### A Very Rough Guide to Getting Started in Data Science: Part II, The Big Picture

Data science to a beginner seems completely overwhelming. Not only are there huge numbers of programming languages, packages and algorithms, but even managing your data is an entire area itself. Some examples are the languages R, Python, Ruby, Perl, Julia, Mathematica, MATLAB/Octave; packages SAS, STATA, SPSS; algorithms linear regression, logistic regression, nested model, neural nets, support vector machines, linear discriminant analysis and deep learning.
For managing your data some people use Excel, or a relational database like MySQL or PostgreSQL. And where do things like big data, NoSQL and Hadoop fit in? And what's gradient descent and why is it important? But perhaps the most difficult part of all is that you actually need to know and understand statistics, too.

It does seem overwhelming, but there's a simple key idea - data science is using data to answer a question. Even if you're only sketching a graph using a stick and a sandbox, you're still doing data science. Your goal for data science should be to continually learn better, more powerful and more efficient ways to answer your questions. My general framework has been strongly influenced by George Pólya's wonderful book "How to Solve It". While it's directed at solving mathematical problems, his approach is helpful for solving problems in general.

"How to Solve It" suggests the following steps when solving a mathematical problem:
1. First, you have to understand the problem.
2. After understanding, then make a plan.
3. Carry out the plan.
4. Review/extend. Look back on your work. How could it be better?
Pólya goes into much greater detail for each step and provides some illustrative examples. It's not the final word on how to approach and solve mathematical problems, but it's very helpful and I highly recommend it. For data science, the analogous steps from my perspective would be:
1. What questions do you want to answer?
2. What data would be helpful to answer these questions? How and where do you get this data?
3. Given the question you want to answer and the data you have, which approaches and models are likely to be useful? This can be very confusing. There are always tradeoffs - underfitting vs overfitting, bias vs variance, simplicity vs complexity, information about where something came from vs what's it doing
4. Perform analysis/fit model.
5. How do you know if your model and analysis are good or bad, and how confident should you be in your predictions and conclusions? A very critical, but commonly treated lightly or even skipped entirely.
6. Given the results, what should you try next?
Let's follow Pólya and do an illustrative example next.

### A Bayes' Solution to Monty Hall

For any problem involving conditional probabilities one of your greatest allies is Bayes' Theorem. Bayes' Theorem says that for two events A and B, the probability of A given B is related to the probability of B given A in a specific way.

Standard notation:

probability of A given B is written $$\Pr(A \mid B)$$
probability of B is written $$\Pr(B)$$

Bayes' Theorem:

Using the notation above, Bayes' Theorem can be written: $\Pr(A \mid B) = \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)}$Let's apply Bayes' Theorem to the Monty Hall problem. If you recall, we're told that behind three doors there are two goats and one car, all randomly placed. We initially choose a door, and then Monty, who knows what's behind the doors, always shows us a goat behind one of the remaining doors. He can always do this as there are two goats; if we chose the car initially, Monty picks one of the two doors with a goat behind it at random.

Assume we pick Door 1 and then Monty sho…

### Mixed Models in R - Bigger, Faster, Stronger

When you start doing more advanced sports analytics you'll eventually starting working with what are known as hierarchical, nested or mixed effects models. These are models that contain both fixed and random effects. There are multiple ways of defining fixed vs random random effects, but one way I find particularly useful is that random effects are being "predicted" rather than "estimated", and this in turn involves some "shrinkage" towards the mean.

Here's some R code for NCAA ice hockey power rankings using a nested Poisson model (which can be found in my hockey GitHub repository):
model <- gs ~ year+field+d_div+o_div+game_length+(1|offense)+(1|defense)+(1|game_id) fit <- glmer(model, data=g, verbose=TRUE, family=poisson(link=log) ) The fixed effects are year, field (home/away/neutral), d_div (NCAA division of the defense), o_div (NCAA division of the offense) and game_length (number of overtime periods); off…

### Gambling to Optimize Expected Median Bankroll

Gambling to optimize your expected bankroll mean is extremely risky, as you wager your entire bankroll for any favorable gamble, making ruin almost inevitable. But what if, instead, we gambled not to maximize the expected bankroll mean, but the expected bankroll median?

Let the probability of winning a favorable bet be $$p$$, and the net odds be $$b$$. That is, if we wager $$1$$ unit and win, we get back $$b$$ units (in addition to our wager). Assume our betting strategy is to wager some fraction $$f$$ of our bankroll, hence $$0 \leq f \leq 1$$. By our assumption, our betting strategy is invariant with respect to the actual size of our bankroll, and so if we were to repeat this gamble $$n$$ times with the same $$p$$ and $$b$$, the strategy wouldn't change. It follows we may assume an initial bankroll of size $$1$$.

Let $$q = 1-p$$. Now, after $$n$$  such gambles our bankroll would have a binomial distribution with probability mass function \[ \Pr(k,n,p) = \binom{n}{k} p^k q^{n-k…