Probabilistic Reasoning in Data Analysis

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

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

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

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

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

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

7. Random Arrivals: Events that do not depend on prior history 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 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 7

8. Poisson Process This form guarantees Since <t>=1/λ 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

9. Consequences of the Poisson process For any time t = t1,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 by so that mean & variance are equal. 9

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

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

13. Poisson Processes in Biology Related studies 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. References del Castillo & Katz, J. Physiol. 124:560-573, 1954. Boyd & Martin, J. Physiol. 132:74-91, 1956. Luria & Delbrück, Genetics 28:491-511, 1943. 13

14. Neuro- muscular Vesicle Release Fluctuations in post-synaptic end plate potential (e.p.p.) as re-investigated by Boyd & Martin (1956) This publication reports on No= 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 fluctuationOver all trials, the mean fluctuation was 0.993 mV. Therefore, the mean arrival is m = 0.993/0.4 = 2.33. 14

15. This implies that the number vesicles is given by: The Gaussian Fit This implies that the number vesicles is given by: 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. Boyd & Martin, J. Physiol. 132:74-91, 1956 15

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

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