Thinking about thinking

Men­tal chronometry is the study of the mind via the mea­su­re­ment of time (i.e. mea­su­ring dif­fe­ren­ces in how fast a par­ti­ci­pant res­ponds to various sti­muli). Men­tal chromometry is a term I use to refer to a para­digm pio­nee­red by Prinz­me­tal et al (1997) that uses the participant’s abi­lity to repro­duce a pre­viously pre­sen­ted color to mea­sure memory. In fact, the idea is rather more gene­ral than the spe­ci­fic case of color memory; the essence of the approach is to use a sti­mu­lus that falls on a con­ti­nuum (color, loca­tion, bright­ness, etc) and use the dis­tri­bu­tion of error on the con­ti­nuum to make infe­ren­ces with regards to memory per­for­mance. Color is a nice sti­mu­lus domain because it can be repre­sen­ted as a cir­cu­lar con­ti­nuum which in turn per­mits appli­ca­tion of a sim­ple per­for­mance model devi­sed by Zhang & Luck (2008) that per­mits esti­ma­tion of two aspects of memory per­for­mance: pro­ba­bi­lity and fide­lity of memory.

Unfor­tu­na­tely, Zhang & Luck fai­led to pro­vide suf­fi­cient back­ground mate­rials on their approach for others to repli­cate the mathe­ma­tics of esti­ma­ting the pro­ba­bi­lity and fide­lity of memory. To ame­lio­rate this over­sight, I deve­lo­ped and tes­ted my own esti­ma­tion algo­rithms. At first I erro­neously (and rather emba­rras­singly) mode­lled per­for­mance as a mix­ture of uni­form and Gaus­sian dis­tri­bu­tions, and this work was briefly repor­ted here. Revie­wers at Beha­vior Research Methods quickly caught my error but I was per­mit­ted to resub­mit a correc­ted manusc­ript appl­ying the pro­per model of per­for­mance as a mix­ture of uni­form and Von Mises dis­tri­bu­tions. In appl­ying these correc­tions I also came across an enhan­ced algo­rithm for esti­ma­ting mix­ture models, the Expectation-Maximization algo­rithm, which turns out to do a much bet­ter job of esti­ma­ting the model’s para­me­ters than straight Maxi­mum Like­lihood Esti­ma­tion, which was used in the ori­gi­nal manuscript.

Here’s a copy of the revi­sed draft manusc­ript, prin­ted in jour­nal layout thanks to LaTeX. For those inte­res­ted in pla­ying with LaTeX, here’s the source.