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Parameter uncertainty in probabilistic forecasting

20 November 2013

One of the interesting things you can do with probabilistic programming is to forecast the future evolution of a timeseries. One thing you need to do this is a model of the system, but you also need to handle the uncertainty correctly. In this blog post, I will show the impact of taking into consideration the uncertainty associated with parameter estimation. I will use Baysig, our purely functional, spatio-temporal, probabilistic programming language, but a similar analysis can be carried out in any other probabilistic programming language. I will use an example from financial data analysis, because stochastic models of financial data have been explored for some time and the data are readily available. However, these principles apply equally to scientific, retail or industrial data.

Here are the data we would like to forecast: about two years worth of daily stock quotes for IBM on the New York Stock Exchange:

"2011-10-02T22:00:00.000Z", 173.29 "2011-10-03T22:00:00.000Z", 174.74 "2011-10-04T22:00:00.000Z", 176.85 "2011-10-05T22:00:00.000Z", 181.69 "2011-10-06T22:00:00.000Z", 182.39 "2011-10-09T22:00:00.000Z", 186.62 "2011-10-10T22:00:00.000Z", 185 "2011-10-11T22:00:00.000Z", 186.12 "2011-10-12T22:00:00.000Z", 186.82 "2011-10-13T22:00:00.000Z", 190.53 "2011-10-16T22:00:00.000Z", 186.59 "2011-10-17T22:00:00.000Z", 178.9 "2011-10-18T22:00:00.000Z", 177.39 "2011-10-19T22:00:00.000Z", 177.25 "2011-10-20T22:00:00.000Z", 181.63 "2011-10-23T22:00:00.000Z", 182.25 "2011-10-24T22:00:00.000Z", 180.36 "2011-10-25T22:00:00.000Z", 181.97 "2011-10-26T22:00:00.000Z", 186.34 "2011-10-27T22:00:00.000Z", 187.45 "2011-10-30T22:00:00.000Z", 184.63 "2011-10-31T22:00:00.000Z", 181.35 "2011-11-01T22:00:00.000Z", 183.92 "2011-11-02T22:00:00.000Z", 187.3 "2011-11-03T22:00:00.000Z", 186.38 "2011-11-06T22:00:00.000Z", 187.32 "2011-11-07T22:00:00.000Z", 187.25 "2011-11-08T22:00:00.000Z", 182.24 "2011-11-09T22:00:00.000Z", 183.35 "2011-11-10T22:00:00.000Z", 187.38 "2011-11-13T22:00:00.000Z", 187.35 "2011-11-14T22:00:00.000Z", 188.75 "2011-11-15T22:00:00.000Z", 186.62 "2011-11-16T22:00:00.000Z", 185.73 "2011-11-17T22:00:00.000Z", 185.24 "2011-11-20T22:00:00.000Z", 181.48 "2011-11-21T22:00:00.000Z", 181.31 "2011-11-22T22:00:00.000Z", 177.95 "2011-11-24T22:00:00.000Z", 177.06 "2011-11-27T22:00:00.000Z", 182.21 "2011-11-28T22:00:00.000Z", 180.94 "2011-11-29T22:00:00.000Z", 188 "2011-11-30T22:00:00.000Z", 189.45 "2011-12-01T22:00:00.000Z", 189.66 "2011-12-04T22:00:00.000Z", 190.84 "2011-12-05T22:00:00.000Z", 192.94 "2011-12-06T22:00:00.000Z", 194.05 "2011-12-07T22:00:00.000Z", 191.58 "2011-12-08T22:00:00.000Z", 194.56 "2011-12-11T22:00:00.000Z", 192.18 "2011-12-12T22:00:00.000Z", 191.15 "2011-12-13T22:00:00.000Z", 188.72 "2011-12-14T22:00:00.000Z", 187.48 "2011-12-15T22:00:00.000Z", 183.57 "2011-12-18T22:00:00.000Z", 182.89 "2011-12-19T22:00:00.000Z", 187.24 "2011-12-20T22:00:00.000Z", 181.47 "2011-12-21T22:00:00.000Z", 182.04 "2011-12-22T22:00:00.000Z", 184.75 "2011-12-26T22:00:00.000Z", 184.95 "2011-12-27T22:00:00.000Z", 183.99 "2011-12-28T22:00:00.000Z", 186.18 "2011-12-29T22:00:00.000Z", 183.88 "2012-01-02T22:00:00.000Z", 186.3 "2012-01-03T22:00:00.000Z", 185.54 "2012-01-04T22:00:00.000Z", 184.66 "2012-01-05T22:00:00.000Z", 182.54 "2012-01-08T22:00:00.000Z", 181.59 "2012-01-09T22:00:00.000Z", 181.31 "2012-01-10T22:00:00.000Z", 182.32 "2012-01-11T22:00:00.000Z", 180.55 "2012-01-12T22:00:00.000Z", 179.16 "2012-01-16T22:00:00.000Z", 180 "2012-01-17T22:00:00.000Z", 181.07 "2012-01-18T22:00:00.000Z", 180.52 "2012-01-19T22:00:00.000Z", 188.52 "2012-01-22T22:00:00.000Z", 189.98 "2012-01-23T22:00:00.000Z", 191.93 "2012-01-24T22:00:00.000Z", 191.73 "2012-01-25T22:00:00.000Z", 190.98 "2012-01-26T22:00:00.000Z", 190.46 "2012-01-29T22:00:00.000Z", 192.5 "2012-01-30T22:00:00.000Z", 192.6 "2012-01-31T22:00:00.000Z", 192.62 "2012-02-01T22:00:00.000Z", 191.53 "2012-02-02T22:00:00.000Z", 193.64 "2012-02-05T22:00:00.000Z", 192.82 "2012-02-06T22:00:00.000Z", 193.35 "2012-02-07T22:00:00.000Z", 192.95 "2012-02-08T22:00:00.000Z", 193.13 "2012-02-09T22:00:00.000Z", 192.42 "2012-02-12T22:00:00.000Z", 192.62 "2012-02-13T22:00:00.000Z", 192.22 "2012-02-14T22:00:00.000Z", 192.25 "2012-02-15T22:00:00.000Z", 193.02 "2012-02-16T22:00:00.000Z", 193.42 "2012-02-20T22:00:00.000Z", 193.39 "2012-02-21T22:00:00.000Z", 193.87 "2012-02-22T22:00:00.000Z", 197.61 "2012-02-23T22:00:00.000Z", 197.76 "2012-02-26T22:00:00.000Z", 197.53 "2012-02-27T22:00:00.000Z", 197.98 "2012-02-28T22:00:00.000Z", 196.73 "2012-02-29T22:00:00.000Z", 197.53 "2012-03-01T22:00:00.000Z", 198.81 "2012-03-04T22:00:00.000Z", 200.66 "2012-03-05T22:00:00.000Z", 197.26 "2012-03-06T22:00:00.000Z", 197.77 "2012-03-07T22:00:00.000Z", 199.81 "2012-03-08T22:00:00.000Z", 200.62 "2012-03-11T22:00:00.000Z", 201 "2012-03-12T22:00:00.000Z", 203.78 "2012-03-13T22:00:00.000Z", 204.72 "2012-03-14T22:00:00.000Z", 206 "2012-03-15T22:00:00.000Z", 206.01 "2012-03-18T22:00:00.000Z", 205.72 "2012-03-19T22:00:00.000Z", 204.25 "2012-03-20T22:00:00.000Z", 204.69 "2012-03-21T22:00:00.000Z", 205.49 "2012-03-22T22:00:00.000Z", 205.48 "2012-03-25T22:00:00.000Z", 207.77 "2012-03-26T22:00:00.000Z", 207.18 "2012-03-27T22:00:00.000Z", 207.29 "2012-03-28T22:00:00.000Z", 208.27 "2012-03-29T22:00:00.000Z", 208.65 "2012-04-01T22:00:00.000Z", 209.47 "2012-04-02T22:00:00.000Z", 209.5 "2012-04-03T22:00:00.000Z", 206.05 "2012-04-04T22:00:00.000Z", 205.47 "2012-04-08T22:00:00.000Z", 204.94 "2012-04-09T22:00:00.000Z", 202.33 "2012-04-10T22:00:00.000Z", 202.58 "2012-04-11T22:00:00.000Z", 205.32 "2012-04-12T22:00:00.000Z", 202.8 "2012-04-15T22:00:00.000Z", 202.72 "2012-04-16T22:00:00.000Z", 207.45 "2012-04-17T22:00:00.000Z", 200.13 "2012-04-18T22:00:00.000Z", 199.51 "2012-04-19T22:00:00.000Z", 199.6 "2012-04-22T22:00:00.000Z", 198.62 "2012-04-23T22:00:00.000Z", 200 "2012-04-24T22:00:00.000Z", 203.57 "2012-04-25T22:00:00.000Z", 205.58 "2012-04-26T22:00:00.000Z", 206.81 "2012-04-29T22:00:00.000Z", 207.08 "2012-04-30T22:00:00.000Z", 208 "2012-05-01T22:00:00.000Z", 208.06 "2012-05-02T22:00:00.000Z", 207.24 "2012-05-03T22:00:00.000Z", 204.99 "2012-05-06T22:00:00.000Z", 203.75 "2012-05-07T22:00:00.000Z", 201.48 "2012-05-08T22:00:00.000Z", 201.23 "2012-05-09T22:00:00.000Z", 200.6 "2012-05-10T22:00:00.000Z", 201.17 "2012-05-13T22:00:00.000Z", 199.44 "2012-05-14T22:00:00.000Z", 199.04 "2012-05-15T22:00:00.000Z", 199.73 "2012-05-16T22:00:00.000Z", 197.89 "2012-05-17T22:00:00.000Z", 195.88 "2012-05-20T22:00:00.000Z", 197.76 "2012-05-21T22:00:00.000Z", 196.82 "2012-05-22T22:00:00.000Z", 196.12 "2012-05-23T22:00:00.000Z", 196.09 "2012-05-24T22:00:00.000Z", 194.3 "2012-05-28T22:00:00.000Z", 196.46 "2012-05-29T22:00:00.000Z", 194.53 "2012-05-30T22:00:00.000Z", 192.9 "2012-05-31T22:00:00.000Z", 189.08 "2012-06-03T22:00:00.000Z", 188.54 "2012-06-04T22:00:00.000Z", 189.2 "2012-06-05T22:00:00.000Z", 193.99 "2012-06-06T22:00:00.000Z", 194.44 "2012-06-07T22:00:00.000Z", 195.14 "2012-06-10T22:00:00.000Z", 192.51 "2012-06-11T22:00:00.000Z", 194.55 "2012-06-12T22:00:00.000Z", 193.1 "2012-06-13T22:00:00.000Z", 195.1 "2012-06-14T22:00:00.000Z", 199.1 "2012-06-17T22:00:00.000Z", 198.29 "2012-06-18T22:00:00.000Z", 198.93 "2012-06-19T22:00:00.000Z", 198.78 "2012-06-20T22:00:00.000Z", 193.39 "2012-06-21T22:00:00.000Z", 193.7 "2012-06-24T22:00:00.000Z", 192.86 "2012-06-25T22:00:00.000Z", 191.95 "2012-06-26T22:00:00.000Z", 193 "2012-06-27T22:00:00.000Z", 191.4 "2012-06-28T22:00:00.000Z", 195.58 "2012-07-01T22:00:00.000Z", 195.83 "2012-07-02T22:00:00.000Z", 195.93 "2012-07-04T22:00:00.000Z", 195.29 "2012-07-05T22:00:00.000Z", 191.41 "2012-07-08T22:00:00.000Z", 189.67 "2012-07-09T22:00:00.000Z", 186.26 "2012-07-10T22:00:00.000Z", 185.25 "2012-07-11T22:00:00.000Z", 183.09 "2012-07-12T22:00:00.000Z", 186.01 "2012-07-15T22:00:00.000Z", 184.79 "2012-07-16T22:00:00.000Z", 183.65 "2012-07-17T22:00:00.000Z", 188.25 "2012-07-18T22:00:00.000Z", 195.34 "2012-07-19T22:00:00.000Z", 192.45 "2012-07-22T22:00:00.000Z", 190.83 "2012-07-23T22:00:00.000Z", 190.34 "2012-07-24T22:00:00.000Z", 191.08 "2012-07-25T22:00:00.000Z", 193.95 "2012-07-26T22:00:00.000Z", 196.39 "2012-07-29T22:00:00.000Z", 196.68 "2012-07-30T22:00:00.000Z", 195.98 "2012-07-31T22:00:00.000Z", 195.18 "2012-08-01T22:00:00.000Z", 194.45 "2012-08-02T22:00:00.000Z", 198.52 "2012-08-05T22:00:00.000Z", 198.76 "2012-08-06T22:00:00.000Z", 199.93 "2012-08-07T22:00:00.000Z", 199.03 "2012-08-08T22:00:00.000Z", 198.42 "2012-08-09T22:00:00.000Z", 199.29 "2012-08-12T22:00:00.000Z", 199.01 "2012-08-13T22:00:00.000Z", 198.29 "2012-08-14T22:00:00.000Z", 198.4 "2012-08-15T22:00:00.000Z", 200.84 "2012-08-16T22:00:00.000Z", 201.22 "2012-08-19T22:00:00.000Z", 200.5 "2012-08-20T22:00:00.000Z", 198.65 "2012-08-21T22:00:00.000Z", 197.25 "2012-08-22T22:00:00.000Z", 195.7 "2012-08-23T22:00:00.000Z", 197.77 "2012-08-26T22:00:00.000Z", 195.69 "2012-08-27T22:00:00.000Z", 194.87 "2012-08-28T22:00:00.000Z", 195.08 "2012-08-29T22:00:00.000Z", 193.37 "2012-08-30T22:00:00.000Z", 194.85 "2012-09-03T22:00:00.000Z", 194.54 "2012-09-04T22:00:00.000Z", 195.04 "2012-09-05T22:00:00.000Z", 199.1 "2012-09-06T22:00:00.000Z", 199.5 "2012-09-09T22:00:00.000Z", 200.95 "2012-09-10T22:00:00.000Z", 203.27 "2012-09-11T22:00:00.000Z", 203.77 "2012-09-12T22:00:00.000Z", 206.36 "2012-09-13T22:00:00.000Z", 206.81 "2012-09-16T22:00:00.000Z", 207.15 "2012-09-17T22:00:00.000Z", 207.07 "2012-09-18T22:00:00.000Z", 206.43 "2012-09-19T22:00:00.000Z", 206.18 "2012-09-20T22:00:00.000Z", 205.98 "2012-09-23T22:00:00.000Z", 205.29 "2012-09-24T22:00:00.000Z", 204.98 "2012-09-25T22:00:00.000Z", 204 "2012-09-26T22:00:00.000Z", 205.91 "2012-09-27T22:00:00.000Z", 207.45 "2012-09-30T22:00:00.000Z", 210.47 "2012-10-01T22:00:00.000Z", 209.84 "2012-10-02T22:00:00.000Z", 210.51 "2012-10-03T22:00:00.000Z", 210.39 "2012-10-04T22:00:00.000Z", 210.59 "2012-10-07T22:00:00.000Z", 209.82 "2012-10-08T22:00:00.000Z", 207.99 "2012-10-09T22:00:00.000Z", 205.82 "2012-10-10T22:00:00.000Z", 205.76 "2012-10-11T22:00:00.000Z", 207.8 "2012-10-14T22:00:00.000Z", 208.93 "2012-10-15T22:00:00.000Z", 211 "2012-10-16T22:00:00.000Z", 200.63 "2012-10-17T22:00:00.000Z", 194.96 "2012-10-18T22:00:00.000Z", 193.36 "2012-10-21T22:00:00.000Z", 194.4 "2012-10-22T22:00:00.000Z", 191.25 "2012-10-23T22:00:00.000Z", 190.72 "2012-10-24T22:00:00.000Z", 191.6 "2012-10-25T22:00:00.000Z", 193.27 "2012-10-30T22:00:00.000Z", 194.53 "2012-10-31T22:00:00.000Z", 197.15 "2012-11-01T22:00:00.000Z", 193.43 "2012-11-04T22:00:00.000Z", 194.14 "2012-11-05T22:00:00.000Z", 195.07 "2012-11-06T22:00:00.000Z", 191.16 "2012-11-07T22:00:00.000Z", 190.1 "2012-11-08T22:00:00.000Z", 189.64 "2012-11-11T22:00:00.000Z", 189.25 "2012-11-12T22:00:00.000Z", 188.32 "2012-11-13T22:00:00.000Z", 185.51 "2012-11-14T22:00:00.000Z", 185.85 "2012-11-15T22:00:00.000Z", 186.94 "2012-11-18T22:00:00.000Z", 190.35 "2012-11-19T22:00:00.000Z", 189.2 "2012-11-20T22:00:00.000Z", 190.29 "2012-11-22T22:00:00.000Z", 193.49 "2012-11-25T22:00:00.000Z", 192.88 "2012-11-26T22:00:00.000Z", 191.23 "2012-11-27T22:00:00.000Z", 191.98 "2012-11-28T22:00:00.000Z", 191.53 "2012-11-29T22:00:00.000Z", 190.07 "2012-12-02T22:00:00.000Z", 189.48 "2012-12-03T22:00:00.000Z", 189.36 "2012-12-04T22:00:00.000Z", 188.65 "2012-12-05T22:00:00.000Z", 189.7 "2012-12-06T22:00:00.000Z", 191.95 "2012-12-09T22:00:00.000Z", 192.62 "2012-12-10T22:00:00.000Z", 194.2 "2012-12-11T22:00:00.000Z", 192.95 "2012-12-12T22:00:00.000Z", 191.99 "2012-12-13T22:00:00.000Z", 191.76 "2012-12-16T22:00:00.000Z", 193.62 "2012-12-17T22:00:00.000Z", 195.69 "2012-12-18T22:00:00.000Z", 195.08 "2012-12-19T22:00:00.000Z", 194.77 "2012-12-20T22:00:00.000Z", 193.42 "2012-12-23T22:00:00.000Z", 192.4 "2012-12-25T22:00:00.000Z", 191.95 "2012-12-26T22:00:00.000Z", 192.71 "2012-12-27T22:00:00.000Z", 189.83 "2012-12-30T22:00:00.000Z", 191.55 "2013-01-01T22:00:00.000Z", 196.35 "2013-01-02T22:00:00.000Z", 195.27 "2013-01-03T22:00:00.000Z", 193.99 "2013-01-06T22:00:00.000Z", 193.14 "2013-01-07T22:00:00.000Z", 192.87 "2013-01-08T22:00:00.000Z", 192.32 "2013-01-09T22:00:00.000Z", 192.88 "2013-01-10T22:00:00.000Z", 194.45 "2013-01-13T22:00:00.000Z", 192.62 "2013-01-14T22:00:00.000Z", 192.5 "2013-01-15T22:00:00.000Z", 192.59 "2013-01-16T22:00:00.000Z", 193.65 "2013-01-17T22:00:00.000Z", 194.47 "2013-01-21T22:00:00.000Z", 196.08 "2013-01-22T22:00:00.000Z", 204.72 "2013-01-23T22:00:00.000Z", 204.42 "2013-01-24T22:00:00.000Z", 204.97 "2013-01-27T22:00:00.000Z", 204.93 "2013-01-28T22:00:00.000Z", 203.9 "2013-01-29T22:00:00.000Z", 203.52 "2013-01-30T22:00:00.000Z", 203.07 "2013-01-31T22:00:00.000Z", 205.18 "2013-02-03T22:00:00.000Z", 203.79 "2013-02-04T22:00:00.000Z", 202.79 "2013-02-05T22:00:00.000Z", 201.02 "2013-02-06T22:00:00.000Z", 199.74 "2013-02-07T22:00:00.000Z", 201.68 "2013-02-10T22:00:00.000Z", 200.16 "2013-02-11T22:00:00.000Z", 200.04 "2013-02-12T22:00:00.000Z", 200.09 "2013-02-13T22:00:00.000Z", 199.65 "2013-02-14T22:00:00.000Z", 200.98 "2013-02-18T22:00:00.000Z", 200.32 "2013-02-19T22:00:00.000Z", 199.31 "2013-02-20T22:00:00.000Z", 198.33 "2013-02-21T22:00:00.000Z", 201.09 "2013-02-24T22:00:00.000Z", 197.51 "2013-02-25T22:00:00.000Z", 199.14 "2013-02-26T22:00:00.000Z", 202.33 "2013-02-27T22:00:00.000Z", 200.83 "2013-02-28T22:00:00.000Z", 202.91 "2013-03-03T22:00:00.000Z", 205.19 "2013-03-04T22:00:00.000Z", 206.53 "2013-03-05T22:00:00.000Z", 208.38 "2013-03-06T22:00:00.000Z", 209.42 "2013-03-07T22:00:00.000Z", 210.38 "2013-03-10T22:00:00.000Z", 210.08 "2013-03-11T22:00:00.000Z", 210.55 "2013-03-12T22:00:00.000Z", 212.06 "2013-03-13T22:00:00.000Z", 215.8 "2013-03-14T22:00:00.000Z", 214.92 "2013-03-17T22:00:00.000Z", 213.21 "2013-03-18T22:00:00.000Z", 213.44 "2013-03-19T22:00:00.000Z", 215.06 "2013-03-20T22:00:00.000Z", 212.26 "2013-03-21T22:00:00.000Z", 212.08 "2013-03-24T22:00:00.000Z", 210.74 "2013-03-25T22:00:00.000Z", 212.36 "2013-03-26T22:00:00.000Z", 210.89 "2013-03-27T22:00:00.000Z", 213.3 "2013-03-31T22:00:00.000Z", 212.38 "2013-04-01T22:00:00.000Z", 214.36 "2013-04-02T22:00:00.000Z", 212.66 "2013-04-03T22:00:00.000Z", 211.31 "2013-04-04T22:00:00.000Z", 209.41 "2013-04-07T22:00:00.000Z", 209.32 "2013-04-08T22:00:00.000Z", 209.22 "2013-04-09T22:00:00.000Z", 212 "2013-04-10T22:00:00.000Z", 212.92 "2013-04-11T22:00:00.000Z", 211.38 "2013-04-14T22:00:00.000Z", 209.26 "2013-04-15T22:00:00.000Z", 212 "2013-04-16T22:00:00.000Z", 209.67 "2013-04-17T22:00:00.000Z", 207.15 "2013-04-18T22:00:00.000Z", 190 "2013-04-21T22:00:00.000Z", 187.83 "2013-04-22T22:00:00.000Z", 191.61 "2013-04-23T22:00:00.000Z", 191.71 "2013-04-24T22:00:00.000Z", 193.95 "2013-04-25T22:00:00.000Z", 194.31 "2013-04-28T22:00:00.000Z", 199.15 "2013-04-29T22:00:00.000Z", 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axisLabels "Time" "IBM price" 
   $ sigPlot (#ibm#close)

I will use one of the simplest continuous-time models for equity prices, geometric Brownian motion1, which forms the basis of the Black-Scholes option pricing model. Here it is as a Bayesian statistical model:

1 dS=strdt+vstdWt
gbmModel = prob
   s_0 ~ improper_uniform_positive
   r   ~ improper_uniform
   v   ~ improper_uniform_positive
   w   ~ wiener
   d s t = s t*r+v*s t*d w t
   return s

The first three lines of the model give the priors for the parameters r (the risk-free interest rate) and v (volatility). The parameter s0 is the initial value of the timeseries. It isn't a variable to be estimated, because it is directly observed from the data as the first time-point. However, it still needs to be introduced for the model to make sense as a generative program; otherwise you wouldn't be able to simulate data from the model.

