The mathematical concept of set diversity is a somewhat neglected topic in current applied decision sciences and optimization. We take this opportunity to discuss the issue.

The problem

Consider the following problem: for a number of items U = {x_1, … x_n} pick a small set of them X = {x_i1, x_i2, ..., x_ik} such that there is a high probability one of the x in X is a “success.” By success I mean some standard business outcome such as making a sale (in the sense of any of: propensity, appetency, up selling, and uplift modeling), clicking an advertisement, adding an account, finding a new medicine, or learning something useful.

This is common in:

Search engines. The user is presented with a page consisting of “top results” with the hope that one of the results is what the user wanted.

Online advertising. The user is presented with a number of advertisements in enticements in the hope that one of them matches user taste.

Science. A number of molecules are simultaneously presented to biological assay hoping that at least one of them is a new drug candidate, or that the simultaneous set of measurements shows us where to experiment further.

Sensor/guard placement. Overlapping areas of coverage don’t make up for uncovered areas.

Machine learning method design. The random forest algorithm requires diversity among its sub-trees to work well. It tries to ensure by both per-tree variable selections and re-sampling (some of these issues discussed here).

In this note we will touch on key applications and some of the theory involved. While our group specializes in practical data science implementations, applications, and training, our researchers experience great joy when they can re-formulate a common problem using known theory/math and the reformulation is game changing (as it is in the case of set-scoring).

Here is a video I made showing how R should not be considered “scarier” than Excel to analysts. One of the takeaway points: it is easier to email R procedures than Excel procedures.

A save of the “email” linking to all code and data is here.

The theory is the recipient of the email already had R, RStudio and the required packages installed from previous use. The package install step is only needed once and is:

install.packages(c('rpart','rpart.plot'))

Then all the steps are (in a more cut/paste friendly format):

“Data Science” is obviously a trendy term making it way through the hype cycle. Either nobody is good enough to be a data scientist (unicorns) or everybody is too good to be a data scientist (or the truth is somewhere in the middle).

And there is a quarter that grumbles that we are merely talking about statistics under a new name (see here and here).

It has always been the case that advances in data engineering (such as punch cards, or data centers) make analysis practical at new scales (though I still suspect Map/Reduce was a plot designed to trick engineers into being excited about ETL and report generation).

In this spirit next week we will write about the sequential analysis solution for A/B-testing, invented in the 1940s by one of the greats of statistics and operations research: Abraham Wald (whom we have written about before).

There is a lot of current interest in various “crypto currencies” such as Bitcoin, but that does not mean there have not been previous combined ledger and token recording systems. Others have noticed the relevance of Crawfurd v The Royal Bank (the case where money became money), and we are going to write about this yet again.

Very roughly: a Bitcoin is a cryptographic secret that is considered to have some value. Bitcoins are individual data tokens, and duplication is prevented through a distributed shared ledger (called the blockchain). As interesting as this is, we want to point out notional value existing both in ledgers and as possessed tokens has quite a long precedent.

This helps us remember that important questions about Bitcoins (such as: are they a currency or a commodity?) will be determined by regulators, courts, and legislators. It will not be a simple inevitable consequence of some detail of implementation as this has never been the case for other forms of value (gold, coins, bank notes, stocks certificates, or bank account balances).

Value has often been recorded in combinations of ledgers and tokens, so many of these issues have been seen before (though they have never been as simple as one would hope). Historically the rules that apply to such systems are subtle, and not completely driven by whether the system primarily resides in ledgers or primarily resides portable tokens. So we shouldn’t expect determinations involving Bitcoin to be simple either.

There remains a bit of a two-way snobbery that Frequentist statistics is what we teach (as so-called objective statistics remain the same no matter who works with them) and Bayesian statistics is what we do (as it tends to directly estimate posterior probabilities we are actually interested in). Nina Zumel hit the nail on the head when she wrote an article explaining the appropriateness of the type of statistical theory depends on the type of question you are trying to answer, not on your personal prejudices.

We will discuss a few more examples that have been in our mind, including one I am calling “baking priors.” This final example will demonstrate some of the advantages of allowing researchers to document their priors.

One of the things I like about R is: because it is not used for systems programming you can expect to install your own current version of R without interference from some system version of R that is deliberately being held back at some older version (for reasons of script compatibility). R is conveniently distributed as a single package (with automated install of additional libraries).

Want to do some data analysis? Install R, load your data, and go. You don’t expect to spend hours on system administration just to get back to your task.

Python, being a popular general purpose language does not have this advantage, but thanks to Anaconda from Continuum Analytics you can skip (or at least delegate) a lot of the system environment imposed pain. With Anaconda trying out Python packages (Jupyter, scikit-learn, pandas, numpy, sympy, cvxopt, bokeh, and more) becomes safe and pleasant. Continue reading Thumbs up for Anaconda

In some of my recent public talks (for example: here and here) I have mentioned a desire for “a deeper theory of fitting and testing.” I thought I would expand on what I meant by this.

