Future topics
1) Network security, Fraud detection (IP address related features), anomaly detection
2) Some automation requirements from Amazon business, What machine learning can do?
3) Lending/ Risk management Some concepts:
Estimation (or estimating): https://en.wikipedia.org/wiki/Estimation
Estimation is the process of finding an estimate, or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is derived from the best information available. Typically, estimation involves "using the value of a statistic derived from a sample to estimate the value of a corresponding population parameter". The sample provides information that can be projected, through various formal or informal processes, to determine a range most likely to describe the missing information. An estimate that turns out to be incorrect will be an overestimate if the estimate exceeded the actual result, and an underestimate if the estimate fell short of the actual result.
Point estimate, for example, is the single number most likely to express the value of the property.
Interval estimate defines a range within which the value of the property can be expected (with a specified degree of confidence) to fall.
Sequential estimation, estimates a parameter by analyzing a sample just large enough to ensure a previously chosen degree of precision. The fundamental technique is to take a sequence of samples, the outcome of each sampling determining the need for another sampling. The procedure is terminated when the desired degree of precision is achieved. On average, fewer total observations will be needed using this procedure than with any procedure using a fixed number of observations.
Statistical inference: https://en.wikipedia.org/wiki/Statistical_inference
Statistical inference is the process of deducing properties of an underlying distribution by analysis of data. Inferential statistical analysis infers properties about a population: this includes testing hypotheses and deriving estimates. The population is assumed to be larger than the observed data set; in other words, the observed data is assumed to be sampled from a larger population.
Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and does not assume that the data came from a larger population
Frequentist inference: it draws conclusions from sample data by emphasizing the frequency or proportion of the data. An alternative name is frequentist statistics. This is the inference framework in which the well-established methodologies of statistical hypothesis testing and confidence intervals are based.
Bayesian inference, sometimes held to include the approach to inference leading to optimal decisions, is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data.
Fiducial inference is one of a number of different types of statistical inference. These are rules, intended for general application, by which conclusions can be drawn from samples of data. In modern statistical practice, attempts to work with fiducial inference have fallen out of fashion in favour of frequentist inference, Bayesian inference and decision theory. However, fiducial inference is important in the history of statistics since its development led to the parallel development of concepts and tools in theoretical statistics that are widely used.
MLE, MAP, Hierarchical Bayes model, and Empirical Bayes model MLE: maximal likelihood estimation ß this is the traditional statistical approach to finding parameters
MAP: Maximum a posteriori (MAP) estimation MAP estimation is the value of the parameter that maximizes the entire posterior distribution (which is calculated using the likelihood). A MAP estimate is the mode of the posterior distribution.
The problem with MLE is that it overfits the data, meaning that the variance of the parameter estimates is high, or the outcome of the parameter estimate is sensitive to random variations in data. To deal with this, it usually helps to add regularization to MLE (i.e., reduce variance by introducing bias into the estimate). In maximum a posteriori (MAP), this regularization is achieved by assuming that the parameters themselves are also (in addition to the data) drawn from a random process. The prior beliefs about the parameters determine what this random process looks like.
It is a design decision as to what prior belief the model has about the parameters, but interestingly, if the prior beliefs are strong, then the observed data have relatively little impact on the parameter estimates, (i.e., low variance but high bias), while if the prior beliefs are weak, then the outcome is more like standard MLE (i.e., low bias but high variance). This leads to two interesting limits: for an infinite amount of data, MAP gives the same result as MLE (as long as the prior is non-zero everywhere in parameter space); for an infinitely weak prior belief (i.e., uniform prior), MAP also gives the same result as MLE.
Conjugate prior: https://en.wikipedia.org/wiki/Conjugate_prior
In Bayesian probability theory, if the posterior distributions p(θ|x) are in the same family as the prior probability distribution p(θ), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function.
For example, the Gaussian family is conjugate to itself (or self-conjugate) with respect to a Gaussian likelihood function: if the likelihood function is Gaussian, choosing a Gaussian prior over the mean will ensure that the posterior distribution is also Gaussian. This means that the Gaussian distribution is a conjugate prior for the likelihood that is also Gaussian. All members of the exponential family have conjugate priors.
Conjugate priors may give intuition, by more transparently showing how a likelihood function updates a prior distribution.
Discussed Problems:
How to test whether two samples come from the same distributions with the same parameters even we do not know the distributions and the parameters.
This is used in “Transfer learning” and model performance monitoring
Suggestions:
1) Empirical methods including drawing histogram and comparing the shape, getting the percentiles and comparing the values
2) Input the two sample sets to 0/1 classifier, if the classifier’s error always very high no matter how you tune the parameters, then it means these two sets are not from two different classes, they should come from the same class
3) Kullback–Leibler divergence:
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