mode is the value that occurs most frequently in a distribution.
what is the value that occurs most frequently in a distribution?
mode
the most likely value of a desired parameter (the mode) could then simply be selected by choosing the sample value that occurs most commonly; this is essentially equivalent to maximum a posteriori estimation of a parameter. (since the parameters are usually continuous, it is often necessary to "bin" the sampled values into one of a finite number of ranges or "bins" in order to get a meaningful estimate of the mode.) more commonly, however, the expected value (mean or average) of the sampled values is chosen; this is a bayes estimator that takes advantage of the additional data about the entire distribution that is available from bayesian sampling, whereas a maximization algorithm such as expectation maximization (em) is capable of only returning a single point from the distribution. for example, for a unimodal distribution the mean (expected value) is usually similar to the mode (most common value), but if the distribution is skewed in one direction, the mean will be moved in that direction, which effectively accounts for the extra probability mass in that direction. (note, however, that if a distribution is multimodal, the expected value may not return a meaningful point, and any of the modes is typically a better choice.)
gibbs sampling
for a random walk with a starting value of "s", and at every iterative step, is moved by a normal distribution having mean "μ" and standard deviation "σ" and failure occurs if it reaches 0 or a negative value. for example, with a starting value of 10, at each iteration, a gaussian random variable having mean 0.1 and standard deviation 1 is added to the value from the previous iteration. in this formula, "s" is 10, "σ" is 1, "μ" is 0.1, and so r is the square root of 1.01, or about 1.005. the mean of the distribution added to the previous value every time is positive, but not nearly as large as the standard deviation, so there is a risk of it falling to negative values before taking off indefinitely toward positive infinity. this formula predicts a probability of failure using these parameters of about 0.1371, or a 13.71% risk of ruin. this approximation becomes more accurate when the number of steps typically expected for ruin to occur, if it occurs, becomes larger; it is not very accurate if the very first step could make or break it. this is because it is an exact solution if the random variable added at each step is not a gaussian random variable but rather a binomial random variable with parameter n=2. however, repeatedly adding a random variable that is not distributed by a gaussian distribution into a running sum in this way asymptotically becomes indistinguishable from adding gaussian distributed random variables, by the law of large numbers.
risk of ruin