In this note I am going to cover a lot of different topics to try and suggest some perspective. I won’t have my usual luxury of fully defining my terms or working concrete examples. Hopefully a number of these ideas (which are related, but don’t seem to easily synthesize together) will be subjects of their own later articles.

Introduction

The focus of this article is: the true goal of predictive analytics is always: to build a model that works well in production. Training and testing procedures are designed to simulate this unknown future model performance, but can be expensive and can also fail.

What we want is a good measure of future model performance, and to apply that measure in picking a model without running deep into Goodhart’s law (“When a measure becomes a target, it ceases to be a good measure.”).

Most common training and testing procedures are destructive in the sense they use up data (data used for one step may not be safely used for another step in an unbiased fashion, example: excess generalization error). In this note I thought I would expand on the ideas for extending statistical efficiency or getting more out of your training while avoiding overfitting.

Here’s a caricature of a data science project: your company or client needs information (usually to make a decision). Your job is to build a model to predict that information. You fit a model, perhaps several, to available data and evaluate them to find the best. Then you cross your fingers that your chosen model doesn’t crash and burn in the real world.

We’ve discussed detecting if your data has a signal. Now: how do you know that your model is good? And how sure are you that it’s better than the models that you rejected?

Notice the Sun in the 4th revolution about the earth. A very pretty, but not entirely reliable model.

In this latest “Statistics as it should be” article, we will systematically look at what to worry about and what to check. This is standard material, but presented in a “data science” oriented manner. Meaning we are going to consider scoring system utility in terms of service to a negotiable business goal (one of the many ways data science differs from pure machine learning).

When fitting and selecting models in a data science project, how do you know that your final model is good? And how sure are you that it’s better than the models that you rejected? In this Part 2 of our four part mini-series “How do you know if your model is going to work?” we develop in-training set measures.

The most tempting procedure is to score your model on the data used to train it. The attraction is this avoids the statistical inefficiency of denying some of your data to the training procedure.

Run it once procedure

A common way to asses score quality is to run your scoring function on the data used to build your model. We might try comparing several models scored by AUC or deviance (normalized to factor out sample size) on their own training data as shown below.

What we have done is take five popular machine learning techniques (random forest, logistic regression, gbm, GAM logistic regression, and elastic net logistic regression) and plotted their performance in terms of AUC and normalized deviance on their own training data. For AUC larger numbers are better, and for deviance smaller numbers are better. Because we have evaluated multiple models we are starting to get a sense of scale. We should suspect an AUC of 0.7 on training data is good (though random forest achieved an AUC on training of almost 1.0), and we should be acutely aware that evaluating models on their own training data has an upward bias (the model has seen the training data, so it has a good chance of doing well on it; or training data is not exchangeable with future data for the purpose of estimating model performance).

There are two more Gedankenexperiment models that any machine data scientist should always have in mind:

The null model (on the graph as “null model”). This is the performance of the best constant model (model that returns the same answer for all datums). In this case it is a model scores each and every row as having an identical 7% chance of churning. This is an important model that you want to better than. It is also a model you are often competing against as a data science as it is the “what if we treat everything in this group the same” option (often the business process you are trying to replace).

The data scientist should always compare their work to the null model on deviance (null model AUC is trivially 0.5) and packages like logistic regression routinely report this statistic.

The best single variable model (on the graph as “best single variable model”). This is the best model built using only one variable or column (in this case using a GAM logistic regression as the modeling method). This is another model the data scientist wants to out perform as it represents the “maybe one of the columns is already the answer case” (if so that would be very good for the business as they could get good predictions without modeling infrastructure).

The data scientist should definitely compare their model to the best single variable model. Until you significantly outperform the best single variable model you have not outperformed what an analyst can find with a single pivot table.

At this point it would be tempting to pick the random forest model as the winner as it performed best on the training data. There are at least two things wrong with this idea:

Here’s a caricature of a data science project: your company or client needs information (usually to make a decision). Your job is to build a model to predict that information. You fit a model, perhaps several, to available data and evaluate them to find the best. Then you cross your fingers that your chosen model doesn’t crash and burn in the real world.

We’ve discussed detecting if your data has a signal. Now: how do you know that your model is good? And how sure are you that it’s better than the models that you rejected?

Notice the Sun in the 4th revolution about the earth. A very pretty, but not entirely reliable model.

In this latest “Statistics as it should be” series, we will systematically look at what to worry about and what to check. This is standard material, but presented in a “data science” oriented manner. Meaning we are going to consider scoring system utility in terms of service to a negotiable business goal (one of the many ways data science differs from pure machine learning).

To organize the ideas into digestible chunks, we are presenting this article as a four part series (to finished in the next 3 Tuesdays). This part (part 1) sets up the specific problem.