1. Probabilistic Reasoning in Data Analysis Lawrence Sirovich Mt. Sinai School of Med.; Rockefeller U.; Courant Inst., NYU lsirovich@rockefeller.edu 1

2. Coin Tossing Urn Model: Urn filled with N b, black, and N w,white marbles, chosen at random with replacement. Experiments with probability More generally … and for all permutations we obtain the binomial probability distribution (slide 3) Table for 3 tosses 2 where N h = number of heads in N trials. Generalize to k colors so that Probability of q heads followed by p tails

3. This is the gamma distribution of rate λ, since = λ -1 Observations on probabilities 3 }, for x discrete or continuous Factorial for noninteger n defined by Therefore is a probability in x, since }, for x discrete or continuous } } } } From and a little thought it is seen that B n (k) is probability of k heads in n trials.

4. Which yields Stirling’s formula: Gaussian probabilty: Miscellanea 4 Hint: Expand log for 1/n small in Hint: Integrand has a max at y = n, and area is concentrated therein Probability distribution functions, pdfs.

5. Binomial probability distribution Binomial distribution: A special case Assume that : Thus, if This is called the Poisson distribution As with all probability distributions 5 and Stirling’s formula are substituted, a pdf in k for t fixed.Hint:

6. Expected Outcomes 6 For pdf P(x) and some function f(x) define expectation Thus, the average or mean is

7. Biological examples: Photon arrivals activating retinal photoreceptors Spontaneous neurotransmitter release events at a synapse Suppose N(t) is the number of events (arrivals) in the time t ; then the arrival rate is estimated by Random Arrivals: Events that do not depend on prior history 7 The approximate number of events for any t is Since the process is memoryless, the pdf in Ƭ satisfies This functional relation can only be satisfied by an exponential

8. Poisson Process This is the probability of waiting times for a Poisson process (not the same as a Poisson Distribution) Average or expected waiting time defined by: 8 This form guarantees Since =1/λ

9. Consequences of the Poisson process 9 For any time t = t 1, the probability of an event in the interval t is A consequence of this is that the probability of no event is Since λdt is the probability of an event, and 1 – λdt a nonevent, in an increment dt in general The previously defined Poisson pdf satisfies this differential equation, which justifies the notation. Recall The variance is given byso that mean & variance are equal.

10. Poisson Processes in Biology A classic, Nobel-worthy, paper Hecht, Shlaer, & Pirenne, J. Gen. Physiol. 25:819-840, 1942. addresses the question: How many photons must be captured by the retina for the subject to correctly perceive an event? In the psychophysical experiment, subjects are exposed to brief 1-ms flashes of light to determine the probability dependence of perception on the brightness of the flash. Quantal content of 1-ms flash It is a reasonable hypothesis that absorption of quanta by the retina obeys a Poisson distribution 10

11. The CumulativePoisson pdf Note: The curves can be distinguished from one another by the steepness, which depends on n. 11 The probability that n or more photons are detected is described by the cumulative pdf:

12. Original data from Hecht, Shlaer, & Pirenne These data are to be analyzed in the Problem Set Hecht, Shlaer, & Pirenne, J. Gen. Physiol. 25:819-840, 1942. 12

13. Poisson Processes in Biology Related studies Boyd & Martin, J. Physiol. 132:74-91, 1956. References Luria & Delbrück, Genetics 28:491-511, 1943. 13 del Castillo & Katz, J. Physiol. 124:560-573, 1954. Next, we consider the work of Bernard Katz (del Castillo & Katz, 1954) who recorded nervous activity at the neuromuscular junction and noted low-level persistent voltage activity, later called mini end-plate potentials. He pursued the origin of this noise and conjectured that it was due to the synaptic release of vesicles of neurotransmitter of uniform size and further hypothesized that the number of arriving vesicles followed a Poisson distribution. The ensuing experiments confirmed his speculations and contributed to his Nobel prize in 1970. The next two slides are based on subsequent verification (Boyd & Martin, 1956), and summarize the experimental and theoretical deliberations that went into this brilliant scientific effort. In a complementary vein, Luria & Delbrück (1943) demonstrated that bacterial mutations were of random origin. In effect, they did this by refuting the hypothesis that that the mutations were governed by Poisson statistics; in the process, they established the genetic basis of bacterial reproduction and were awarded the Nobel prize for this work in 1969.

14. Neuro- muscular Vesicle Release Fluctuations in post-synaptic end plate potential (e.p.p.) as re- investigated by Boyd & Martin (1956) 14 This publication reports on N o = 198 trials measuring postsynaptic epps in response to a single presynaptic neural impulse. The inset to the figure below is the histogram of spontaneous activity-no upstream impulse. Under Katz’s hypotheses, this implies that a single vesicle produces a 0.4-mV fluctuation Over all trials, the mean fluctuation was 0.993 mV. Therefore, the mean arrival is m = 0.993/0.4 = 2.33.

15. The Gaussian Fit 15 Boyd & Martin, J. Physiol. 132:74-91, 1956 The continuous curve in the previous slide was created by a Gaussian fit, as explained in the figure legend below, that allows for the inclusion of the side bars that are seen in the above histogram. This implies that the number vesicles is given by:

16. Theory vs. Experiment Read this paper to see how beautifully all the data agree with the hypothesis that evoked release follows a Poisson distribution k012345678 Poisson194452402411521 Experiment184455362512521 Comparison between theory and experiment is summarized in the following table

17. www.sciencesignaling.org Slides from a lecture in the course Systems Biology—Biomedical Modeling Citation: L. Sirovich, Probabilistic reasoning in data analysis. Sci. Signal. 4, tr14 (2011).