Lecture Notes on Gaussian Discriminant Analysis, Naive Bayes and EM Algorithm Feng Li fli@sdu.edu.cn Shandong University, China 1 Bayes’ Theorem and Inference Bayes’ theorem is stated mathematically almost always do better than GDA. In practice, logistic regression is used more often than GDA 4 Naive Bayes 4.1 Assumption Again, we assume that the m training data are denoted by {x(i), y(i)}i=1,··· j ∈ [n] and y ∈ [k]. In Naive Bayes (NB) model, the feature and label can be represented by random variables {Xj}j∈[n] and Y , respectively. Furthermore, for ∀j ̸= j′, Naive Bayes assumes Xj and Xj′ are

Lecture 5: Gaussian Discriminant Analysis, Naive Bayes and EM Algorithm Feng Li Shandong University fli@sdu.edu.cn September 27, 2023 Feng Li (SDU) GDA, NB and EM September 27, 2023 1 / 122 Outline Outline 1 Probability Theory Review 2 A Warm-Up Case 3 Gaussian Discriminate Analysis 4 Naive Bayes 5 Expectation-Maximization (EM) Algorithm Feng Li (SDU) GDA, NB and EM September 27, 2023 2 / 122 indicates if the i-th email is a spam Feng Li (SDU) GDA, NB and EM September 27, 2023 62 / 122 Naive Bayes Training data (x(i), y(i))i=1,··· ,m x(i) is a n-dimensional vector Each feature x(i) j ∈

improvements by using the SelectQuery.naive() query method. See the documentation for details on this optimization. stats_query = Stat.select().naive() # note we are calling "naive()" stats_qr = stats_query.execute() queries can get a performance boost (especially when iterating over large result sets) by calling naive(). This method simply patches all attributes directly from the cursor onto the model. For simple queries posted on %s' % (entry.title, entry.blog.title) And here is how you would do the same if using a naive query: # very similar query to the above -- main difference is we're # aliasing the blog title to

improvements by using the SelectQuery.naive() query method. See the documentation for details on this optimization. stats_query = Stat.select().naive() # note we are calling "naive()" stats_qr = stats_query.execute() queries can get a performance boost (especially when iterating over large result sets) by calling naive(). This method simply patches all attributes directly from the cursor onto the model. For simple queries posted on %s' % (entry.title, entry.blog.title) And here is how you would do the same if using a naive query: # very similar query to the above -- main difference is we're # aliasing the blog title to

improvements by using the SelectQuery.naive() query method. See the documentation for details on this optimization. stats_query = Stat.select().naive() # note we are calling "naive()" stats_qr = stats_query.execute() queries can get a performance boost (especially when iterating over large result sets) by calling naive(). This method simply patches all attributes directly from the cursor onto the model. For simple queries posted on %s’ % (entry.title, entry.blog.title) And here is how you would do the same if using a naive query: # very similar query to the above -- main difference is we’re # aliasing the blog title to

improvements by using the SelectQuery.naive() query method. See the documentation for details on this optimization. stats_query = Stat.select().naive() # note we are calling "naive()" stats_qr = stats_query.execute() queries can get a performance boost (especially when iterating over large result sets) by calling naive(). This method simply patches all attributes directly from the cursor onto the model. For simple queries if using a naive query: # very similar query to the above -- main difference is we’re # aliasing the blog title to "blog_title" tweets = Tweet.select(Tweet, User.username).join(User).naive() for tweet

improvements by using the SelectQuery.naive() query method. See the documentation for details on this optimization. stats_query = Stat.select().naive() # note we are calling "naive()" stats_qr = stats_query.execute() queries can get a performance boost (especially when iterating over large result sets) by calling naive(). This method simply patches all attributes directly from the cursor onto the model. For simple queries posted on %s’ % (entry.title, entry.blog.title) And here is how you would do the same if using a naive query: # very similar query to the above -- main difference is we’re # aliasing the blog title to

incremented with each revision. • All dates are converted to UTC when serializing. Even timezone naive values, which are treated as UTC with an offset of 0. In [275]: from pandas.io.json import build_table_schema Datetime data types Using SQLAlchemy, to_sql() is capable of writing datetime data that is timezone naive or timezone aware. However, the resulting data stored in the database ultimately depends on the supported written as timezone naive timestamps that are in local time with respect to the timezone. read_sql_table() is also capable of reading datetime data that is timezone aware or naive. When reading TIMESTAMP

incremented with each revision. • All dates are converted to UTC when serializing. Even timezone naive values, which are treated as UTC with an offset of 0. In [296]: from pandas.io.json import build_table_schema Datetime data types Using SQLAlchemy, to_sql() is capable of writing datetime data that is timezone naive or timezone aware. However, the resulting data stored in the database ultimately depends on the supported written as timezone naive timestamps that are in local time with respect to the timezone. read_sql_table() is also capable of reading datetime data that is timezone aware or naive. When reading TIMESTAMP

incremented with each revision. • All dates are converted to UTC when serializing. Even timezone naive values, which are treated as UTC with an offset of 0. In [293]: from pandas.io.json import build_table_schema Datetime data types Using SQLAlchemy, to_sql() is capable of writing datetime data that is timezone naive or timezone aware. However, the resulting data stored in the database ultimately depends on the supported written as timezone naive timestamps that are in local time with respect to the timezone. read_sql_table() is also capable of reading datetime data that is timezone aware or naive. When reading TIMESTAMP