Let and  and let kernel k be defined as follows:

which transformation does this kernel correspond to?

Mila wishes to build a machine learning model to classify PET images as either containing cancerous tumours, or not. Mila performs the following steps:

·        First, she splits her data into training/test sets.

·        Next, Mila creates 100 subsets of her training data by resampling images from the training data, with replacement, until each dataset contains 200 images.

·        Then, she trains 100 models (one for each subset, and each model is a logistic regression) 

·        Finally, Mila runs each of the 100 models through the test set, and uses as her final prediction for each test image the majority vote of the 100 models.

Mila’s experiment is an example of the ensemble method known as:

Mila wishes to build a machine learning model to classify PET images as either containing cancerous tumours, or not. Mila chooses to use decision trees as her base model, and performs the following steps:

·        First, she splits her data into training/test sets.

·        Next, Mila creates 100 subsets of her training data by resampling images from the training data, with replacement, until each dataset contains 200 images.

·        Then, for each subset, she chooses to only look at a random sample of 40 pixels (same pixels for images in the same subset)

·        Then, she trains 100 models (one for each subset)

·        Finally, Mila runs each of the 100 models through the test set, and uses as her final prediction for each test image the majority vote of the 100 models.

Mila’s experiment is an example of the ensemble method known as:

There are two jars (jar A and jar B).

Jar A is composed of 50% red balls, and 50% blue balls. Jar B is composed of 60% red balls, and 40% blue balls.

You play the following game: you toss a biased coin (which has probability of heads 0.8), and if it comes up heads you pick a ball randomly from jar A, otherwise you pick a ball randomly from jar B.

You play the game and end up with a blue ball, what is the probability that this blue ball came from jar A?

Compute the derivative of

In a casino in Liverpool, there are two slot machines: if you play on the first one (machine A), you win 10% of the time. If you play on the second one (machine B), you win 20% of the time. You do not know which machine is which.

Your strategy: you assume the machines have an equal chance of being the better one, you select one of the two at random and put a coin in it. You lose the first bet. What is the probability that you selected machine B?

Let denote the outcome of tossing a special 6 sided dice - this dice has 6 faces like a normal dice, but it is twice as likely to land on an even face as it is to land on an odd face. Compute the following probability: