python problem

STAT535: Statistical Computing Spring 2019

Problem Set 3: Simulation

P. Flaherty

Objective This problem set will introduce simulation and visualization. This builds on your previous problem set, so make sure that your word game passes all tests. You will build a computer player for the word game and then use that player to offer hints to the user. You will visualize the quality of the hints provided by the computer player.

Collaboration You may work with other students. However, each student must writeup and hand in their assignment individually. Be sure to indicate whom you worked with in the comments of your submission.

Logistics For each of problem 1 and 2 you will submit 2 files: the python file for ps3 and a file called ps2.py containing your problem set 2 implementation. Submit your solution to question 1 as a file named ps3a.py using the provided framework. Submit your solution to question 2 as a file named ps3b.py using the provided framework. Submit your solution to question 3 by modifying the provided jupyter notebook and exporting a pdf format file. The file should be named ps3c.pdf.

1. Computer Player In this problem you will write a computer player for the word game that you implemented in problem set 2. The computer player will generate candidate words using Monte Carlo simulation. You will explore the relationship between the Monte Carlo parameters and the quality of the hints provided.

(a) Implement choose_word. Using the framework provided, implement the function choose_word which takes a hand and N – the number of Monte Carlo candidates to generate and returns the best scoring word and the score of that word.

The function should only return valid words. If no valid word can be found, it should return None for the word and 0 for the score.

(b) Implement play_mc_hand. Using the framework provided, implement the function play_hand using the choose_word function. It should take the hand and N. It should return (1) a list containing the words played and (2) the total score of the sequence of words played.

(c) Update play_hand. Update the branching logic in play_hand so that if the user enters ’ ?’ the function choose_word is run with N = 100. It should simply print the suggested word and the corresponding score for the user and then allow them to continue play. Make sure you handle the situation where a valid word is not found – this is not an error condition.

(d) Implement play_n_mc_hand. using the framework provided, implement play_n_mc_hand. This function takes (1) hand, (2) N, (3) n, where n is the number of times the same hand is played and each word is selected from N candidate Monte Carlo samples. The function should return (1) list of word lists (2) a list of scores.

2. Reproducible Monte Carlo

1

2

You will write unit tests for your Monte Carlo computer player. The Monte Carlo player ran- domly samples plays, so you will need to set the random number generator seed for reproducible results.

Implement test_mc_player. Write a function called test_mc_player. The function should test the following hands three times and produce the same score each time.

1. helloworld

2. UMasswins

3. statisticscomputing

The function should take three parameters: (1) the hand, (2) the number of Monte Carlo samples, and (3) the random number seed. It should return True if the test is passed and False if not.

3. Visualizing Hands Is the best hand a far-and-away best or is it only marginally better than many other hands? In order to answer this question we need a way to visualize the set of possible scores from our computer players. There are two variables in the MC player function - N: the number of MC samples to choose for each round and n the number of times to play the hand.

(a) Write a function called plot_mc_hand that takes the hand helloworld, a number of Monte Carlo samples N, and a number of times to play the hand n and generates a plot that shows a histogram of the scores produced by the Monte Carlo samples of playing the hand. be sure to add a title to your plot so we know what you are showing.

(b) Show how the distribution of scores for N ∈ {1, 10, 100} and n ∈ {10, 100}. (c) What do you notice about the maximum score and the distribution of scores for the hand

as these MC parameters change?