It might be argued that the parameter r should be fixed to the actual interest rate. Here, I have left it as a parameter to be estimated. I have also allowed it to be negative, with which you may or may not agree.

The remainder of the program samples a wiener-process sample path w and then defines s as the solution to the stochastic differential equation corresponding to the geometric Brownian motion.

Then, comrades, let's estimate!

params <* estimate gbmModel (#ibm#close)
params ⇒ {
   r => 1.0e-4 ± 4.0e-4,
   v => 0.01123 ± 0.00035,
}

The estimate gives us a probability distribution over the parameters in the model given the data2. It is a joint distribution, meaning that we also have information about the correlations in the parameter estimates.

2The parameter estimates are expressed in daily units, which is why they are very small
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0.0103836 0.0112536 0.0115561 0.0117864 0.0112188 0.0113361 0.011393 0.0113624 0.0121338 0.0111786 0.0111217 0.0114412 0.0108968 0.0109931 0.0112111 0.0106993 0.0112797 0.0105394 0.0113875 0.0109113 0.0108583 0.011496 0.0113895 0.0108717 0.0110708 0.011124 0.0111076 0.0111473 0.0105783 0.0109114 0.0110196 0.0110519 0.0109825 0.0107458 0.0115171 0.0111865 0.0109747 0.0110062 0.0113312 0.011413 0.0113551 0.0106631

In this case, it seems as if the parameter estimates are mostly independent. The parameters are probabilistic estimates which means that they cannot be collapsed into single values. Except when we cheat, which I will do here to show you how bad an idea it is.

Do this

First we will do the forecast the correct way, which takes into account both the uncertainty in the estimated parameters and the stochastic evolution of the system in the future. The long way of doing that, which has the advantage of showing what is going on, is to update the probabilistic model with the parameter estimates. The quick but correct way of doing that is what we call fast-forwarding the model

ffModel <* fastForward gbmModel params (#ibm#close)

Unfortunately, that doesn't really tell you what is going on. It is is equivalent to3:

3 r#f: get field f in record r
ffModel1 <* update gbmModel $ prob 
                 p ~ params
                 return {r=> p#r;v=>p#v; s_0 => sigLast (#ibm#close)} 

We derive a new probabilistic model from gbmModelUp where the parameters r and v are set to draws from the parameter estimates and the initial value is set to the last value in the observed timeseries. There's an even longer way of doing that if this all seems like too much magic:

ffModel2 = prob
   p   ~ params
   s_0 = sigLast (#ibm#close)
   r   = p#r
   v   = p#v
   w   ~ wiener
   d s t = s t*r+v*s t*d w t
   return s

Here is the resulting forecast, plotted together with the historical data:

"2011-10-02T22:00:00.000Z", 173.29 "2011-10-03T22:00:00.000Z", 174.74 "2011-10-04T22:00:00.000Z", 176.85 "2011-10-05T22:00:00.000Z", 181.69 "2011-10-06T22:00:00.000Z", 182.39 "2011-10-09T22:00:00.000Z", 186.62 "2011-10-10T22:00:00.000Z", 185 "2011-10-11T22:00:00.000Z", 186.12 "2011-10-12T22:00:00.000Z", 186.82 "2011-10-13T22:00:00.000Z", 190.53 "2011-10-16T22:00:00.000Z", 186.59 "2011-10-17T22:00:00.000Z", 178.9 "2011-10-18T22:00:00.000Z", 177.39 "2011-10-19T22:00:00.000Z", 177.25 "2011-10-20T22:00:00.000Z", 181.63 "2011-10-23T22:00:00.000Z", 182.25 "2011-10-24T22:00:00.000Z", 180.36 "2011-10-25T22:00:00.000Z", 181.97 "2011-10-26T22:00:00.000Z", 186.34 "2011-10-27T22:00:00.000Z", 187.45 "2011-10-30T22:00:00.000Z", 184.63 "2011-10-31T22:00:00.000Z", 181.35 "2011-11-01T22:00:00.000Z", 183.92 "2011-11-02T22:00:00.000Z", 187.3 "2011-11-03T22:00:00.000Z", 186.38 "2011-11-06T22:00:00.000Z", 187.32 "2011-11-07T22:00:00.000Z", 187.25 "2011-11-08T22:00:00.000Z", 182.24 "2011-11-09T22:00:00.000Z", 183.35 "2011-11-10T22:00:00.000Z", 187.38 "2011-11-13T22:00:00.000Z", 187.35 "2011-11-14T22:00:00.000Z", 188.75 "2011-11-15T22:00:00.000Z", 186.62 "2011-11-16T22:00:00.000Z", 185.73 "2011-11-17T22:00:00.000Z", 185.24 "2011-11-20T22:00:00.000Z", 181.48 "2011-11-21T22:00:00.000Z", 181.31 "2011-11-22T22:00:00.000Z", 177.95 "2011-11-24T22:00:00.000Z", 177.06 "2011-11-27T22:00:00.000Z", 182.21 "2011-11-28T22:00:00.000Z", 180.94 "2011-11-29T22:00:00.000Z", 188 "2011-11-30T22:00:00.000Z", 189.45 "2011-12-01T22:00:00.000Z", 189.66 "2011-12-04T22:00:00.000Z", 190.84 "2011-12-05T22:00:00.000Z", 192.94 "2011-12-06T22:00:00.000Z", 194.05 "2011-12-07T22:00:00.000Z", 191.58 "2011-12-08T22:00:00.000Z", 194.56 "2011-12-11T22:00:00.000Z", 192.18 "2011-12-12T22:00:00.000Z", 191.15 "2011-12-13T22:00:00.000Z", 188.72 "2011-12-14T22:00:00.000Z", 187.48 "2011-12-15T22:00:00.000Z", 183.57 "2011-12-18T22:00:00.000Z", 182.89 "2011-12-19T22:00:00.000Z", 187.24 "2011-12-20T22:00:00.000Z", 181.47 "2011-12-21T22:00:00.000Z", 182.04 "2011-12-22T22:00:00.000Z", 184.75 "2011-12-26T22:00:00.000Z", 184.95 "2011-12-27T22:00:00.000Z", 183.99 "2011-12-28T22:00:00.000Z", 186.18 "2011-12-29T22:00:00.000Z", 183.88 "2012-01-02T22:00:00.000Z", 186.3 "2012-01-03T22:00:00.000Z", 185.54 "2012-01-04T22:00:00.000Z", 184.66 "2012-01-05T22:00:00.000Z", 182.54 "2012-01-08T22:00:00.000Z", 181.59 "2012-01-09T22:00:00.000Z", 181.31 "2012-01-10T22:00:00.000Z", 182.32 "2012-01-11T22:00:00.000Z", 180.55 "2012-01-12T22:00:00.000Z", 179.16 "2012-01-16T22:00:00.000Z", 180 "2012-01-17T22:00:00.000Z", 181.07 "2012-01-18T22:00:00.000Z", 180.52 "2012-01-19T22:00:00.000Z", 188.52 "2012-01-22T22:00:00.000Z", 189.98 "2012-01-23T22:00:00.000Z", 191.93 "2012-01-24T22:00:00.000Z", 191.73 "2012-01-25T22:00:00.000Z", 190.98 "2012-01-26T22:00:00.000Z", 190.46 "2012-01-29T22:00:00.000Z", 192.5 "2012-01-30T22:00:00.000Z", 192.6 "2012-01-31T22:00:00.000Z", 192.62 "2012-02-01T22:00:00.000Z", 191.53 "2012-02-02T22:00:00.000Z", 193.64 "2012-02-05T22:00:00.000Z", 192.82 "2012-02-06T22:00:00.000Z", 193.35 "2012-02-07T22:00:00.000Z", 192.95 "2012-02-08T22:00:00.000Z", 193.13 "2012-02-09T22:00:00.000Z", 192.42 "2012-02-12T22:00:00.000Z", 192.62 "2012-02-13T22:00:00.000Z", 192.22 "2012-02-14T22:00:00.000Z", 192.25 "2012-02-15T22:00:00.000Z", 193.02 "2012-02-16T22:00:00.000Z", 193.42 "2012-02-20T22:00:00.000Z", 193.39 "2012-02-21T22:00:00.000Z", 193.87 "2012-02-22T22:00:00.000Z", 197.61 "2012-02-23T22:00:00.000Z", 197.76 "2012-02-26T22:00:00.000Z", 197.53 "2012-02-27T22:00:00.000Z", 197.98 "2012-02-28T22:00:00.000Z", 196.73 "2012-02-29T22:00:00.000Z", 197.53 "2012-03-01T22:00:00.000Z", 198.81 "2012-03-04T22:00:00.000Z", 200.66 "2012-03-05T22:00:00.000Z", 197.26 "2012-03-06T22:00:00.000Z", 197.77 "2012-03-07T22:00:00.000Z", 199.81 "2012-03-08T22:00:00.000Z", 200.62 "2012-03-11T22:00:00.000Z", 201 "2012-03-12T22:00:00.000Z", 203.78 "2012-03-13T22:00:00.000Z", 204.72 "2012-03-14T22:00:00.000Z", 206 "2012-03-15T22:00:00.000Z", 206.01 "2012-03-18T22:00:00.000Z", 205.72 "2012-03-19T22:00:00.000Z", 204.25 "2012-03-20T22:00:00.000Z", 204.69 "2012-03-21T22:00:00.000Z", 205.49 "2012-03-22T22:00:00.000Z", 205.48 "2012-03-25T22:00:00.000Z", 207.77 "2012-03-26T22:00:00.000Z", 207.18 "2012-03-27T22:00:00.000Z", 207.29 "2012-03-28T22:00:00.000Z", 208.27 "2012-03-29T22:00:00.000Z", 208.65 "2012-04-01T22:00:00.000Z", 209.47 "2012-04-02T22:00:00.000Z", 209.5 "2012-04-03T22:00:00.000Z", 206.05 "2012-04-04T22:00:00.000Z", 205.47 "2012-04-08T22:00:00.000Z", 204.94 "2012-04-09T22:00:00.000Z", 202.33 "2012-04-10T22:00:00.000Z", 202.58 "2012-04-11T22:00:00.000Z", 205.32 "2012-04-12T22:00:00.000Z", 202.8 "2012-04-15T22:00:00.000Z", 202.72 "2012-04-16T22:00:00.000Z", 207.45 "2012-04-17T22:00:00.000Z", 200.13 "2012-04-18T22:00:00.000Z", 199.51 "2012-04-19T22:00:00.000Z", 199.6 "2012-04-22T22:00:00.000Z", 198.62 "2012-04-23T22:00:00.000Z", 200 "2012-04-24T22:00:00.000Z", 203.57 "2012-04-25T22:00:00.000Z", 205.58 "2012-04-26T22:00:00.000Z", 206.81 "2012-04-29T22:00:00.000Z", 207.08 "2012-04-30T22:00:00.000Z", 208 "2012-05-01T22:00:00.000Z", 208.06 "2012-05-02T22:00:00.000Z", 207.24 "2012-05-03T22:00:00.000Z", 204.99 "2012-05-06T22:00:00.000Z", 203.75 "2012-05-07T22:00:00.000Z", 201.48 "2012-05-08T22:00:00.000Z", 201.23 "2012-05-09T22:00:00.000Z", 200.6 "2012-05-10T22:00:00.000Z", 201.17 "2012-05-13T22:00:00.000Z", 199.44 "2012-05-14T22:00:00.000Z", 199.04 "2012-05-15T22:00:00.000Z", 199.73 "2012-05-16T22:00:00.000Z", 197.89 "2012-05-17T22:00:00.000Z", 195.88 "2012-05-20T22:00:00.000Z", 197.76 "2012-05-21T22:00:00.000Z", 196.82 "2012-05-22T22:00:00.000Z", 196.12 "2012-05-23T22:00:00.000Z", 196.09 "2012-05-24T22:00:00.000Z", 194.3 "2012-05-28T22:00:00.000Z", 196.46 "2012-05-29T22:00:00.000Z", 194.53 "2012-05-30T22:00:00.000Z", 192.9 "2012-05-31T22:00:00.000Z", 189.08 "2012-06-03T22:00:00.000Z", 188.54 "2012-06-04T22:00:00.000Z", 189.2 "2012-06-05T22:00:00.000Z", 193.99 "2012-06-06T22:00:00.000Z", 194.44 "2012-06-07T22:00:00.000Z", 195.14 "2012-06-10T22:00:00.000Z", 192.51 "2012-06-11T22:00:00.000Z", 194.55 "2012-06-12T22:00:00.000Z", 193.1 "2012-06-13T22:00:00.000Z", 195.1 "2012-06-14T22:00:00.000Z", 199.1 "2012-06-17T22:00:00.000Z", 198.29 "2012-06-18T22:00:00.000Z", 198.93 "2012-06-19T22:00:00.000Z", 198.78 "2012-06-20T22:00:00.000Z", 193.39 "2012-06-21T22:00:00.000Z", 193.7 "2012-06-24T22:00:00.000Z", 192.86 "2012-06-25T22:00:00.000Z", 191.95 "2012-06-26T22:00:00.000Z", 193 "2012-06-27T22:00:00.000Z", 191.4 "2012-06-28T22:00:00.000Z", 195.58 "2012-07-01T22:00:00.000Z", 195.83 "2012-07-02T22:00:00.000Z", 195.93 "2012-07-04T22:00:00.000Z", 195.29 "2012-07-05T22:00:00.000Z", 191.41 "2012-07-08T22:00:00.000Z", 189.67 "2012-07-09T22:00:00.000Z", 186.26 "2012-07-10T22:00:00.000Z", 185.25 "2012-07-11T22:00:00.000Z", 183.09 "2012-07-12T22:00:00.000Z", 186.01 "2012-07-15T22:00:00.000Z", 184.79 "2012-07-16T22:00:00.000Z", 183.65 "2012-07-17T22:00:00.000Z", 188.25 "2012-07-18T22:00:00.000Z", 195.34 "2012-07-19T22:00:00.000Z", 192.45 "2012-07-22T22:00:00.000Z", 190.83 "2012-07-23T22:00:00.000Z", 190.34 "2012-07-24T22:00:00.000Z", 191.08 "2012-07-25T22:00:00.000Z", 193.95 "2012-07-26T22:00:00.000Z", 196.39 "2012-07-29T22:00:00.000Z", 196.68 "2012-07-30T22:00:00.000Z", 195.98 "2012-07-31T22:00:00.000Z", 195.18 "2012-08-01T22:00:00.000Z", 194.45 "2012-08-02T22:00:00.000Z", 198.52 "2012-08-05T22:00:00.000Z", 198.76 "2012-08-06T22:00:00.000Z", 199.93 "2012-08-07T22:00:00.000Z", 199.03 "2012-08-08T22:00:00.000Z", 198.42 "2012-08-09T22:00:00.000Z", 199.29 "2012-08-12T22:00:00.000Z", 199.01 "2012-08-13T22:00:00.000Z", 198.29 "2012-08-14T22:00:00.000Z", 198.4 "2012-08-15T22:00:00.000Z", 200.84 "2012-08-16T22:00:00.000Z", 201.22 "2012-08-19T22:00:00.000Z", 200.5 "2012-08-20T22:00:00.000Z", 198.65 "2012-08-21T22:00:00.000Z", 197.25 "2012-08-22T22:00:00.000Z", 195.7 "2012-08-23T22:00:00.000Z", 197.77 "2012-08-26T22:00:00.000Z", 195.69 "2012-08-27T22:00:00.000Z", 194.87 "2012-08-28T22:00:00.000Z", 195.08 "2012-08-29T22:00:00.000Z", 193.37 "2012-08-30T22:00:00.000Z", 194.85 "2012-09-03T22:00:00.000Z", 194.54 "2012-09-04T22:00:00.000Z", 195.04 "2012-09-05T22:00:00.000Z", 199.1 "2012-09-06T22:00:00.000Z", 199.5 "2012-09-09T22:00:00.000Z", 200.95 "2012-09-10T22:00:00.000Z", 203.27 "2012-09-11T22:00:00.000Z", 203.77 "2012-09-12T22:00:00.000Z", 206.36 "2012-09-13T22:00:00.000Z", 206.81 "2012-09-16T22:00:00.000Z", 207.15 "2012-09-17T22:00:00.000Z", 207.07 "2012-09-18T22:00:00.000Z", 206.43 "2012-09-19T22:00:00.000Z", 206.18 "2012-09-20T22:00:00.000Z", 205.98 "2012-09-23T22:00:00.000Z", 205.29 "2012-09-24T22:00:00.000Z", 204.98 "2012-09-25T22:00:00.000Z", 204 "2012-09-26T22:00:00.000Z", 205.91 "2012-09-27T22:00:00.000Z", 207.45 "2012-09-30T22:00:00.000Z", 210.47 "2012-10-01T22:00:00.000Z", 209.84 "2012-10-02T22:00:00.000Z", 210.51 "2012-10-03T22:00:00.000Z", 210.39 "2012-10-04T22:00:00.000Z", 210.59 "2012-10-07T22:00:00.000Z", 209.82 "2012-10-08T22:00:00.000Z", 207.99 "2012-10-09T22:00:00.000Z", 205.82 "2012-10-10T22:00:00.000Z", 205.76 "2012-10-11T22:00:00.000Z", 207.8 "2012-10-14T22:00:00.000Z", 208.93 "2012-10-15T22:00:00.000Z", 211 "2012-10-16T22:00:00.000Z", 200.63 "2012-10-17T22:00:00.000Z", 194.96 "2012-10-18T22:00:00.000Z", 193.36 "2012-10-21T22:00:00.000Z", 194.4 "2012-10-22T22:00:00.000Z", 191.25 "2012-10-23T22:00:00.000Z", 190.72 "2012-10-24T22:00:00.000Z", 191.6 "2012-10-25T22:00:00.000Z", 193.27 "2012-10-30T22:00:00.000Z", 194.53 "2012-10-31T22:00:00.000Z", 197.15 "2012-11-01T22:00:00.000Z", 193.43 "2012-11-04T22:00:00.000Z", 194.14 "2012-11-05T22:00:00.000Z", 195.07 "2012-11-06T22:00:00.000Z", 191.16 "2012-11-07T22:00:00.000Z", 190.1 "2012-11-08T22:00:00.000Z", 189.64 "2012-11-11T22:00:00.000Z", 189.25 "2012-11-12T22:00:00.000Z", 188.32 "2012-11-13T22:00:00.000Z", 185.51 "2012-11-14T22:00:00.000Z", 185.85 "2012-11-15T22:00:00.000Z", 186.94 "2012-11-18T22:00:00.000Z", 190.35 "2012-11-19T22:00:00.000Z", 189.2 "2012-11-20T22:00:00.000Z", 190.29 "2012-11-22T22:00:00.000Z", 193.49 "2012-11-25T22:00:00.000Z", 192.88 "2012-11-26T22:00:00.000Z", 191.23 "2012-11-27T22:00:00.000Z", 191.98 "2012-11-28T22:00:00.000Z", 191.53 "2012-11-29T22:00:00.000Z", 190.07 "2012-12-02T22:00:00.000Z", 189.48 "2012-12-03T22:00:00.000Z", 189.36 "2012-12-04T22:00:00.000Z", 188.65 "2012-12-05T22:00:00.000Z", 189.7 "2012-12-06T22:00:00.000Z", 191.95 "2012-12-09T22:00:00.000Z", 192.62 "2012-12-10T22:00:00.000Z", 194.2 "2012-12-11T22:00:00.000Z", 192.95 "2012-12-12T22:00:00.000Z", 191.99 "2012-12-13T22:00:00.000Z", 191.76 "2012-12-16T22:00:00.000Z", 193.62 "2012-12-17T22:00:00.000Z", 195.69 "2012-12-18T22:00:00.000Z", 195.08 "2012-12-19T22:00:00.000Z", 194.77 "2012-12-20T22:00:00.000Z", 193.42 "2012-12-23T22:00:00.000Z", 192.4 "2012-12-25T22:00:00.000Z", 191.95 "2012-12-26T22:00:00.000Z", 192.71 "2012-12-27T22:00:00.000Z", 189.83 "2012-12-30T22:00:00.000Z", 191.55 "2013-01-01T22:00:00.000Z", 196.35 "2013-01-02T22:00:00.000Z", 195.27 "2013-01-03T22:00:00.000Z", 193.99 "2013-01-06T22:00:00.000Z", 193.14 "2013-01-07T22:00:00.000Z", 192.87 "2013-01-08T22:00:00.000Z", 192.32 "2013-01-09T22:00:00.000Z", 192.88 "2013-01-10T22:00:00.000Z", 194.45 "2013-01-13T22:00:00.000Z", 192.62 "2013-01-14T22:00:00.000Z", 192.5 "2013-01-15T22:00:00.000Z", 192.59 "2013-01-16T22:00:00.000Z", 193.65 "2013-01-17T22:00:00.000Z", 194.47 "2013-01-21T22:00:00.000Z", 196.08 "2013-01-22T22:00:00.000Z", 204.72 "2013-01-23T22:00:00.000Z", 204.42 "2013-01-24T22:00:00.000Z", 204.97 "2013-01-27T22:00:00.000Z", 204.93 "2013-01-28T22:00:00.000Z", 203.9 "2013-01-29T22:00:00.000Z", 203.52 "2013-01-30T22:00:00.000Z", 203.07 "2013-01-31T22:00:00.000Z", 205.18 "2013-02-03T22:00:00.000Z", 203.79 "2013-02-04T22:00:00.000Z", 202.79 "2013-02-05T22:00:00.000Z", 201.02 "2013-02-06T22:00:00.000Z", 199.74 "2013-02-07T22:00:00.000Z", 201.68 "2013-02-10T22:00:00.000Z", 200.16 "2013-02-11T22:00:00.000Z", 200.04 "2013-02-12T22:00:00.000Z", 200.09 "2013-02-13T22:00:00.000Z", 199.65 "2013-02-14T22:00:00.000Z", 200.98 "2013-02-18T22:00:00.000Z", 200.32 "2013-02-19T22:00:00.000Z", 199.31 "2013-02-20T22:00:00.000Z", 198.33 "2013-02-21T22:00:00.000Z", 201.09 "2013-02-24T22:00:00.000Z", 197.51 "2013-02-25T22:00:00.000Z", 199.14 "2013-02-26T22:00:00.000Z", 202.33 "2013-02-27T22:00:00.000Z", 200.83 "2013-02-28T22:00:00.000Z", 202.91 "2013-03-03T22:00:00.000Z", 205.19 "2013-03-04T22:00:00.000Z", 206.53 "2013-03-05T22:00:00.000Z", 208.38 "2013-03-06T22:00:00.000Z", 209.42 "2013-03-07T22:00:00.000Z", 210.38 "2013-03-10T22:00:00.000Z", 210.08 "2013-03-11T22:00:00.000Z", 210.55 "2013-03-12T22:00:00.000Z", 212.06 "2013-03-13T22:00:00.000Z", 215.8 "2013-03-14T22:00:00.000Z", 214.92 "2013-03-17T22:00:00.000Z", 213.21 "2013-03-18T22:00:00.000Z", 213.44 "2013-03-19T22:00:00.000Z", 215.06 "2013-03-20T22:00:00.000Z", 212.26 "2013-03-21T22:00:00.000Z", 212.08 "2013-03-24T22:00:00.000Z", 210.74 "2013-03-25T22:00:00.000Z", 212.36 "2013-03-26T22:00:00.000Z", 210.89 "2013-03-27T22:00:00.000Z", 213.3 "2013-03-31T22:00:00.000Z", 212.38 "2013-04-01T22:00:00.000Z", 214.36 "2013-04-02T22:00:00.000Z", 212.66 "2013-04-03T22:00:00.000Z", 211.31 "2013-04-04T22:00:00.000Z", 209.41 "2013-04-07T22:00:00.000Z", 209.32 "2013-04-08T22:00:00.000Z", 209.22 "2013-04-09T22:00:00.000Z", 212 "2013-04-10T22:00:00.000Z", 212.92 "2013-04-11T22:00:00.000Z", 211.38 "2013-04-14T22:00:00.000Z", 209.26 "2013-04-15T22:00:00.000Z", 212 "2013-04-16T22:00:00.000Z", 209.67 "2013-04-17T22:00:00.000Z", 207.15 "2013-04-18T22:00:00.000Z", 190 "2013-04-21T22:00:00.000Z", 187.83 "2013-04-22T22:00:00.000Z", 191.61 "2013-04-23T22:00:00.000Z", 191.71 "2013-04-24T22:00:00.000Z", 193.95 "2013-04-25T22:00:00.000Z", 194.31 "2013-04-28T22:00:00.000Z", 199.15 "2013-04-29T22:00:00.000Z", 202.54 "2013-04-30T22:00:00.000Z", 199.63 "2013-05-01T22:00:00.000Z", 202.39 "2013-05-02T22:00:00.000Z", 204.51 "2013-05-05T22:00:00.000Z", 202.78 "2013-05-06T22:00:00.000Z", 203.63 "2013-05-07T22:00:00.000Z", 204.82 "2013-05-08T22:00:00.000Z", 203.24 "2013-05-09T22:00:00.000Z", 204.47 "2013-05-12T22:00:00.000Z", 202.47 "2013-05-13T22:00:00.000Z", 203.21 "2013-05-14T22:00:00.000Z", 203.32 "2013-05-15T22:00:00.000Z", 204.69 "2013-05-16T22:00:00.000Z", 208.44 "2013-05-19T22:00:00.000Z", 207.6 "2013-05-20T22:00:00.000Z", 208.65 "2013-05-21T22:00:00.000Z", 206.99 "2013-05-22T22:00:00.000Z", 206.16 "2013-05-23T22:00:00.000Z", 205.72 "2013-05-27T22:00:00.000Z", 207.78 "2013-05-28T22:00:00.000Z", 207.92 "2013-05-29T22:00:00.000Z", 209.36 "2013-05-30T22:00:00.000Z", 208.02 "2013-06-02T22:00:00.000Z", 208.95 "2013-06-03T22:00:00.000Z", 206.19 "2013-06-04T22:00:00.000Z", 202.74 "2013-06-05T22:00:00.000Z", 203.8 "2013-06-06T22:00:00.000Z", 206.35 "2013-06-09T22:00:00.000Z", 205.02 "2013-06-10T22:00:00.000Z", 203.98 "2013-06-11T22:00:00.000Z", 201.2 "2013-06-12T22:00:00.000Z", 203.77 "2013-06-13T22:00:00.000Z", 202.2 "2013-06-16T22:00:00.000Z", 203.04 "2013-06-17T22:00:00.000Z", 204.87 "2013-06-18T22:00:00.000Z", 201.94 "2013-06-19T22:00:00.000Z", 197.35 "2013-06-20T22:00:00.000Z", 195.46 "2013-06-23T22:00:00.000Z", 193.54 "2013-06-24T22:00:00.000Z", 194.98 "2013-06-25T22:00:00.000Z", 194.86 "2013-06-26T22:00:00.000Z", 195.65 "2013-06-27T22:00:00.000Z", 191.11 "2013-06-30T22:00:00.000Z", 191.28 "2013-07-01T22:00:00.000Z", 191.5 "2013-07-02T22:00:00.000Z", 193.25 "2013-07-04T22:00:00.000Z", 194.93 "2013-07-07T22:00:00.000Z", 194.98 "2013-07-08T22:00:00.000Z", 191.3 "2013-07-09T22:00:00.000Z", 192.25 "2013-07-10T22:00:00.000Z", 192.8 "2013-07-11T22:00:00.000Z", 192.07 "2013-07-14T22:00:00.000Z", 194 "2013-07-15T22:00:00.000Z", 193.85 "2013-07-16T22:00:00.000Z", 194.55 "2013-07-17T22:00:00.000Z", 197.99 "2013-07-18T22:00:00.000Z", 193.54 "2013-07-21T22:00:00.000Z", 194.09 "2013-07-22T22:00:00.000Z", 194.98 "2013-07-23T22:00:00.000Z", 196.61 "2013-07-24T22:00:00.000Z", 197.22 "2013-07-25T22:00:00.000Z", 197.35 "2013-07-28T22:00:00.000Z", 196.21 "2013-07-29T22:00:00.000Z", 196.01 "2013-07-30T22:00:00.000Z", 195.04 "2013-07-31T22:00:00.000Z", 195.81 "2013-08-01T22:00:00.000Z", 195.16 "2013-08-04T22:00:00.000Z", 195.5 "2013-08-05T22:00:00.000Z", 190.99 "2013-08-06T22:00:00.000Z", 188.56 "2013-08-07T22:00:00.000Z", 187.93 "2013-08-08T22:00:00.000Z", 187.82 "2013-08-11T22:00:00.000Z", 189.09 "2013-08-12T22:00:00.000Z", 188.42 "2013-08-13T22:00:00.000Z", 187.53 "2013-08-14T22:00:00.000Z", 185.79 "2013-08-15T22:00:00.000Z", 185.34 "2013-08-18T22:00:00.000Z", 184.23 "2013-08-19T22:00:00.000Z", 184.56 "2013-08-20T22:00:00.000Z", 184.86 "2013-08-21T22:00:00.000Z", 185.19 "2013-08-22T22:00:00.000Z", 185.42 "2013-08-25T22:00:00.000Z", 184.74 "2013-08-26T22:00:00.000Z", 182.74 "2013-08-27T22:00:00.000Z", 182.16 "2013-08-28T22:00:00.000Z", 182.64 "2013-08-29T22:00:00.000Z", 182.27 "2013-09-02T22:00:00.000Z", 183.96 "2013-09-03T22:00:00.000Z", 183.13 "2013-09-04T22:00:00.000Z", 184.15 "2013-09-05T22:00:00.000Z", 183.03 "2013-09-08T22:00:00.000Z", 184.98 "2013-09-09T22:00:00.000Z", 186.6 "2013-09-10T22:00:00.000Z", 190.7 "2013-09-11T22:00:00.000Z", 190.73 "2013-09-12T22:00:00.000Z", 192.17 "2013-09-15T22:00:00.000Z", 193.15 "2013-09-16T22:00:00.000Z", 192.16 "2013-09-17T22:00:00.000Z", 194.42 "2013-09-18T22:00:00.000Z", 193.39 "2013-09-19T22:00:00.000Z", 190.02 "2013-09-22T22:00:00.000Z", 190.99 "2013-09-23T22:00:00.000Z", 189.97 "2013-09-24T22:00:00.000Z", 189.47 "2013-09-25T22:00:00.000Z", 190.22 "2013-09-26T22:00:00.000Z", 186.92 "2013-09-29T22:00:00.000Z", 185.18 "2013-09-30T22:00:00.000Z", 186.38 "2013-10-01T22:00:00.000Z", 184.96 "2013-10-02T22:00:00.000Z", 183.86 "2013-10-03T22:00:00.000Z", 184.1 "2013-10-06T22:00:00.000Z", 182.01 "2013-10-07T22:00:00.000Z", 178.72 "2013-10-08T22:00:00.000Z", 181.32 "2013-10-09T22:00:00.000Z", 184.77 "2013-10-10T22:00:00.000Z", 186.16 "2013-10-13T22:00:00.000Z", 186.97 "2013-10-14T22:00:00.000Z", 184.66 "2013-10-15T22:00:00.000Z", 186.73 "2013-10-16T22:00:00.000Z", 174.83 "2013-10-17T22:00:00.000Z", 173.78 "2013-10-20T22:00:00.000Z", 172.86 "2013-10-21T22:00:00.000Z", 174.97 "2013-10-22T22:00:00.000Z", 175.77 "2013-10-23T22:00:00.000Z", 177.8 "2013-10-24T22:00:00.000Z", 176.85 "2013-10-27T22:00:00.000Z", 177.35 "2013-10-28T22:00:00.000Z", 182.12 "2013-10-29T22:00:00.000Z", 180.15 "2013-10-30T22:00:00.000Z", 179.21 "2013-10-31T22:00:00.000Z", 179.23 "2013-11-03T22:00:00.000Z", 180.27 "2013-11-04T22:00:00.000Z", 177.85 "2013-11-05T22:00:00.000Z", 179.19 "2013-11-06T22:00:00.000Z", 180
axisLabels "Time" "IBM price" $
  over [sigPlot (#ibm#close),
        psigPlot ffModel]

