Difficult computational statistics 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?

 

[Ray Vickson]

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]

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!