Python is a great programming language. It's primary merits are readability and the numerous packages that are available online. However, being an interpreted language the speed of execution is always an issue. Here is a simple example of a piece of code written in Python that tries to add two numbers from a grid. #!/usr/bin/python def overlap_pp(x,y): count = 0 for i in range(x): for j in range(y): count += i + j return count for _ in range(1000): q = overlap_pp(500,500) If you run the above script (saved as n.py) on the command line terminal with the time command you should see some numbers like the below time ./n.py real 0m22.379s user 0m22.363s sys 0m0.407s The process running on one processor took about 22 seconds to complete the run. Now lets do something seemingly magical. Lets add a decorator. Decorators are python speak for a mechanism to do wrapper functions. The decorator we are going to add is '@jit'. The code looks l...
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This write up is about an estimator. A statistical estimator, is used when you already have a model in mind, data at hand and want to estimate some parameters needed for the model. For example, you want to predict how many runs a batter would make in a game given recent history \(x_1,x_2,x_3,\ldots\) . If we assume (we are making a choice of a model now) that the scores come from a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) then the probability density function for a given value \(x\) is The likelihood that a series of points \(x_1,x_2,x_3,\ldots\) come from such a distribution can be expressed as Next is basic calculus. Take the logarithm on both sides, set the partial derivative w.r.t. \(\mu\) to zero yields (excluding the algebra) To verify, you also need to check the second derivative's sign to see if its negative to ensure that it is indeed a maxima you have found. So a simple estimator would be to use the average ...
I chanced on this excellent puzzle on the net that tends to reveal a cognitive bias in our heads against Bayesian reasoning. The puzzle statement is quite simple You are given four cards. Each card has a letter on one side and number on the other side. You are told the statement "If there is a D on one side, there is a 5 on the other side". Which two cards would you flip over to validate the statement? The original article is here , think hard before you click through for an answer :)
Follow @ProbabilityPuz Q: You are about to play a game wherein you flip a biased coin. The coin falls heads with probability \(p\) and tails with \(1 - p\) where \(p \le \frac{1}{2}\). You are forced to play by selecting heads so the game is biased against you. For every toss you make, your opponent gets to toss too. The winner of this game is the one who wins the toss the most. You, however get to choose the number of rounds that get played. Can you ever hope to win? Machine Learning: The Art and Science of Algorithms that Make Sense of Data A: At a first look, it might appear that the odds are stacked against you as you are forced to play by choosing heads. You would think that your chances or winning decrease as you play more and more. But, surprisingly there is a way to choose the optimal number of tosses (remember, you get to choose the number of times this game is played). To see how, lets crank out some numbers. If you get to toss the coin \(n\) times, then t...
The following are some of the best books to own to learn Monte Carlo methods for sampling and estimation problems Monte Carlo Methods in Statistics (Springer) This is a good book which discusses both Bayesian methods from a practical point of view as well as theoretical point of view with integrations. The explanations given are also fairly comprehensive. There are also a fair amount of examples in this text. Overall, this is an excellent book to own if you want to understand Monte Carlo sampling methods and algorithms at an intermediate to graduate level. Explorations in Monte Carlo Methods (Undergraduate Texts in Mathematics) This is good book to own to get you started on Monte Carlo methods. It starts with fairly simple and basic examples and illustrations. The mathematics used is also fairly basic. Buy this if you are at an undergraduate level and want to get into using Monte Carlo methods but have only a basic knowledge of statistics and probability. Monte Carlo Methods in...
Follow @ProbabilityPuz Q: You are conducting a survey and want to ask an embarrassing yes/no question to subjects. The subjects wouldn't answer that embarrassing question honestly unless they are guaranteed complete anonymity. How would you conduct the survey? Machine Learning: The Art and Science of Algorithms that Make Sense of Data A: One way to do this is to assign a fair coin to the subject and ask them to toss it in private. If it came out heads then answer the question truthfully else toss the coin a second time and record the result (heads = yes, tails = no). With some simple algebra you can estimate the proportion of users who have answered the question with a yes. Assume total population surveyed is \(X\). Let \(Y\) subjects have answered with a "yes". Let \(p\) be the sort after proportion. The tree diagram below shows the user flow. The total expected number of "yes" responses can be estimated as $$ \frac{pX}{2} +...
Follow @ProbabilityPuz A class of analysis that has piqued the interest of mathematicians across millennia are Diophantine equations . Diophantine equations are polynomials with multiple variables and seek integer solutions. A special case of Diophantine equations is the Pell's equation . The name is a bit of a misnomer as Euler mistakenly attributed it to the mathematician John Pell. The problem seeks integer solutions to the polynomial $$ x^{2} - Dy^{2} = 1 $$ Several ancient mathematicians have attempted to study and find generic solutions to Pell's equation. The best known algorithm is the Chakravala algorithm discovered by Bhaskara circa 1114 AD. Bhaskara implicitly credits Brahmagupta (circa 598 AD) for it initial discovery, though some credit it to Jayadeva too. Several Sanskrit words used to describe the algorithm appear to have changed in the 500 years between the two implying other contributors. The Chakravala technique is simple and implementing it ...