on 03-Mar-2017 (Fri)

The concept of representing a distribution by a large representative sample is foundational for the approach we take to Bayesian analysis of complex models

What is new in the present application is that the population from which we are sampling is a mathematically defined distribution, such as a posterior probability distributio

Here is a summary of our algorithm for moving from one position to another. We are currently at position θ current . We then propose to move one position right or one position left. The specific proposal is determined by flipping a coin, which can result in 50% heads (move right) or 50% tails (move left). The range of possible proposed moves, and the probability of proposing each, is called the proposal distribution. In the present algorithm, the proposal distribution is very simple: It has only two values with 50-50 probabilities. Having proposed a move, we then decide whether or not to accept it. The acceptance decision is based on the value of the target distribution at the proposed position, relative to the value of the target distribution at our current position. Specifically, if the target distribution is greater at the proposed position than at our current position, then we definitely accept the proposed move: We always move higher if we can. On the other hand, if the target position is less at the proposed position than at our current position, we accept the move probabilistically: We move to the proposed position with probability p move =P(θ proposed )/P(θ current ),whereP(θ) is the value of the target distribution at θ .We can combine these two possibilities, of the target distribution being higher or lower at the proposed position than at our current position, into a single expression for the probability of moving to the proposed position: p move = min P(θ proposed ) P(θ current ) ,1 (7.1) Notice that Equation 7.1 says that when P(θ proposed )>P(θ current ),thenp move =1. Notice also that the target distribution, P(θ), does not need to be normalized, which means it does not need to sum to 1 as a probability distribution must. This is because what matters for our choice is the ratio, P(θ proposed )/P(θ current ), not the absolute magnitude of P(θ)

Notice what we must be able to do in the random-walk process: • We must be able to generate a random value from the proposal distribution, to create θ proposed . • We must be able to evaluate the target distribution at any proposed position, to compute P(θ proposed )/P(θ current ). • We must be able to generate a random value from a uniform distribution, to accept or reject the proposal according to p move

Suppose we are at position θ. The probability of moving to θ + 1, denoted p(θ → θ + 1), is the probability of proposing that move times the probability of accepting it if proposed, which is p(θ → θ + 1) =0.5 · min ( P(θ + 1)/P(θ ),1 )

On the other hand, if we are presently at position θ + 1, the probability of moving to θ is the probability of proposing that move times the probability of accepting it if proposed, which is p(θ + 1 → θ)=0.5 · min ( P(θ)/P(θ + 1),1 ) .

The ratio of the transition probabilities is \(\frac{p(θ → θ +1)}{ p(θ +1 → θ)} = \frac{0.5 min ( P(θ +1)/P(θ),1 )}{ 0.5 min ( P(θ)/P(θ +1),1 )}\) =
\(\frac {1} {P(\theta)/P(\theta+1)} \) (if P(\theta+1) > P(\theta))

\(\frac {P(\theta)/P(\theta+1)} {1}\) (if P(\theta+1) < P(\theta))

= \(\frac{P(\theta+1)} {P(\theta)}\)

When the vector of position probabilities is the target distribution, it stays that way on the next time step! In other words, the position probabilities are stable at the target distribution

Sample values from the target distribution are generated by taking a random walk through the parameter space. The walk starts at some arbitrary point, specified by the user. The starting point should be someplace where P(θ ) is nonzero. The random walk progresses at each time step by proposing a move to a new position in parameter space and then deciding whether or not to accept the proposed move. Proposal distributions can take on many different forms, with the goal being to use a proposal distribution that efficiently explores the regions of the parameter space where P(θ ) has most of its mass.

Having generated a proposed new position, the algorithm then decides whether or not to accept the proposal. The decision rule is exactly what was already specified in Equation 7.1. In detail, this is accomplished by computing the ratio p move =P(θ proposed )/P(θ current ).

Here’s how to get handles: The functions that draw graphics objects can also be used to return the handle of the object drawn, e.g., x = 0:pi/20:2*pi; hsin = plot(x, sin(x)) hold on hx = xlabel(’x’)

gcf gets the handle of the current figure, e.g., hf = gcf

gca gets the handle of the current axes

gco gets the handle of the current graphics object, which is the last graph- ics object created or clicked on. For example, draw the sine graph above and get its handle hsin. Click on the graph in the figure win- dow. Then enter the command ho = gco

Handle Graphics objects are the basic elements used in MATLAB graphics. The objects are arranged in a parent-child inheritance structure

To see all the property names of an object and their current values use get(h) where h is the object’s handle

You can change any property value with the set function: set(handle, ‘PropertyName’, PropertyValue)

The command set(handle) lists all the possible property values (where appro- priate)

If a graphics object has a number of children the get command used with the children property returns a vector of the children’s handles

The answer is that the handles of children of the axes are returned in the reverse order in which they are created

If you are desperate and don’t know the handles of any of your graphics ob- jects you can use the findobj function to get the handle of an object with a property value that uniquely identifies it

you can specify the parent of an object when you create it

From the concentrations of the ligand, antibody, and antibody–ligand complex at equilibrium, we can calculate a convenient measure of the strength of bind- ing—the equilibrium constant (K)

The equilibrium constant for a reaction in which two molecules (A and B) bind to each other to form a complex (AB) has units of liters/mole, and half of the binding sites will be occupied by ligand when that ligand’s concentration (in moles/liter) reaches a value that is equal to 1/K