Monte Carlo Models, Chains and Monte Carlo Simulation

By pjain      Published March 30, 2020, 2:30 a.m. in blog AI-Analytics-Data   

Part of series - AI, Data Science, Analytics, Visualization Series Guide - Series

Markov Models and Monte Carlo - Business Review

Overview

What

Applications of Markov models

  • Peoples actions based on weather - eg if average ice creams per customer can be 1,2,3 it can relate to temperature.
  • The stock market - based on sentiment, related news, other countries' stock futures, etc.
  • Tweet generators! Chatbot suggestions for replies.
  • You could use a corpus to create and generate new content based on a Markov Model.

Markov Models and Monte Carlo - Technical Review

Random variables, processes, states and probabilities

  • Consider rolling a dice, or customers selecting an item from a menu. This is an example of a random process with associated probability.

Domain - possible states of outcome - eg for dice domain(O) = {1,2,3,3,5,6}

Probability Distribution - in case of dice should be an uniform distribution P(o) - 1/6 ∀ ∈ domain(O)

  • Domain Random Distributions
  • discrete random variables - in dice and menus cases since domain of variables are fixed choices
  • continous random variables - in some cases eg analog height of children it could range say 1'-7'.

  • Random processes

  • Sets of random variables that represent random variables over a period of time are also known as random processes. For example a stock price over a day can range over a series of values.
  • Random processes are often time-based but can depend on some other event.

  • Deterministic processes - if we have a definite output value eg position of a dropped ball given initial height and gravitational constant, then we know y(t) by that. So even though range can be continuous i.e. this is a random deterministic process.

  • Stochastic random processes - however, most vary due to many factors so are NOT determined eg stock price can fluctuate and not explained very easily - but depend on overall market, news of related stocks, or earnings release or news or product announcements or just sentiment.

Inference on random processes is more interesting if we can infer from that. For example candlestick charting reduces the entire days to a pattern eg hammer, red-down-day, green-up-day, etc. One might be able to state, that a breakout above say a 50-day moving average is a high probability that next day will also be an up day.

Markov Models and processes

In probability theory, a Markov model is a stochastic model used to model randomly changing systems where it is assumed that future states depend only on the current state not on the events that occurred before it. This assumption is called the Markov property. Generally, this assumption enables reasoning and computation with the model that would otherwise be intractable.

Markov model - Transition Probabilities diagrams, Histograms

  • For every key token -> value or set of possible depending on domain you have
    • Probabilities of state Transitions
    • Histogram of tokens either overall for corpus or

Markov model Order

Increasing the size of the window is known as bringing the Markov Model to a “higher order”. The current examples we have worked with have been first order markov models. If we use a second order Markov Model our window size would be two! Similarly, for a third order → window size of three.

ORDERs and Randomness. Lower order Markov models help you generate more random outputs. But, if you have 2 or 3 order markov models then you need a HUGE corpus of 100k+ tokens or even 1m+. Otherwise just given a couple of tokens third is likely to be highly limited.

Markov chains or discrete-time Markov processes

Continuous-time Markov chains

Markov Chains and Markov chain model

You need to learn some of the ways to represent a Markov chain like a state diagram and transition matrix.

Once you have an analysis of the probabilities of state transitions, you can construct a Markov chain model ie. which can generate predictive models.

Hidden Markov Models HMM

Monte Carlo Simulation

Resources

Business, Applications, Understanding Impact

Intermediate Books

  • Handbook of Markov Chain Monte Carlo by Steve Brooks, Andrew Gelman, Galin L. Jones, Xiao-Li Meng

Monte Carlo Simulation

Intermediate Simulation

  • Markov Chain Monte Carlo Stochastic Simulation for Bayesian Inference by Dani Gamerman, Hedibert Freitas Lopes

Advanced Markov Methods

  • Advanced Markov Chain Monte Carlo Methods_ Learning from Past Samples - hv

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