*).
188+
$\rightarrow$<i>word</i>).
190189
Replace each rule of the form (${\it X}$
191-
$\rightarrow$…*word* …)
190+
$\rightarrow$…<i>word</i> …)
192191
with (${\it X}$
193192
$\rightarrow$…${\it W'}$ …)
194193
and (${\it W'}$
195-
$\rightarrow$*word*),
194+
$\rightarrow$<i>word</i>),
196195
using a new symbol ${\it W'}$.<br />
197196

198197
4. A rule (${\it X}$
199198
$\rightarrow$${\it Y}$)
200199
is not allowed in CNF; it must be (${\it X}$
201200
$\rightarrow$${\it Y}$
202201
${\it Z}$) or (${\it X}$
203-
$\rightarrow$*word*).
202+
$\rightarrow$<i>word</i>).
204203
Replace each rule of the form (${\it X}$
205204
$\rightarrow$${\it Y}$)
206205
with a set of rules of the form (${\it X}$
@@ -241,9 +240,8 @@ <h3 class="masthead-title">
241240
In this exercise you will transform $\large \varepsilon_0$ into
242241
Chomsky Normal Form (CNF). There are five steps: (a) Add a new start
243242
symbol, (b) Eliminate $\epsilon$ rules, (c) Eliminate multiple words on
244-
right-hand sides, (d) Eliminate rules of the form<br>
245-
(${\it X}$<br>
246-
$\rightarrow$${\it Y}$),<br>
243+
right-hand sides, (d) Eliminate rules of the form
244+
(${\it X} \rightarrow$${\it Y}$),
247245
(e) Convert long right-hand sides into binary rules.<br>
248246

249247
1. The start symbol, $S$, can occur only on the left-hand side in CNF.
@@ -258,21 +256,21 @@ <h3 class="masthead-title">
258256

259257
3. A word can appear on the right-hand side in a rule only of the form
260258
(${\it X}$
261-
$\rightarrow$*word*).
259+
$\rightarrow$<i>word</i>).
262260
Replace each rule of the form (${\it X}$
263-
$\rightarrow$…*word* …)
261+
$\rightarrow$…<i>word</i> …)
264262
with (${\it X}$
265263
$\rightarrow$…${\it W'}$ …)
266264
and (${\it W'}$
267-
$\rightarrow$*word*),
265+
$\rightarrow$<i>word</i>),
268266
using a new symbol ${\it W'}$.<br>
269267

270268
4. A rule (${\it X}$
271269
$\rightarrow$${\it Y}$)
272270
is not allowed in CNF; it must be (${\it X}$
273271
$\rightarrow$${\it Y}$
274272
${\it Z}$) or (${\it X}$
275-
$\rightarrow$*word*).
273+
$\rightarrow$<i>word</i>).
276274
Replace each rule of the form (${\it X}$
277275
$\rightarrow$${\it Y}$)
278276
with a set of rules of the form (${\it X}$

‎_site/nlp-english-exercises/ex_14/index.html

+2-2
Original file line numberDiff line numberDiff line change
@@ -171,7 +171,7 @@ <h3 class="masthead-title">
171171
the language $a^nb^nc^n$. The allowable values for augmentation
172172
variables are 1 and $SUCCESSOR(n)$, where $n$ is a value. The rule for a sentence
173173
in this language is<br />
174-
$$S(n) \rightarrow}}A(n) B(n) C(n) \ .$$
174+
$$S(n) \rightarrow A(n) B(n) C(n) \ .$$
175175
Show the rule(s) for each of ${\it A}$,
176176
${\it B}$, and ${\it C}$.
177177
</div>
@@ -199,7 +199,7 @@ <h3 class="masthead-title">
199199
the language $a^nb^nc^n$. The allowable values for augmentation
200200
variables are 1 and $SUCCESSOR(n)$, where $n$ is a value. The rule for a sentence
201201
in this language is<br>
202-
$$S(n) \rightarrow}}A(n) B(n) C(n) \ .$$
202+
$$S(n) \rightarrow A(n) B(n) C(n) \ .$$
203203
Show the rule(s) for each of ${\it A}$,
204204
${\it B}$, and ${\it C}$.
205205
</p>

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