Don't do this

The incorrect way4 of forecasting is to ignore the uncertainty in the parameter estimates. I will use the expect operator to collapse the parameter estimates. This operator is usually only defensibly used in fully probabilistic programming when making decisions.

4One of the incorrect ways. There are many ways of screwing this up
paramVal = { r => expect (params <#> r); 
             v => expect (params <#> v);
             s_0 => sigLast (#ibm#close)}
paramVal ⇒ {r => 0.0001;v => 0.0112;s_0 => 180.0000}

Then we will use fastForward again to forecast without parameter uncertainty. We have to use return to turn paramVal into a probability distribution over the parameters, which is what fastForward expects.

ffModelIncorrect <* fastForward gbmModel (return paramVal) (#ibm#close)
"2011-10-02T22:00:00.000Z", 173.29 "2011-10-03T22:00:00.000Z", 174.74 "2011-10-04T22:00:00.000Z", 176.85 "2011-10-05T22:00:00.000Z", 181.69 "2011-10-06T22:00:00.000Z", 182.39 "2011-10-09T22:00:00.000Z", 186.62 "2011-10-10T22:00:00.000Z", 185 "2011-10-11T22:00:00.000Z", 186.12 "2011-10-12T22:00:00.000Z", 186.82 "2011-10-13T22:00:00.000Z", 190.53 "2011-10-16T22:00:00.000Z", 186.59 "2011-10-17T22:00:00.000Z", 178.9 "2011-10-18T22:00:00.000Z", 177.39 "2011-10-19T22:00:00.000Z", 177.25 "2011-10-20T22:00:00.000Z", 181.63 "2011-10-23T22:00:00.000Z", 182.25 "2011-10-24T22:00:00.000Z", 180.36 "2011-10-25T22:00:00.000Z", 181.97 "2011-10-26T22:00:00.000Z", 186.34 "2011-10-27T22:00:00.000Z", 187.45 "2011-10-30T22:00:00.000Z", 184.63 "2011-10-31T22:00:00.000Z", 181.35 "2011-11-01T22:00:00.000Z", 183.92 "2011-11-02T22:00:00.000Z", 187.3 "2011-11-03T22:00:00.000Z", 186.38 "2011-11-06T22:00:00.000Z", 187.32 "2011-11-07T22:00:00.000Z", 187.25 "2011-11-08T22:00:00.000Z", 182.24 "2011-11-09T22:00:00.000Z", 183.35 "2011-11-10T22:00:00.000Z", 187.38 "2011-11-13T22:00:00.000Z", 187.35 "2011-11-14T22:00:00.000Z", 188.75 "2011-11-15T22:00:00.000Z", 186.62 "2011-11-16T22:00:00.000Z", 185.73 "2011-11-17T22:00:00.000Z", 185.24 "2011-11-20T22:00:00.000Z", 181.48 "2011-11-21T22:00:00.000Z", 181.31 "2011-11-22T22:00:00.000Z", 177.95 "2011-11-24T22:00:00.000Z", 177.06 "2011-11-27T22:00:00.000Z", 182.21 "2011-11-28T22:00:00.000Z", 180.94 "2011-11-29T22:00:00.000Z", 188 "2011-11-30T22:00:00.000Z", 189.45 "2011-12-01T22:00:00.000Z", 189.66 "2011-12-04T22:00:00.000Z", 190.84 "2011-12-05T22:00:00.000Z", 192.94 "2011-12-06T22:00:00.000Z", 194.05 "2011-12-07T22:00:00.000Z", 191.58 "2011-12-08T22:00:00.000Z", 194.56 "2011-12-11T22:00:00.000Z", 192.18 "2011-12-12T22:00:00.000Z", 191.15 "2011-12-13T22:00:00.000Z", 188.72 "2011-12-14T22:00:00.000Z", 187.48 "2011-12-15T22:00:00.000Z", 183.57 "2011-12-18T22:00:00.000Z", 182.89 "2011-12-19T22:00:00.000Z", 187.24 "2011-12-20T22:00:00.000Z", 181.47 "2011-12-21T22:00:00.000Z", 182.04 "2011-12-22T22:00:00.000Z", 184.75 "2011-12-26T22:00:00.000Z", 184.95 "2011-12-27T22:00:00.000Z", 183.99 "2011-12-28T22:00:00.000Z", 186.18 "2011-12-29T22:00:00.000Z", 183.88 "2012-01-02T22:00:00.000Z", 186.3 "2012-01-03T22:00:00.000Z", 185.54 "2012-01-04T22:00:00.000Z", 184.66 "2012-01-05T22:00:00.000Z", 182.54 "2012-01-08T22:00:00.000Z", 181.59 "2012-01-09T22:00:00.000Z", 181.31 "2012-01-10T22:00:00.000Z", 182.32 "2012-01-11T22:00:00.000Z", 180.55 "2012-01-12T22:00:00.000Z", 179.16 "2012-01-16T22:00:00.000Z", 180 "2012-01-17T22:00:00.000Z", 181.07 "2012-01-18T22:00:00.000Z", 180.52 "2012-01-19T22:00:00.000Z", 188.52 "2012-01-22T22:00:00.000Z", 189.98 "2012-01-23T22:00:00.000Z", 191.93 "2012-01-24T22:00:00.000Z", 191.73 "2012-01-25T22:00:00.000Z", 190.98 "2012-01-26T22:00:00.000Z", 190.46 "2012-01-29T22:00:00.000Z", 192.5 "2012-01-30T22:00:00.000Z", 192.6 "2012-01-31T22:00:00.000Z", 192.62 "2012-02-01T22:00:00.000Z", 191.53 "2012-02-02T22:00:00.000Z", 193.64 "2012-02-05T22:00:00.000Z", 192.82 "2012-02-06T22:00:00.000Z", 193.35 "2012-02-07T22:00:00.000Z", 192.95 "2012-02-08T22:00:00.000Z", 193.13 "2012-02-09T22:00:00.000Z", 192.42 "2012-02-12T22:00:00.000Z", 192.62 "2012-02-13T22:00:00.000Z", 192.22 "2012-02-14T22:00:00.000Z", 192.25 "2012-02-15T22:00:00.000Z", 193.02 "2012-02-16T22:00:00.000Z", 193.42 "2012-02-20T22:00:00.000Z", 193.39 "2012-02-21T22:00:00.000Z", 193.87 "2012-02-22T22:00:00.000Z", 197.61 "2012-02-23T22:00:00.000Z", 197.76 "2012-02-26T22:00:00.000Z", 197.53 "2012-02-27T22:00:00.000Z", 197.98 "2012-02-28T22:00:00.000Z", 196.73 "2012-02-29T22:00:00.000Z", 197.53 "2012-03-01T22:00:00.000Z", 198.81 "2012-03-04T22:00:00.000Z", 200.66 "2012-03-05T22:00:00.000Z", 197.26 "2012-03-06T22:00:00.000Z", 197.77 "2012-03-07T22:00:00.000Z", 199.81 "2012-03-08T22:00:00.000Z", 200.62 "2012-03-11T22:00:00.000Z", 201 "2012-03-12T22:00:00.000Z", 203.78 "2012-03-13T22:00:00.000Z", 204.72 "2012-03-14T22:00:00.000Z", 206 "2012-03-15T22:00:00.000Z", 206.01 "2012-03-18T22:00:00.000Z", 205.72 "2012-03-19T22:00:00.000Z", 204.25 "2012-03-20T22:00:00.000Z", 204.69 "2012-03-21T22:00:00.000Z", 205.49 "2012-03-22T22:00:00.000Z", 205.48 "2012-03-25T22:00:00.000Z", 207.77 "2012-03-26T22:00:00.000Z", 207.18 "2012-03-27T22:00:00.000Z", 207.29 "2012-03-28T22:00:00.000Z", 208.27 "2012-03-29T22:00:00.000Z", 208.65 "2012-04-01T22:00:00.000Z", 209.47 "2012-04-02T22:00:00.000Z", 209.5 "2012-04-03T22:00:00.000Z", 206.05 "2012-04-04T22:00:00.000Z", 205.47 "2012-04-08T22:00:00.000Z", 204.94 "2012-04-09T22:00:00.000Z", 202.33 "2012-04-10T22:00:00.000Z", 202.58 "2012-04-11T22:00:00.000Z", 205.32 "2012-04-12T22:00:00.000Z", 202.8 "2012-04-15T22:00:00.000Z", 202.72 "2012-04-16T22:00:00.000Z", 207.45 "2012-04-17T22:00:00.000Z", 200.13 "2012-04-18T22:00:00.000Z", 199.51 "2012-04-19T22:00:00.000Z", 199.6 "2012-04-22T22:00:00.000Z", 198.62 "2012-04-23T22:00:00.000Z", 200 "2012-04-24T22:00:00.000Z", 203.57 "2012-04-25T22:00:00.000Z", 205.58 "2012-04-26T22:00:00.000Z", 206.81 "2012-04-29T22:00:00.000Z", 207.08 "2012-04-30T22:00:00.000Z", 208 "2012-05-01T22:00:00.000Z", 208.06 "2012-05-02T22:00:00.000Z", 207.24 "2012-05-03T22:00:00.000Z", 204.99 "2012-05-06T22:00:00.000Z", 203.75 "2012-05-07T22:00:00.000Z", 201.48 "2012-05-08T22:00:00.000Z", 201.23 "2012-05-09T22:00:00.000Z", 200.6 "2012-05-10T22:00:00.000Z", 201.17 "2012-05-13T22:00:00.000Z", 199.44 "2012-05-14T22:00:00.000Z", 199.04 "2012-05-15T22:00:00.000Z", 199.73 "2012-05-16T22:00:00.000Z", 197.89 "2012-05-17T22:00:00.000Z", 195.88 "2012-05-20T22:00:00.000Z", 197.76 "2012-05-21T22:00:00.000Z", 196.82 "2012-05-22T22:00:00.000Z", 196.12 "2012-05-23T22:00:00.000Z", 196.09 "2012-05-24T22:00:00.000Z", 194.3 "2012-05-28T22:00:00.000Z", 196.46 "2012-05-29T22:00:00.000Z", 194.53 "2012-05-30T22:00:00.000Z", 192.9 "2012-05-31T22:00:00.000Z", 189.08 "2012-06-03T22:00:00.000Z", 188.54 "2012-06-04T22:00:00.000Z", 189.2 "2012-06-05T22:00:00.000Z", 193.99 "2012-06-06T22:00:00.000Z", 194.44 "2012-06-07T22:00:00.000Z", 195.14 "2012-06-10T22:00:00.000Z", 192.51 "2012-06-11T22:00:00.000Z", 194.55 "2012-06-12T22:00:00.000Z", 193.1 "2012-06-13T22:00:00.000Z", 195.1 "2012-06-14T22:00:00.000Z", 199.1 "2012-06-17T22:00:00.000Z", 198.29 "2012-06-18T22:00:00.000Z", 198.93 "2012-06-19T22:00:00.000Z", 198.78 "2012-06-20T22:00:00.000Z", 193.39 "2012-06-21T22:00:00.000Z", 193.7 "2012-06-24T22:00:00.000Z", 192.86 "2012-06-25T22:00:00.000Z", 191.95 "2012-06-26T22:00:00.000Z", 193 "2012-06-27T22:00:00.000Z", 191.4 "2012-06-28T22:00:00.000Z", 195.58 "2012-07-01T22:00:00.000Z", 195.83 "2012-07-02T22:00:00.000Z", 195.93 "2012-07-04T22:00:00.000Z", 195.29 "2012-07-05T22:00:00.000Z", 191.41 "2012-07-08T22:00:00.000Z", 189.67 "2012-07-09T22:00:00.000Z", 186.26 "2012-07-10T22:00:00.000Z", 185.25 "2012-07-11T22:00:00.000Z", 183.09 "2012-07-12T22:00:00.000Z", 186.01 "2012-07-15T22:00:00.000Z", 184.79 "2012-07-16T22:00:00.000Z", 183.65 "2012-07-17T22:00:00.000Z", 188.25 "2012-07-18T22:00:00.000Z", 195.34 "2012-07-19T22:00:00.000Z", 192.45 "2012-07-22T22:00:00.000Z", 190.83 "2012-07-23T22:00:00.000Z", 190.34 "2012-07-24T22:00:00.000Z", 191.08 "2012-07-25T22:00:00.000Z", 193.95 "2012-07-26T22:00:00.000Z", 196.39 "2012-07-29T22:00:00.000Z", 196.68 "2012-07-30T22:00:00.000Z", 195.98 "2012-07-31T22:00:00.000Z", 195.18 "2012-08-01T22:00:00.000Z", 194.45 "2012-08-02T22:00:00.000Z", 198.52 "2012-08-05T22:00:00.000Z", 198.76 "2012-08-06T22:00:00.000Z", 199.93 "2012-08-07T22:00:00.000Z", 199.03 "2012-08-08T22:00:00.000Z", 198.42 "2012-08-09T22:00:00.000Z", 199.29 "2012-08-12T22:00:00.000Z", 199.01 "2012-08-13T22:00:00.000Z", 198.29 "2012-08-14T22:00:00.000Z", 198.4 "2012-08-15T22:00:00.000Z", 200.84 "2012-08-16T22:00:00.000Z", 201.22 "2012-08-19T22:00:00.000Z", 200.5 "2012-08-20T22:00:00.000Z", 198.65 "2012-08-21T22:00:00.000Z", 197.25 "2012-08-22T22:00:00.000Z", 195.7 "2012-08-23T22:00:00.000Z", 197.77 "2012-08-26T22:00:00.000Z", 195.69 "2012-08-27T22:00:00.000Z", 194.87 "2012-08-28T22:00:00.000Z", 195.08 "2012-08-29T22:00:00.000Z", 193.37 "2012-08-30T22:00:00.000Z", 194.85 "2012-09-03T22:00:00.000Z", 194.54 "2012-09-04T22:00:00.000Z", 195.04 "2012-09-05T22:00:00.000Z", 199.1 "2012-09-06T22:00:00.000Z", 199.5 "2012-09-09T22:00:00.000Z", 200.95 "2012-09-10T22:00:00.000Z", 203.27 "2012-09-11T22:00:00.000Z", 203.77 "2012-09-12T22:00:00.000Z", 206.36 "2012-09-13T22:00:00.000Z", 206.81 "2012-09-16T22:00:00.000Z", 207.15 "2012-09-17T22:00:00.000Z", 207.07 "2012-09-18T22:00:00.000Z", 206.43 "2012-09-19T22:00:00.000Z", 206.18 "2012-09-20T22:00:00.000Z", 205.98 "2012-09-23T22:00:00.000Z", 205.29 "2012-09-24T22:00:00.000Z", 204.98 "2012-09-25T22:00:00.000Z", 204 "2012-09-26T22:00:00.000Z", 205.91 "2012-09-27T22:00:00.000Z", 207.45 "2012-09-30T22:00:00.000Z", 210.47 "2012-10-01T22:00:00.000Z", 209.84 "2012-10-02T22:00:00.000Z", 210.51 "2012-10-03T22:00:00.000Z", 