Bayesian games

DNA test

A suspect has a prior probability \(P(G)\) of being guilty. A DNA test can return a match or no match, with different conditional probabilities depending on whether the suspect is guilty. After observing one or more independent test outcomes, Bayes' rule updates the probability of guilt.

Prior probability \(P(G)\)

Outcome	Guilty \(G\)	Not guilty \(G^c\)
Match \(M\)
No match \(M^c\)

Number of matches

Number of no matches

Posterior probability of guilt

0.909

After observing 1 match and 0 no matches.

Modeling

Solution

Prior

1.0%

Posterior

90.9%

Polling with a Dirichlet prior

Enter party counts from a poll and prior Dirichlet parameters. The posterior is another Dirichlet distribution.

Number of parties \(n\)

Party	Poll count \(y_i\)	Prior \(\alpha_i\)

Posterior means and winning probabilities

Bars show posterior mean support. Whiskers show one marginal posterior standard deviation. The right column estimates the posterior probability that each party has the largest support.

Modeling

Solution

Posterior parameters and posterior mean

Baggage delivery

You are waiting for one checked-in bag. Other passengers may have checked in random numbers of bags, and each checked-in bag may be forgotten by the airline. You have seen \(k\) bags come out, but yours has not appeared.

Passengers \(n\)

Mean bags per passenger \(\lambda\)

Forgetting probability \(\rho\)

Bags already out \(k\)

End of delivery announced

Probability your bag was forgotten

0.000

After 20 bags are out and yours has not appeared.

Posterior probability versus \(k\)

Modeling

Solution

Ocean crash

Pieces of a crashed plane are found at different times. The mean wind drift is known, but random variation grows with time. Use the debris locations to infer the unknown crash location.

Wind vector \(v=[v_x,v_y]\)

Base variance \(\sigma_0^2\)

Variance growth \(q\)

Noise correlation \(r\)

Prior mean \(m_0=[m_{0,x},m_{0,y}]\)

Prior SD \(\tau\)

Number of pieces

Piece	Time \(t_i\)	\(x_i\)	\(y_i\)

Posterior mean crash location

(0, 0)

Posterior uncertainty is shown by the density image in the corrected plane.

Debris locations and posterior crash location

Uncorrected locations, draggable

Drift-corrected locations and the posterior distribution