# Difficult computational statistics problem

• Bazzinga
In summary, the conversation discusses a computational statistics problem involving a Hidden Markov Model. The problem involves a penny and a dime, each with their own probability of showing a head when tossed. The state of the coin is not observable, but only the outcome of the toss is known. The conversation concludes with suggestions for tutorials and resources for estimating the probabilities involved in the problem.
Bazzinga
I've got a tricky computational statistics problem and I was wondering if anyone could help me solve it.

Okay, so in your left pocket is a penny and in your right pocket is a dime. On a fair toss, the probability of showing a head is p for the penny and d for the dime. You randomly chooses a coin to begin, toss it, and report the outcome (heads or tails) without revealing which coin was tossed. Then you decide whether to use the same coin for the next toss, or to switch to the other coin. You switch coins with probability s, and use the same coin with probability (1 - s). The outcome of the second toss is reported, again not reveling the coin used.

I have a sequence of heads and tails data based on these flips, so how would I go about estimating p, d, and s?

Bazzinga said:
I've got a tricky computational statistics problem and I was wondering if anyone could help me solve it.

Okay, so in your left pocket is a penny and in your right pocket is a dime. On a fair toss, the probability of showing a head is p for the penny and d for the dime. You randomly chooses a coin to begin, toss it, and report the outcome (heads or tails) without revealing which coin was tossed. Then you decide whether to use the same coin for the next toss, or to switch to the other coin. You switch coins with probability s, and use the same coin with probability (1 - s). The outcome of the second toss is reported, again not reveling the coin used.

I have a sequence of heads and tails data based on these flips, so how would I go about estimating p, d, and s?

What you are describing is a so-called Hidden Markov Model. Here, the underlying state (dime or penny) follows a Markov chain with transition probability matrix
$$\mathbb{P}= \pmatrix{1-s & s \\ s & 1-s}$$
However, the state is not observable---only the outcomes (H or T) of tossing the coins can be observed.

There are several useful tutorials available on-line: see, eg.,
http://di.ubi.pt/~jpaulo/competence/tutorials/hmm-tutorial-1.pdf or
http://www.cs.ubc.ca/~murphyk/Bayes/rabiner.pdf

This last source has a brief treatment of your problem, as an illustrative example.

Bazzinga
Ray Vickson said:
What you are describing is a so-called Hidden Markov Model. Here, the underlying state (dime or penny) follows a Markov chain with transition probability matrix
$$\mathbb{P}= \pmatrix{1-s & s \\ s & 1-s}$$
However, the state is not observable---only the outcomes (H or T) of tossing the coins can be observed.

There are several useful tutorials available on-line: see, eg.,
http://di.ubi.pt/~jpaulo/competence/tutorials/hmm-tutorial-1.pdf or
http://www.cs.ubc.ca/~murphyk/Bayes/rabiner.pdf

This last source has a brief treatment of your problem, as an illustrative example.

Great I'll take a look at those! Thanks!

## 1. What is a difficult computational statistics problem?

A difficult computational statistics problem is a complex statistical problem that requires advanced mathematical and computational techniques to solve. These problems often involve large datasets, intricate statistical models, and complex algorithms.

## 2. How do you approach a difficult computational statistics problem?

Approaching a difficult computational statistics problem requires a systematic approach. This typically involves understanding the problem, identifying the relevant data and variables, selecting appropriate statistical techniques, and applying them using programming languages or software.

## 3. What are some common challenges faced when solving difficult computational statistics problems?

There are several common challenges that researchers and scientists face when solving difficult computational statistics problems. These include data cleaning and preprocessing, selecting appropriate models and parameters, dealing with missing data and outliers, and interpreting the results accurately.

## 4. How can one improve their skills in solving difficult computational statistics problems?

To improve skills in solving difficult computational statistics problems, one can take courses or workshops on advanced statistical methods and programming languages, read research papers and books, participate in online forums and discussions, and practice solving different types of statistical problems.

## 5. What are some real-world examples of difficult computational statistics problems?

Some real-world examples of difficult computational statistics problems include predicting stock market trends, analyzing climate change data, developing models for predicting disease outbreaks, and analyzing social media data to understand user behavior. These problems require a combination of statistical knowledge, computational skills, and domain expertise to solve successfully.

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