210.39 "2012-10-04T22:00:00.000Z", 210.59 "2012-10-07T22:00:00.000Z", 209.82 "2012-10-08T22:00:00.000Z", 207.99 "2012-10-09T22:00:00.000Z", 205.82 "2012-10-10T22:00:00.000Z", 205.76 "2012-10-11T22:00:00.000Z", 207.8 "2012-10-14T22:00:00.000Z", 208.93 "2012-10-15T22:00:00.000Z", 211 "2012-10-16T22:00:00.000Z", 200.63 "2012-10-17T22:00:00.000Z", 194.96 "2012-10-18T22:00:00.000Z", 193.36 "2012-10-21T22:00:00.000Z", 194.4 "2012-10-22T22:00:00.000Z", 191.25 "2012-10-23T22:00:00.000Z", 190.72 "2012-10-24T22:00:00.000Z", 191.6 "2012-10-25T22:00:00.000Z", 193.27 "2012-10-30T22:00:00.000Z", 194.53 "2012-10-31T22:00:00.000Z", 197.15 "2012-11-01T22:00:00.000Z", 193.43 "2012-11-04T22:00:00.000Z", 194.14 "2012-11-05T22:00:00.000Z", 195.07 "2012-11-06T22:00:00.000Z", 191.16 "2012-11-07T22:00:00.000Z", 190.1 "2012-11-08T22:00:00.000Z", 189.64 "2012-11-11T22:00:00.000Z", 189.25 "2012-11-12T22:00:00.000Z", 188.32 "2012-11-13T22:00:00.000Z", 185.51 "2012-11-14T22:00:00.000Z", 185.85 "2012-11-15T22:00:00.000Z", 186.94 "2012-11-18T22:00:00.000Z", 190.35 "2012-11-19T22:00:00.000Z", 189.2 "2012-11-20T22:00:00.000Z", 190.29 "2012-11-22T22:00:00.000Z", 193.49 "2012-11-25T22:00:00.000Z", 192.88 "2012-11-26T22:00:00.000Z", 191.23 "2012-11-27T22:00:00.000Z", 191.98 "2012-11-28T22:00:00.000Z", 191.53 "2012-11-29T22:00:00.000Z", 190.07 "2012-12-02T22:00:00.000Z", 189.48 "2012-12-03T22:00:00.000Z", 189.36 "2012-12-04T22:00:00.000Z", 188.65 "2012-12-05T22:00:00.000Z", 189.7 "2012-12-06T22:00:00.000Z", 191.95 "2012-12-09T22:00:00.000Z", 192.62 "2012-12-10T22:00:00.000Z", 194.2 "2012-12-11T22:00:00.000Z", 192.95 "2012-12-12T22:00:00.000Z", 191.99 "2012-12-13T22:00:00.000Z", 191.76 "2012-12-16T22:00:00.000Z", 193.62 "2012-12-17T22:00:00.000Z", 195.69 "2012-12-18T22:00:00.000Z", 195.08 "2012-12-19T22:00:00.000Z", 194.77 "2012-12-20T22:00:00.000Z", 193.42 "2012-12-23T22:00:00.000Z", 192.4 "2012-12-25T22:00:00.000Z", 191.95 "2012-12-26T22:00:00.000Z", 192.71 "2012-12-27T22:00:00.000Z", 189.83 "2012-12-30T22:00:00.000Z", 191.55 "2013-01-01T22:00:00.000Z", 196.35 "2013-01-02T22:00:00.000Z", 195.27 "2013-01-03T22:00:00.000Z", 193.99 "2013-01-06T22:00:00.000Z", 193.14 "2013-01-07T22:00:00.000Z", 192.87 "2013-01-08T22:00:00.000Z", 192.32 "2013-01-09T22:00:00.000Z", 192.88 "2013-01-10T22:00:00.000Z", 194.45 "2013-01-13T22:00:00.000Z", 192.62 "2013-01-14T22:00:00.000Z", 192.5 "2013-01-15T22:00:00.000Z", 192.59 "2013-01-16T22:00:00.000Z", 193.65 "2013-01-17T22:00:00.000Z", 194.47 "2013-01-21T22:00:00.000Z", 196.08 "2013-01-22T22:00:00.000Z", 204.72 "2013-01-23T22:00:00.000Z", 204.42 "2013-01-24T22:00:00.000Z", 204.97 "2013-01-27T22:00:00.000Z", 204.93 "2013-01-28T22:00:00.000Z", 203.9 "2013-01-29T22:00:00.000Z", 203.52 "2013-01-30T22:00:00.000Z", 203.07 "2013-01-31T22:00:00.000Z", 205.18 "2013-02-03T22:00:00.000Z", 203.79 "2013-02-04T22:00:00.000Z", 202.79 "2013-02-05T22:00:00.000Z", 201.02 "2013-02-06T22:00:00.000Z", 199.74 "2013-02-07T22:00:00.000Z", 201.68 "2013-02-10T22:00:00.000Z", 200.16 "2013-02-11T22:00:00.000Z", 200.04 "2013-02-12T22:00:00.000Z", 200.09 "2013-02-13T22:00:00.000Z", 199.65 "2013-02-14T22:00:00.000Z", 200.98 "2013-02-18T22:00:00.000Z", 200.32 "2013-02-19T22:00:00.000Z", 199.31 "2013-02-20T22:00:00.000Z", 198.33 "2013-02-21T22:00:00.000Z", 201.09 "2013-02-24T22:00:00.000Z", 197.51 "2013-02-25T22:00:00.000Z", 199.14 "2013-02-26T22:00:00.000Z", 202.33 "2013-02-27T22:00:00.000Z", 200.83 "2013-02-28T22:00:00.000Z", 202.91 "2013-03-03T22:00:00.000Z", 205.19 "2013-03-04T22:00:00.000Z", 206.53 "2013-03-05T22:00:00.000Z", 208.38 "2013-03-06T22:00:00.000Z", 209.42 "2013-03-07T22:00:00.000Z", 210.38 "2013-03-10T22:00:00.000Z", 210.08 "2013-03-11T22:00:00.000Z", 210.55 "2013-03-12T22:00:00.000Z", 212.06 "2013-03-13T22:00:00.000Z", 215.8 "2013-03-14T22:00:00.000Z", 214.92 "2013-03-17T22:00:00.000Z", 213.21 "2013-03-18T22:00:00.000Z", 213.44 "2013-03-19T22:00:00.000Z", 215.06 "2013-03-20T22:00:00.000Z", 212.26 "2013-03-21T22:00:00.000Z", 212.08 "2013-03-24T22:00:00.000Z", 210.74 "2013-03-25T22:00:00.000Z", 212.36 "2013-03-26T22:00:00.000Z", 210.89 "2013-03-27T22:00:00.000Z", 213.3 "2013-03-31T22:00:00.000Z", 212.38 "2013-04-01T22:00:00.000Z", 214.36 "2013-04-02T22:00:00.000Z", 212.66 "2013-04-03T22:00:00.000Z", 211.31 "2013-04-04T22:00:00.000Z", 209.41 "2013-04-07T22:00:00.000Z", 209.32 "2013-04-08T22:00:00.000Z", 209.22 "2013-04-09T22:00:00.000Z", 212 "2013-04-10T22:00:00.000Z", 212.92 "2013-04-11T22:00:00.000Z", 211.38 "2013-04-14T22:00:00.000Z", 209.26 "2013-04-15T22:00:00.000Z", 212 "2013-04-16T22:00:00.000Z", 209.67 "2013-04-17T22:00:00.000Z", 207.15 "2013-04-18T22:00:00.000Z", 190 "2013-04-21T22:00:00.000Z", 187.83 "2013-04-22T22:00:00.000Z", 191.61 "2013-04-23T22:00:00.000Z", 191.71 "2013-04-24T22:00:00.000Z", 193.95 "2013-04-25T22:00:00.000Z", 194.31 "2013-04-28T22:00:00.000Z", 199.15 "2013-04-29T22:00:00.000Z", 202.54 "2013-04-30T22:00:00.000Z", 199.63 "2013-05-01T22:00:00.000Z", 202.39 "2013-05-02T22:00:00.000Z", 204.51 "2013-05-05T22:00:00.000Z", 202.78 "2013-05-06T22:00:00.000Z", 203.63 "2013-05-07T22:00:00.000Z", 204.82 "2013-05-08T22:00:00.000Z", 203.24 "2013-05-09T22:00:00.000Z", 204.47 "2013-05-12T22:00:00.000Z", 202.47 "2013-05-13T22:00:00.000Z", 203.21 "2013-05-14T22:00:00.000Z", 203.32 "2013-05-15T22:00:00.000Z", 204.69 "2013-05-16T22:00:00.000Z", 208.44 "2013-05-19T22:00:00.000Z", 207.6 "2013-05-20T22:00:00.000Z", 208.65 "2013-05-21T22:00:00.000Z", 206.99 "2013-05-22T22:00:00.000Z", 206.16 "2013-05-23T22:00:00.000Z", 205.72 "2013-05-27T22:00:00.000Z", 207.78 "2013-05-28T22:00:00.000Z", 207.92 "2013-05-29T22:00:00.000Z", 209.36 "2013-05-30T22:00:00.000Z", 208.02 "2013-06-02T22:00:00.000Z", 208.95 "2013-06-03T22:00:00.000Z", 206.19 "2013-06-04T22:00:00.000Z", 202.74 "2013-06-05T22:00:00.000Z", 203.8 "2013-06-06T22:00:00.000Z", 206.35 "2013-06-09T22:00:00.000Z", 205.02 "2013-06-10T22:00:00.000Z", 203.98 "2013-06-11T22:00:00.000Z", 201.2 "2013-06-12T22:00:00.000Z", 203.77 "2013-06-13T22:00:00.000Z", 202.2 "2013-06-16T22:00:00.000Z", 203.04 "2013-06-17T22:00:00.000Z", 204.87 "2013-06-18T22:00:00.000Z", 201.94 "2013-06-19T22:00:00.000Z", 197.35 "2013-06-20T22:00:00.000Z", 195.46 "2013-06-23T22:00:00.000Z", 193.54 "2013-06-24T22:00:00.000Z", 194.98 "2013-06-25T22:00:00.000Z", 194.86 "2013-06-26T22:00:00.000Z", 195.65 "2013-06-27T22:00:00.000Z", 191.11 "2013-06-30T22:00:00.000Z", 191.28 "2013-07-01T22:00:00.000Z", 191.5 "2013-07-02T22:00:00.000Z", 193.25 "2013-07-04T22:00:00.000Z", 194.93 "2013-07-07T22:00:00.000Z", 194.98 "2013-07-08T22:00:00.000Z", 191.3 "2013-07-09T22:00:00.000Z", 192.25 "2013-07-10T22:00:00.000Z", 192.8 "2013-07-11T22:00:00.000Z", 192.07 "2013-07-14T22:00:00.000Z", 194 "2013-07-15T22:00:00.000Z", 193.85 "2013-07-16T22:00:00.000Z", 194.55 "2013-07-17T22:00:00.000Z", 197.99 "2013-07-18T22:00:00.000Z", 193.54 "2013-07-21T22:00:00.000Z", 194.09 "2013-07-22T22:00:00.000Z", 194.98 "2013-07-23T22:00:00.000Z", 196.61 "2013-07-24T22:00:00.000Z", 197.22 "2013-07-25T22:00:00.000Z", 197.35 "2013-07-28T22:00:00.000Z", 196.21 "2013-07-29T22:00:00.000Z", 196.01 "2013-07-30T22:00:00.000Z", 195.04 "2013-07-31T22:00:00.000Z", 195.81 "2013-08-01T22:00:00.000Z", 195.16 "2013-08-04T22:00:00.000Z", 195.5 "2013-08-05T22:00:00.000Z", 190.99 "2013-08-06T22:00:00.000Z", 188.56 "2013-08-07T22:00:00.000Z", 187.93 "2013-08-08T22:00:00.000Z", 187.82 "2013-08-11T22:00:00.000Z", 189.09 "2013-08-12T22:00:00.000Z", 188.42 "2013-08-13T22:00:00.000Z", 187.53 "2013-08-14T22:00:00.000Z", 185.79 "2013-08-15T22:00:00.000Z", 185.34 "2013-08-18T22:00:00.000Z", 184.23 "2013-08-19T22:00:00.000Z", 184.56 "2013-08-20T22:00:00.000Z", 184.86 "2013-08-21T22:00:00.000Z", 185.19 "2013-08-22T22:00:00.000Z", 185.42 "2013-08-25T22:00:00.000Z", 184.74 "2013-08-26T22:00:00.000Z", 182.74 "2013-08-27T22:00:00.000Z", 182.16 "2013-08-28T22:00:00.000Z", 182.64 "2013-08-29T22:00:00.000Z", 182.27 "2013-09-02T22:00:00.000Z", 183.96 "2013-09-03T22:00:00.000Z", 183.13 "2013-09-04T22:00:00.000Z", 184.15 "2013-09-05T22:00:00.000Z", 183.03 "2013-09-08T22:00:00.000Z", 184.98 "2013-09-09T22:00:00.000Z", 186.6 "2013-09-10T22:00:00.000Z", 190.7 "2013-09-11T22:00:00.000Z", 190.73 "2013-09-12T22:00:00.000Z", 192.17 "2013-09-15T22:00:00.000Z", 193.15 "2013-09-16T22:00:00.000Z", 192.16 "2013-09-17T22:00:00.000Z", 194.42 "2013-09-18T22:00:00.000Z", 193.39 "2013-09-19T22:00:00.000Z", 190.02 "2013-09-22T22:00:00.000Z", 190.99 "2013-09-23T22:00:00.000Z", 189.97 "2013-09-24T22:00:00.000Z", 189.47 "2013-09-25T22:00:00.000Z", 190.22 "2013-09-26T22:00:00.000Z", 186.92 "2013-09-29T22:00:00.000Z", 185.18 "2013-09-30T22:00:00.000Z", 186.38 "2013-10-01T22:00:00.000Z", 184.96 "2013-10-02T22:00:00.000Z", 183.86 "2013-10-03T22:00:00.000Z", 184.1 "2013-10-06T22:00:00.000Z", 182.01 "2013-10-07T22:00:00.000Z", 178.72 "2013-10-08T22:00:00.000Z", 181.32 "2013-10-09T22:00:00.000Z", 184.77 "2013-10-10T22:00:00.000Z", 186.16 "2013-10-13T22:00:00.000Z", 186.97 "2013-10-14T22:00:00.000Z", 184.66 "2013-10-15T22:00:00.000Z", 186.73 "2013-10-16T22:00:00.000Z", 174.83 "2013-10-17T22:00:00.000Z", 173.78 "2013-10-20T22:00:00.000Z", 172.86 "2013-10-21T22:00:00.000Z", 174.97 "2013-10-22T22:00:00.000Z", 175.77 "2013-10-23T22:00:00.000Z", 177.8 "2013-10-24T22:00:00.000Z", 176.85 "2013-10-27T22:00:00.000Z", 177.35 "2013-10-28T22:00:00.000Z", 182.12 "2013-10-29T22:00:00.000Z", 180.15 "2013-10-30T22:00:00.000Z", 179.21 "2013-10-31T22:00:00.000Z", 179.23 "2013-11-03T22:00:00.000Z", 180.27 "2013-11-04T22:00:00.000Z", 177.85 "2013-11-05T22:00:00.000Z", 179.19 "2013-11-06T22:00:00.000Z", 180
axisLabels "Time" "IBM price" $ 
  over [sigPlot (#ibm#close), 
        psigPlot ffModelIncorrect]

Why?

It is difficult to look at these plots and make any firm conclusions about them. It might be more helpful to plot histograms over future prices.

futurePrice s = s 999

This difference is amplified when we calculate risk or options pricing. To calculate the risk that the price drops below $100, we define a function from the future price to a Boolean value

tooLow price = price  < 100

and then transform the probability distribution over future prices using this function to create a boolean probability distribution representing this risk.

fmap (tooLow . futurePrice) ffModel ⇒ 14%
fmap (tooLow . futurePrice) ffModelIncorrect ⇒ 6%

The risk is much greater when taking parameter uncertainty into account.

An option is a contract that allows the buyer to buy or sell an asset at a point in the future at a fixed price. For instance, if the price dropping below $100 presents an unacceptable risk for you, you may want to buy an option that allows you to sell the shares at a price of $100 in the future. The fair value of such an option is the expected payoff of such an option. If the price is greater than $100, then the option is worthless because you will not exercise it. If, however, the future price is lower than $100, then it is worth the difference between the strike price ($100) and its real value. You should pay $20 for the right to sell something worth $80 for $100, and this applies to a future sale as well.

payoff strike price = if price > strike then 0 else strike - price 
expect $ fmap (payoff 100 . futurePrice) ffModel ⇒ 1.0841
expect $ fmap (payoff 100 . futurePrice) ffModelIncorrect ⇒ 0.4245

That is a big difference in the option price.

So does this matter? If you have a lot of data, the uncertainty may be small and the expectation is a good approximation to the full probability distribution. Propagating the full uncertainty is likely to matter the most when you are making forecasts based on sparse data. As we have seen, the impact of parameter uncertainty is likely to be greatest when considering tail events in risk assessment or derivatives pricing.